SYMMETRY, REDUCTION, AND STABILITY IN HAMILTONIAN …Foreword This thesis presents some...

SYMMETRY, REDUCTION,

AND STABILITY IN

HAMILTONIAN SYSTEMS

Juan Pablo Ortega Lahuerta

Advisor: Tudor S. Ratiu

University of California, Santa CruzSanta Cruz, California

June, 1998

Copyright c! by

Juan Pablo Ortega Lahuerta

1998

A mis padres,

Joaquın y Marıa Teresa,

quienes, con carino infinito y generosidad sin lımites,han dedicado su vida a que mis hermanos y yo

podamos hacer rendir nuestros “talentos”.

Foreword

This thesis presents some contributions to the study of Hamiltoniansystems with symmetry. The use of the symmetries of a system inthe description of its dynamical evolution has a long history that goesback to the founders of classical mechanics. The main idea behind allthe attempts made to take advantage of symmetries, consists of sym-plifiying the kinematical framework of the system (the phase space)by means of a procedure called generically reduction that has evolvedalong the years. As long as the system in question does not presentsingularities, there is a standard formulation of this procedure, dueto Marsden and Weinstein [MW74], by means of which one obtainsby reduction smaller smooth Hamiltonian systems out of the originalHamiltonian system. In the presence of singularities, the situationis not so clear since several approaches lying in di!erent categorieshave been developed in parallel, all of them satisfactory and gener-alizing of the standard Marsden–Weinstein reduction procedure. Wewill show in this thesis, by constructing a singular reduction diagram,that there is no antagonism between di!erent approaches to singularreduction. In fact, the reduction diagram will show that there aremorphisms that put into relation all these approaches in a satisfac-tory category theoretical way. The use of one scheme or the otherwill be given by our preferences in the choice of category at the timeof treating a particular problem

Our approach to singular reduction is not historically faithful. Wehave chosen to construct the Poisson singular reduced spaces, an alge-braic reduction method due to Arms, Cushman, and Gotay [ACG91],by using a generalization of the Poisson Reduction Theorem of Mars-den and Ratiu [MR86] to the singular case, and some global modelsto depict the symplectic reduced spaces, due to Sjamaar and Ler-man [SL91], and to Bates and Lerman [BL97]. The traditional ap-proach based on the Marle–Guillemin–Sternberg (MGS) normal form

vi Foreword

is also described. Indeed, Chapter 3 gives a self contained presen-tation of the MGS normal form as well as some of its preliminaryapplications, not only to kinematics, but also to dynamics, morespecifically, the equations that govern the dynamics induced by agroup invariant Hamiltonian function (we call them the reconstruc-tion equations) are presented in detail.

All these global and local reduction tools are first applied in thestudy of the persistence of non degenerate relative equilibria andrelative periodic orbits, that is, the study of the existence of relativecritical elements near a given one, that share the same symmetry.

Chapters 5 and 6 conjugate the ideas previously introduced onsymmetric Hamiltonian systems with energetics techniques, that is,the use of the conserved quantities of the system (of symmetric originor not) to elaborate arguments that allow us to conclude stability.The actual implementation of this strategy is based on the use ofpenalty functions in a way first devised by Patrick [Pat92]. The reachof the stability results that we present has not yet been determined.However, there are hints that indicate that the combination of thesemethods with classical Hamiltonian techniques may yield applicationsto systems of certain relevance.

In the last chapter we take the first steps towards a theory ofbifurcation of symmetric Hamiltonian systems that tries to adapt thevery well known results in this field for general dynamical systems, tothe idiosyncracies of the Hamiltonian framework. As a first o!springof this approach we will obtain a theorem that fully describes thebifurcations coming out of a non degenerate relative equilibrium withrespect to an Abelian Hamiltonian symmetry.

Acknowledgments

Capıtulo III: Donde se cuenta la graciosamanera que tuvo don Quijote enarmarse caballero.Cervantes, Don Quijote de la Mancha

First, I would like to thank Tudor Ratiu, my teacher, advisor,and mentor. His tutoring, advice and expertise are responsible formy transformation from a student into a researcher. His unbridledenergy and enthusiasm have made the last three years a thrilling ridein which the advisor–student relation has turned into a collaboration,whose o!spring is this thesis, that has been written using the editorial“we” not for stylistic purposes but as a reflection of the way it wascreated.

I also thank Professor Jerrold Marsden. I have benefitted greatlyfrom numerous discussions with him. It is during my visits to Caltechwhen some of the problems treated in this thesis arose for the firsttime, out of his astonishing sense of perspective and mathematicaltaste. I thank him for his generosity with his time, his hospitality,and his gorgeous picture of the rigid body that decorates one of thepages of this thesis.

There are a number of others who were crucial in my developmentas a physicist/mathematician:

• The excellent professors and lecturers that I had in my almamater, the Universidad de Zaragoza, especially Jose LuisAlonso, who opened my eyes to the beauty of physics and ac-cepted graciously my drift into more mathematical waters, andPepın Carinena, who introduced me to the symplectic world.

viii Acknowledgments

• The faculty of the Mathematics Department at UCSC who wereable to recycle a theoretical physicist into a mathematician de-cent enough to pass the prelims and even made him enjoy it,especially Viktor Ginzburg, Debbie Lewis, and Richard Mont-gomery whose door was always open for help and advice.

• As to this thesis, the collaboration with Debbie Lewis and Pas-cal Chossat reflected in the last chapter has been a real privi-lege.

• In the course of my research, I profitted in one way or an-other from discussions with the following people: L. Bates, M.Castrillon, R. Cushman, M. Czachor, F. Diacu, C. Dong, F.Fasso, A. Fischer, V. Ginzburg, A. Hernandez, R. Hernandez,R. Lauterbach, E. Lerman, J. Montaldi, G. Patrick, M. Roberts,and A. Weinstein.

• My fellow graduate students, especially P. Birtea, C. Castilho,M. Hoyle, S. Kouranbaeva, and D. Ray. Their help and friend-ship are greatly appreciated.

This is a good place to thank my family, whose continued love andsupport have been a crucial factor in my development not only as ascientist, but also as a person. Thanks go to my parents, to whomthis thesis is dedicated, and also to Joaquın, Alba, Jesus, Cristina,Domingo, Marıa, Pepe, Elvira, Aurelio, Marıa–Pilar(es), Fernando,Marta and the rest of the family and friends who during the last fouryears, among other things, have flooded the cyber space with dailyelectronic support.

A very special place goes to my wife, whose unconditional loveand good heart amaze me every single day. I am specially gratefulfor her graciously sharing her husband with math and understandinghis eccentricities and mental absences.

This thesis has been possible thanks to the financial support ofseveral institutions:

• The Spanish taxpayer who, through the Ministerio de Edu-cacion y Ciencia paid for all my undergraduate studies and myfirst year as a graduate student.

Acknowledgments ix

• The Fulbright Commission for the Cultural Exchange BetweenSpain and the USA, with funds provided by the Banco CentralHispano. The guidance and support of Liz Anderson and Clau-dia Constanzo are greatly appreciated, as well as the humblingexample of my fellow Fulbrighters.

• The Institute of International Education, with funds providedby the United States International Agency. The advice receivedfrom Arthur Austin during my visit to New York, and the ef-ficient and continued support provided by Kate Leiva and hersta! in San Francisco are deeply appreciated.

• Rotary Foundation of Rotary International, with funds pro-vided by the Rotarians of the District 2210 (North Spain). Spe-cial thanks go to D. Jose Marıa Moncasi and to Sr. Casajuanafor taking a personal interest in my application and for believingin a project that, at that point was exactly that, just a project.I also thank the hospitality of my host district in North Califor-nia (5170) who made the fulfillment of my duties as a RotaryAmbassadorial Scholar a very pleasurable experience.

• The Mathematics Department at UCSC, that gave me a jobafter I ran out of scholarships, and funded some of my trips toconferences. Special thanks go to JoAnn Mcfarland and EllenMorrison for their e"ciency and charm.

• The French C.N.R.S. and the Societe Mathematique de France,for the funding provided for my attendance to the Seminairesud Rhodanien in December 1997 where I presented for the firsttime in front of an audience of specialists some of the results ofthis thesis..

• The University of California, Los Angeles, for the funding pro-vided for my attendance to the meeting Dynamical Systems:Quo Vadis?.

x Acknowledgments

Agradecimientos

Capıtulo III: Donde se cuenta la graciosamanera que tuvo don Quijote enarmarse caballero.Cervantes, Don Quijote de la Mancha

En primer lugar, me gustarıa dar las gracias a Tudor Ratiu, mimaestro, director de tesis y mentor. Su consejo y experiencia sonresponsables de mi transformacion de estudiante a investigador. Suenergıa y entusiasmo han convertido los ultimos tres anos en unaaventura apasionante, en los que la relacion director–estudiante seha convertido en una colaboracion cuyo fruto es esta tesis doctoral,la cual ha sido redactada utilizando el “we” editorial no por razonesestilısticas, sino como un reflejo de la manera en que ha sido creada.

Gracias tambien al Profesor Jerrold Marsden, con quien he tenidoel placer de discutir en numerosas ocasiones. Fue durante mis visitasa Caltech cuando algunos de los problemas tratados en esta tesissurgieron por primera vez, gracias a su extraordinario sentido dela perspectiva y buen gusto. Le agradezco la generosidad que hamostrado con su tiempo, su hospitalidad y su estupendo dibujo delsolido rıgido que adorna una de las paginas de esta tesis.

Hay un conjunto de personas que han jugado un papel crucial enmi desarrollo como fısico/matematico:

• Los excelentes profesores y ayudantes que tuve en mi almamater, la Universidad de Zaragoza, en especial Jose LuisAlonso, quien abrio mis ojos a la belleza de la fısica y no se tomomuy a mal cuando empece a “flirtear” con las matematicas, yPepın Carinena, quien me introdujo en el mundo simplectico.

xii Agradecimientos

• La plantilla del Departamento de Matematicas de la UCSCquienes se las apanaron para reciclar a un fısico teorico en unmatematico lo suficientemente bueno como para pasar los “pre-lims”, y consiguieron que incluso lo disfrutase. En lo que serefiere a esta tesis, la colaboracion con Debbie Lewis, PascalChossat y Richard Montgomery han sido un verdadero privile-gio.

• En el curso del desarrollo de esta tesis me beneficie de unamanera u otra de discusiones con las personas siguientes: L.Bates, M. Castrillon, R. Cushman, M. Czachor, F. Diacu, C.Dong, F. Fasso, A. Fischer, V. Ginzburg, A. Hernandez, R.Hernandez, R. Lauterbach, E. Lerman, J. Montaldi, G. Patrick,M. Roberts, y A. Weinstein.

• Mis companeros de clase, especialmente P. Birtea, C. Castilho,M. Hoyle, S. Kouranbaeva y D. Ray, cuya ayuda y amistad hasido enormemente valiosa.

Este es un buen lugar para agradecer a mi familia, cuyo apoyoincondicional ha sido crucial en mi formacion no solo como inves-tigador, sino tambien como persona. En particular a mis padres,a quien esta tesis esta dedicada, asi como a Joaquın, Alba, Jesus,Cristina, Domingo, Marıa, Pepe, Elvira, Aurelio, Marıa–Pilar(es),Fernando, Marta y el resto de la familia y amigos quienes, durantelos ultimo cuatro anos, entre otras cosas, han inundado el ciberespaciocon apoyo electronico diario.

Me gustarıa dedicar un lugar muy especial a mi mujer, cuyo amorincondicional y corazon de oro me sorprenden dıa tras dıa. Es deagradecer en especial la gracia con la que ha sabido compartir a sumarido con esta tesis y la comprension que ha mostrado con susexcentricidades y “ausencias mentales”.

Esta tesis ha sido posible gracias al apoyo financiero de variasinstituciones:

• El contribuyente espanol quien, a traves del Ministerio de Edu-cacion y Ciencia pago por mi estancia en la Universidad deZaragoza.

Agradecimientos xiii

• La Comision Fulbright para el Intercambio Cultural entreEspana y los Estados Unidos de America, con fondos pro-porcionados por el Banco Central Hispano. La orientaciony el apoyo ofrecidos por Liz Anderson y Claudia Constanzo,ası como el imponente ejemplo dado por mis companeros Ful-brighters, son muy apreciados.

• El Institute of International Education, con fondos proporciona-dos por la United States International Agency. Agradezco es-pecialmente el consejo prestado por Arthur Austin durante mivisita a Nueva York y el eficiente apoyo de Kate Leiva y suscolaboradores en la oficina de San Francisco.

• Rotary Foundation de Rotary International, con fondos pro-porcionados por los rotarios del Distrito 2210. Me gustarıamencionar especialmente a D. Jose Marıa Moncasi y al Sr,Casajuana por su interes personal en mi solicitud y por creer enel proyecto que les presente cuando no era mas que eso, un sim-ple proyecto. Agradezco tambien la hospitalidad de mi distritoanfitrion en California del Norte (5170) quienes convirtieron elcumplimiento de mis deberes como Embajador de Buena Vo-luntad en una experiencia realmente agradable.

• El Departamento de Matematicas de la UCSC, el cual me dioun trabajo tras la finalizacion de mis becas, y financio algunosde mis viajes a conferencias. Un agradecimiento especial es paraJo Ann Mcfarland y Ellen Morrison por su eficacia y apoyo.

• El C.N.R.S. frances y la Societe Mathematique de France, por lafinanciacion de mi estancia en Lumini en uno de los SeminairesSud Rhodanien.

• La Universidad de California en Los Angeles por la financiacionde mi asistencia a la reunion Dynamical Systems: Quo Vadis?

xiv Agradecimientos

Contents

Foreword v

Acknowledgments vii

Agradecimientos xi

1 Preliminary Concepts 11.1 Hamiltonian Systems . . . . . . . . . . . . . . . . . . . 11.2 Critical Elements and Stability . . . . . . . . . . . . . 41.3 Actions of Lie Groups . . . . . . . . . . . . . . . . . . 101.4 Hamiltonian Actions and Momentum Maps . . . . . . 191.5 A First Approach to Reduction Theory . . . . . . . . 251.6 Relative Critical Elements . . . . . . . . . . . . . . . . 41

2 Singular Symplectic Reduction 472.1 Stratified Poisson Reduction by Foliations . . . . . . . 482.2 Singular Poisson, Point, and Orbit Reduction . . . . . 582.3 Free Actions in a Singular World . . . . . . . . . . . . 662.4 Singular Symplectic Point and Orbit Reduction . . . . 78

3 The Marle–Guillemin–Sternberg Normal Form 893.1 Proper Actions, Tubes, and Slices . . . . . . . . . . . . 903.2 Hamiltonian Tubes: the Marle–Guillemin–Sternberg

Normal Form . . . . . . . . . . . . . . . . . . . . . . . 943.3 The Reconstruction Equations . . . . . . . . . . . . . 1063.4 The MGS Normal Form and Reduction . . . . . . . . 111

4 Persistence and Smoothness of Relative Critical Ele-ments 1254.1 Singular Relative Critical Elements . . . . . . . . . . . 126

xvi Agradecimientos

4.2 Persistence of Singular Relative Critical Elements . . . 1324.3 Manifolds of Relative Equilibria . . . . . . . . . . . . . 144

5 Stability of Equilibria and Relative Equilibria 1495.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1495.2 Hessians and Patrick’s Lemma . . . . . . . . . . . . . 1505.3 The Energy–Momentum Method . . . . . . . . . . . . 1525.4 Block Diagonalization of the Stability Form . . . . . . 1705.5 An Example: the Stability of the Sleeping Lagrange Top182

6 Stability of Periodic and Relative Periodic Orbits 1896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1896.2 Orbital Stability and the Energy–Integrals Method . . 190

6.2.1 The Energy–Integrals Method . . . . . . . . . . 1916.2.2 The Energy–Integrals Method and Integrable

Systems . . . . . . . . . . . . . . . . . . . . . . 1976.3 Applications of the Energy–Integrals Method . . . . . 200

6.3.1 The Stability of the Closed Rigid Body Orbits 2006.3.2 Stability of the Orbits of the Bidimensional

Harmonic Oscillator . . . . . . . . . . . . . . . 2046.3.3 Stability of the Elliptical Orbits of the Three

Dimensional Kepler Problem . . . . . . . . . . 2076.3.4 Orbital Stability Via the Energy–Integrals

Method in a Non Integrable System . . . . . . 2136.4 Stability of Regular Relative Periodic Orbits . . . . . . 2156.5 Applications of the Symmetric Energy–Integrals Method223

6.5.1 The S1–Stability of the Precessing Orbits of theSpherical Pendulum . . . . . . . . . . . . . . . 223

6.5.2 Stability of the Nutating Motion of a LagrangeTop. . . . . . . . . . . . . . . . . . . . . . . . . 229

6.5.3 Stability of the Bounded Manev Orbits. ThePrecessing Orbit of Mercury. . . . . . . . . . . 235

6.6 The Symmetric Energy–Integrals Method. The Gen-eral Case . . . . . . . . . . . . . . . . . . . . . . . . . 242

6.7 Block Diagonalization and Reduced Periodic Orbits . 251

7 Bifurcation of Hamiltonian Relative Equilibria 2597.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2597.2 The Witt–Artin Decomposition . . . . . . . . . . . . . 2607.3 The Bifurcation Method . . . . . . . . . . . . . . . . . 269

Contents xvii

7.4 Bifurcation in Hamiltonian Systems with AbelianSymmetries . . . . . . . . . . . . . . . . . . . . . . . . 279

Bibliography 287

Index 300

xviii Contents

Chapter 1

Preliminary Concepts

Capıtulo XXIV: Donde se cuentan mil zarandajastan impertinentes como necesarias al verdaderoentendimiento desta grande historia.Cervantes, Don Quijote de la Mancha, II

In this chapter we shall introduce the concepts of symplectic andPoisson manifolds and discuss briefly Hamiltonian dynamics on them,as well as many standard constructions related to Hamiltonian actionsof Lie groups. It is not the intention to give a systematic treatmentof this topics, since they are already presented in many books, but togive an account of results and notations that will be needed in whatfollows.

1.1 Hamiltonian Systems

A symplectic manifold is a pair (M, !), where M is a manifoldand ! is a closed non degenerate two–form on M . If we drop thenon degeneracy condition, we say that (M, !) is a presymplec-tic manifold. A Hamiltonian dynamical system is a triple(M, !, h), where (M, !) is a symplectic manifold and h " C!(M)is the Hamiltonian function of the system. The nondegeneracyof the symplectic form ! associates naturally to each Hamiltoniansystem a Hamiltonian vector field Xh " X(M), defined by

iXh! = dh.

2 Chapter 1. Preliminary Concepts

Theorem 1.1.1 (Darboux) Let (M, !) be a symplectic manifoldof dimension 2n. For each point m " M , there is a chart (U, "),with "(z) = (q1, . . . , qn, p1, . . . , pn), in which ! is written as

!U =n!

i=1

dqi # dpi.

Coordinates in which ! takes the above form are called canonicalcoordinates.

Notice that in canonical coordinates, the integral curves corre-sponding to the Hamiltonian vector field Xh, are determined by thewell–known Hamilton’s equations, that is

dqi

dt=#h

#pi

dpi

dt= $ #h

#qi. (1.1.1)

Definition 1.1.1 Let f, g " C!(M). The Poisson bracket ofthese two functions, is the function {f, g} " C!(M) defined by

{f, g}(z) = !(z)(Xf (z), Xg(z)) = Xg[f ](z).

Note that in canonical coordinates, the Poisson bracket takes thetraditional form

{f, g} =n!

i=1

#f

#qi

#g

#pi$ #g

#qi

#f

#pi.

Once we have fixed a symplectic structure on M , the set C!(M)is endowed with a real Lie algebra structure with the product de-fined by the Poisson bracket. This property leads to the followinggeneralization:

Definition 1.1.2 A Poisson manifold is a pair (M, {·, ·}), whereM is a manifold and {·, ·} is a bilinear operation on C!(M) suchthat (C!(M), {·, ·}) is a Lie algebra, and {·, ·} is a derivation (thatis, the Leibniz identity holds) in each argument. We will also say thatthe pair (C!(M), {·, ·}) is a Poisson algebra.

The natural isomorphism between derivations on C!(M) andvector fields in M (see [Mat72, page 73]) allows the definition ofa Hamiltonian vector field Xh, for each h " C!(M), by the relation

Xh = {·, h}.

§ 1.1. Hamiltonian Systems 3

The elements of the center of the Lie algebra (C!(M), {·, ·}) arecalled Casimir functions. We will denote the center or Casimirsubalgebra by C(M). The triplet (M, {·, ·}, h) is called a Poissondynamical system . Obviously, any Hamiltonian system is a Pois-son algebra with the bracket given by the Poisson bracket associ-ated to the symplectic structure. The converse relation is given bythe Symplectic Stratification Theorem, (see [W83a], or [MR94])which states that any Poisson manifold (M, {·, ·}) is partitioned intosymplectic leaves, that is, sets of points that can be linked to eachother by a finite number of smooth curves, each of which is a piece ofan integral curve of a locally defined Hamiltonian vector field. Thesymplectic leaves are connected immersed symplectic manifolds of Mby the inclusion map, whose Poisson bracket is that of M . The tan-gent space at m to a leaf consists of all vectors that are equal to thevalue of some Hamiltonian vector field at m. The symplectic leavesare invariant under the flow of any Hamiltonian vector field.

The derivation property of the Poisson bracket implies that forany two functions f, g " C!(M), the value of the bracket {f, g}(z)at an arbitrary point z " M (and therefore Xf (z) as well), dependson f only through df(z) (see [AMR, Theorem 4.2.16] for a justifi-cation of this argument) which allows us to define a contravariantantisymmetric two–tensor B " #2(T "M) by

B(z)($z, %z) = {f, g}(z),

where df(z) = $z and dg(z) = %z " T "z M . This tensor receives the

name of Poisson tensor.Given a Poisson dynamical system (M, {·, ·}, h), we define its in-

tegrals of the motion or conserved quantities as the centralizerof h in (C!(M), {·, ·}) that is, the subalgebra of (C!(M), {·, ·})consisting of the functions f " C!(M) such that {f, h} = 0. Notethat the terminology is justified by the fact that {f, h} encodes theevolution of f along the integral curves of Xh which, in this case isconstant, that is

f = Xh[f ] = {f, h} = 0.

Definition 1.1.3 A smooth mapping " : M1 %M2, between the twoPoisson manifolds (M1, {·, ·}1) and (M2, {·, ·}2) is called canonicalor Poisson if for all g, h " C!(M2)

""{g, h}2 = {""g, ""g}1.


For future reference we state a result whose proof can be foundin [MR94, Proposition 10.5.2].

Proposition 1.1.1 Let " : M1 % M2 be a Poisson map betweentwo Poisson manifolds (M1, {·, ·}1) and (M2, {·, ·}2), and let h "C!(M2). If F 2

t is the flow of Xh and F 1t is the flow of Xh#!, then

F 2t & " = " & F 1

t , and T" &Xh#! = Xh & ".

1.2 Critical Elements and Stability

The critical elements of a vector field X " X(M) are its equilib-rium points and its periodic points or periodic orbits.

An equilibrium point of the vector field X " X(M) is a pointm " M such that Ft(m) = m for all t " R, where Ft denotes theflow associated to X. In the Hamiltonian context, equilibria arealso referred to as critical points, since the nondegeneracy of thesymplectic form guarantees that m " M is an equilibrium for theHamiltonian vector field Xh if and only if m is a critical point of theHamiltonian function h, that is dh(m) = 0. Notice that this is notin general the case for Poisson systems.

A point m " M is said to be periodic if there is a constant& > 0 such that for any time t, Ft+" (m) = Ft(m). The periodof m is the smallest positive & satisfying this condition. The set' = {Ft(m) | t " R} is usually referred to as the periodic orbit 'that goes through the point m. It is easy to see that every point of aperiodic orbit is periodic with the same positive period; thus we mayspeak of the period of the periodic orbit '.

Two important tools in the study of periodic orbits are the localtransversal sections and the Poincare maps. We briefly review theseconcepts and their principal properties below (see [AM78]).

Definition 1.2.1 Let X " X(M) be a vector field on the manifoldM . A local transversal section of X at m " M is a submanifoldS ' M of codimension one with m " S and such that for all s " S,X(s) is not contained in TsS. Therefore TsM = TsS ( span{X(s)}.

Let F : DX ' M ) R % M be the flow of X which includes aclosed orbit ' through m with period & ; DX := {(z, t) " M ) R |Ft(z) is defined}, is the domain of the flow, an open subset of M)R.A Poincare map of ' is a mapping $ : W0 %W1 where

§ 1.2. Critical Elements and Stability 5

(PM1) W0, W1 ' S are open neighborhoods of m " S and $ is adi!eomorphism;

(PM2) there is a continuous function ( : W0 % R, called the periodfunction, such that (s, & $ ((s)) " DX , and $(s) = F (s, & $((s)) for all s "W0;

(PM3) if t " (0, & $ ((s)), then F (s, t) /"W0.

Figure 1.2.1: Poincare section and Poincare map.

A fundamental theorem guarantees the existence and uniquenessof Poincare maps for closed orbits in arbitrary dynamical systems.By uniqueness we mean that if $$ : W $

0 % W $1 is another Poincare

map constructed using the section S$ through m$ " ', then $ and$$ are locally conjugate, that is, there are open neighborhoods W2

of m " S, W $2 of m$ " S$, and a di!eomorphism H : W2 % W $

2, such


that W2 'W0 *W1, W $2 'W $

0 *W $1, and the diagram

$%1(W2) *W2!$% W2 *$(W2)

H""#

""#H

W $2

!!$% S$

commutes.If the manifold M is symplectic, with symplectic form !, and

the vector field X is a Hamiltonian dynamical system associated tothe function h " C!(M) (we will denote X by Xh in this case)then these additional structures allow us to choose the elements ofDefinition 1.2.1 with the properties stated in the following theorem(which is proposition 8.1.3 in [AM78]). Note that if ' is a closed orbitof Xh, then we may assume that ' lies in a regular energy surface %e

of h since near ', dh must be nonzero.

Theorem 1.2.1 Let (M, !) be a symplectic manifold, h " C!(M),and ' a closed orbit of Xh lying in the regular energy surface %e.Then, there exists a local transversal section S at m " ' and aPoincare map $ : W0 %W1 on S, such that the following hold:

(i) (W0, !0) and (W1, !1) are contact manifolds where !j = i"j!, ij :Wj )%M being the natural inclusion and j " {0, 1}; this meansthat the rank of !0 and !1 is one less than dimW0 = dimW1.

(ii) $ is a canonical transformation; that is, $ preserves h, andthere is a function ( " C!(W0) such that $"!1 = !0$d(#dh;moreover, ( is the period function of the Poincare map describedin Definition 1.2.1.

(iii) There exist an * > 0 and regular energy surfaces %e! fore$ " (e $ *, e + *), such that (Se! := S * %e! , !e!) is a sym-plectic submanifold of M of codimension two and $ |W0&Se!

is a symplectomorphism onto W1 * Se!, where !e! = i"! andi : Se! )%M is the natural inclusion.

We will devote part of our e!orts to study the stability of criticalelements hence, in what follows we give precise information on thekind of stability that we will be searching.

In the case of equilibria we will focus on a definition that datesback to Lyapunov [Lia07].


Definition 1.2.2 Let X " X(M) be a vector field on the manifoldM , and let m " M be an equilibrium of (M, X). We say that m isstable or Lyapunov stable, if for any open neighborhood U of m inM , there is an open neighborhood V of m such that if Ft is the flowassociated to X, then Ft(z) " U , for any z " V and for all t > 0.

Figure 1.2.2: Lyapunov stability of an equilibrium.

Intuitively, stability of an equilibrium implies that all the solu-tions that start su"ciently close to the equilibrium remain near theequilibrium for all future time or, equivalently, continuity of the inte-gral curves of X around m for all t > 0, on their dependence on theinitial conditions (see Figure 1.2.2). The importance for applicationsof this concept is evident: one can predict approximately the positionof the system for all time just by knowing the initial conditions.

We will construct stability criteria for critical elements in Hamil-tonian systems that is, conditions verifiable in terms of computablequantities that will imply stability. In the case of equilibria there isa very well known stability criterion that we present in what follows.The study of similar criteria for periodic orbits will be the subject of


Chapter 6.

Theorem 1.2.2 (Lagrange) Let (M, !, h) be a Hamiltonian dy-namical system, and m " M be an equilibrium of Xh. Suppose thatd2h(m) is definite; that is, for all v " TmM , v += 0

d2h(m)(v, v) >< 0.

Then, m is stable.

Proof By the Morse Lemma (see Lemma 5.2.1), the level surfaces ofh in a neighborhood of m, are di!eomorphic to concentric spheres. Byconservation of energy, the motion near the equilibrium takes placeon these spheres, which guarantees that it remains in a neighborhoodof m. !

The condition described in the previous theorem is not necessaryfor stability. A classical counterexample is the restricted three–bodyproblem where the quadratic part of the Hamiltonian is not definiteand one may have stability. In this case the stability criterion isprovided by the KAM Theorem (See [M68, M73, A61, A63], andreferences therein).

Regarding the periodic orbits, we will be interested in the no-tion of stability expressed in the following definition, introduced byBirkho! [Bi27].

Definition 1.2.3 Let X " X(M) be a vector field on the manifoldM , and let ' be a periodic orbit of (M, X) such that m " '. We saythat ' is orbitally stable, or that m is a stable periodic point, iffor any open neighborhood U of ' in M , there is an open neighborhoodV of m such that if Ft is the flow associated to X, then Ft(z) " U ,for any z " V and for all t > 0.

The main tool to establish this sort of stability in particular ex-amples is a classical Theorem due to Lyapunov. This result uses theeigenvalues of the linearized Poincare map at a point of the periodicorbit, or characteristic multipliers and states that if the charac-teristic multipliers of a periodic orbit lie strictly inside the unit disc,then the periodic orbit is asymptotically stable , a stability defini-tion due to Poisson, stronger than orbital stability (see, for example,[AM78], [AMR] or [Ro95]) . However, it is worth remarking that


Figure 1.2.3: Birkho!’s orbital stability.

this stability criterion is useless in the case of Hamiltonian systemsbecause of the Symplectic Eigenvalue Theorem. Indeed, if Xh isa Hamiltonian vector field on the symplectic manifold M , then thetime t–map Ft given by the flow of Xh is a symplectic di!eomorphism.Since F" (m) = m for m " ', where & is the period of ', it followsthat TmF" is a linear symplectic isomorphism on TmM , whose setof eigenvalues equals the union of all characteristic multipliers with{1}. Therefore, by the Symplectic Eigenvalue Theorem, if + is aneigenvalue of TmF" then +,+%1, and +%1 are eigenvalues as well, allhaving the same multiplicity, hence the characteristic multiplier 1always occurs with odd multiplicity at least once. Thus, it is im-possible in this case for all characteristic multipliers to lie strictly inthe unit disc. An intuitive reason behind this argument is Liouville’sTheorem on the conservation of volume by Hamiltonian flows, whichmakes asymptotic stability impossible in this case (the asymptoticstability implies that the flow contracts the phase space winding itaround the periodic orbit). Finding a criterion for the stability of


periodic orbits similar in philosophy to Theorem 1.2.2, will be one ofthe subjects of Chapter 6.

1.3 Actions of Lie Groups

The existence of symmetries in a dynamical system facilitates thesearch for its solutions. This is particularly apparent in the Hamil-tonian case where, if the symmetry fulfills certain requirements, itleads to the existence of conserved quantities that, generically reducethe dimensionality of the problem. How to formalize these ideas isthe subject of the next three sections, where we put together all theresults on Hamiltonian actions that we will need, for future reference.Proofs are provided only when they are not readily available in theliterature.

The first step in the modern mathematical formulation of theseconcepts consists of rephrasing the physical symmetries of the me-chanical system in terms of Lie groups acting on its phase space,leaving invariant the Hamiltonian function.

Let us recall that a Lie group G, is a smooth manifold which isa group such that multiplication is a smooth map. We will denotethe Lie algebra corresponding to G by g or by Lie(G), and its dualby g".

Definition 1.3.1 Let M be a manifold and G be a Lie group. A leftaction of a Lie group G on M is a smooth mapping & : G)M %M(very often we will denote &(g, z) , g · z) such that:

(i) &(e, z) = z, for all z "M .

(ii) &(g, &(h, z)) = &(gh, z) for all g, h " G and z "M .

A right action is a map ' : M)G%M , such that '(z, e) = z,for all z " M , and '('(z, g), h) = '(z, gh) for all g, h " G andz "M . We will often call the triple (M, G, &) a G–space. If M is avector space and G acts linearly on M , that is, &g " GL(M) for allg " G then, the action is said to be a representation of G on M .

Example 1.3.1 (Translations and conjugations) Left transla-tion, that is, Lg : G % G, defined by h -% gh induces a left actionof G on itself. Right translation, Rg : G % G, h -% hg, defines aright action. The inner automorphism Ig : G% G, defined as the

§ 1.3. Actions of Lie Groups 11

composition Ig := Rg"1 &Lg induces a left action of G on itself calledconjugation. "

Example 1.3.2 (Adjoint and coadjoint action) The di!eren-tial at the identity of the conjugation mapping produces a left actionof G on g called the adjoint representation, that is

Adg := TeIg : g $% g.

If Ad"g : g" % g" is the dual of Adg then, the map

& : G) g" $% g"

(g, ,) -$% Ad"g"1,,

defines a left action of G on g" called the coadjoint representationof G on g". "

The infinitesimal generator -M " X(M) defined by - " g is

-M (m) =d

dt

$$$$t=0

&exp t#(m).

The isotropy subgroup or stabilizer of an element m " M is theclosed subgroup Gm ' G defined by

Gm := {g " G |&g(m) = m}.

whose Lie algebra gm, equals

gm = {- " g | -M (m) = 0}.

The orbit of the element m "M under the group action & is the set

G · m , Om := {&g(m) | g " G}.

In general Om is an immersed submanifold of M by the inclusionmap, di!eomorphic to G/Gm by the map g · m -% [g]. Its tangentspace at each point is given by

g · m := Tm(G · m) = {-M (m) | - " g}.

Recall that the group action on M is said to be transitive if thereis only one orbit, and free if the isotropy of every element in M


consists only of the identity element. The set of smooth G–invariantfunctions on M will be denoted by C!(M)G, that is,

C!(M)G := {f " C!(M) | f & &g = f, for all g " G}.

A mapping " : M1 % M2, between two G–spaces, M1 and M2, issaid to be G–equivariant provided that for any g " G and z " M1,the mapping " satisfies the identity

"(g · z) = g · "(z).

The following lemma is a trivial consequence of the decompositionof the di!erential of a mapping in terms of its partial derivatives(see [AMR, Proposition 2.4.12 (ii)]).

Lemma 1.3.1 (Leibniz rule for group actions) Let m(t) be acurve in M through m "M and - " g. Then

(i) ddt

$$t=0

exp t- · m(t) = -M (m) + ddt

$$t=0

m(t).

(ii) If, analogously, g(t) is a curve in G through g " G then

d

dt

$$$$t=0

g(t) · m(t) =d

dt

$$$$t=0

g(t) · m + g · d

dt

$$$$t=0

m(t)

.

If H and K are closed subgroups of G such that H ' K ' G, wewill denote by

N(H) = {n " G |nHn%1 = H}NK(H) = {n " K |nHn%1 = H} = N(H) *K

the normalizers of H in G and K respectively.The following lemma will be of much importance in what follows.

Lemma 1.3.2 Let K be a Lie group and let N be a closed normalsubgroup with corresponding Lie algebras k and n. Then for all - " kand all n " N , we have

Adn- $ - " n .

Conversely, let K be a Lie group and H be a closed subgroup withcorresponding Lie algebras k and h. If - " k satisfies

Adh- $ - " h

for all h " H, then - lies in the Lie algebra of the normalizer N(H)of H.


Proof Let ADn : K % K denote the inner automorphism for n " N ,defined by ADn(m) = nmn%1. Since Adn is the derivative of ADn atthe identity, we get for any - " k,

Adn- $ - =d

dt

$$$$t=0

ADn(exp(t-)) exp($t-)

=d

dt

$$$$t=0

(n[exp(t-)]n%1) exp($t-)

=d

dt

$$$$t=0

n[exp(t-)n%1 exp($t-)].

Since N is a normal subgroup of K, exp(t-)n%1 exp($t-) is a curvein N (passing through the point n%1 at t = 0), so the result is someelement in n.

Conversely, let H be a closed subgroup of K and assume thatAdh- $ - " h for all h " H. By taking the derivative relative to hat the identity, it follows that [., -] " h for all . " h. By the Baker–Campbell–Hausdor! formula (see, for example, [KMS93], page 40),exp(tAdh-) exp($t-) = exp(t(Adh- $ -) + O(t2)), where O(t2) is aconvergent series each of whose terms is some iterated bracket of Adh-and - in some order, but always applied to [Adh-, -] = [Adh-$-, -] "h since Adh- $ - " h. Thus each time one takes a bracket with - theresult is in h and each time one takes the bracket with Adh-, one addsand subtracts a - to get two terms: the first, a bracket with - whichlies in h, the second, a bracket with Adh-$- " h, which again lies in h,because both elements in the bracket are in h. The conclusion is thateach term in this series lies in h and hence h exp(t-)h%1 exp($t-) =exp(tAdh-) exp($t-) = exp(t(Adh- $ -) + O(t2)) " H for all h " Hand all t " R. Therefore exp(t-)H exp($t-) ' H for all t " R whichsays that exp(t-) " N(H) for all t " R, that is, - is in the Lie algebraof N(H). !

Most of the group actions we will be interested in will fall in thecategory given by the following

Definition 1.3.2 The action & : G )M % M is called proper ifthe following condition is satisfied: given two convergent sequences,{mn} and {gn ·mn} in M , there exists a convergent subsequence {gnk}in G.

Example 1.3.3 Compact Group Actions Suppose that G is acompact Lie group acting on the manifold M . Since the group G


is by itself a di!erentiable manifold, it is metrizable and thereforecompactness is equivalent to sequential compactness, that is, everysequence has a convergent subsequence. In these conditions, the re-quirement described by Definition 1.3.2 is satisfied trivially. "

Example 1.3.4 The Euclidean Group The special Euclideangroup SE(n) is defined as

SE(n) = {(A, a) | A " SO(n), and a " Rn},

with the composition law given by

(A, a) · (B, b) = (AB, Ab + a).

The group SE(n) acts on Rn by

(A, a) · z = Az + b.

This action of SE(n) on Rn is proper. Indeed, let zn % z1 and(An, an)·zn % z2 be two convergent sequences. Since {An} ' SO(n),and SO(n) is a compact Lie group, by the remarks made in the pre-vious example, the sequence {An} admits a convergent subsequenceAnk % A. Since clearly Ankznk % Az1 and Ankznk + ank % z2,then ank % z2 $ Az1 and, therefore (Ank , ank) % (A, z2 $ Az1) is aconvergent subsequence of (An, an), as required. "

Note that if H is a closed subgroup of G and N is a H-invariantsubmanifold of M , the restricted action of H on N is also proper.Properness is a very powerful assumption in a group action becauseit guarantees that some of the technically important properties ofcompact groups are still valid, as can be seen in the following propo-sition.

Proposition 1.3.1 Let G be a Lie group acting properly on the man-ifold M. Then

(i) For any m " M , the isotropy subgroup Gm is compact and theorbits of G in M are closed and embedded submanifolds of M .

(ii) The orbit space M/G is a Hausdor! topological space.

(iii) If the action is free, M/G is a smooth manifold, and the canon-ical projection / : M % M/G defines on M the structure of asmooth left principal G–bundle.


(iv) If all the isotropy subgroups of the elements of M under the G–action are conjugate to a given one, say H ' G, then M/Gis a smooth manifold, and the canonical projection / : M %M/G defines on M the structure of a smooth locally trivial fiberbundle with structural group N(H)/H and fiber G/H.

(v) There exists on M a G–invariant Riemannian metric.

(vi) Smooth G–invariant functions separate the orbits of G.

(vii) There exists a G–invariant partition of unity subordinate to anyG–invariant open cover of M .

(viii) Let N be any G–invariant submanifold of M and supposethat f " C!(N) is constant on each G–orbit. Then thereis a smooth G–invariant extension F of f to M , that is,F " C!(M)G and F |N= f .

Proof See for example [Pal61, GS84b, BL97, AMR, CB97, ACG91,AM78]. !

One more feature that proper actions share with compact groupsis the existence of slices; we postpone the introduction of this conceptuntil Chapter 3. Also, the properness of the action guarantees thesmoothness of the subsets of M introduced in the following

Proposition 1.3.2 Let G be a Lie group acting properly on the man-ifold M. Let H and K be closed subgroups of G such that H ' K ' G.The connected components of the sets

M(H) = {z "M | Gz is conjugate to H}MK

(H) = {z "M | Gz is conjugate to H in K}

MH = {z "M | H ' Gz}MH = {z "M | H = Gz} = MH *M(H)

are submanifolds of M . MH is an open submanifold of MH . M(H)

is called the (H)–orbit type manifold and MH the H–fixed pointsmanifold. If M is symplectic, MH and MH are symplectic subman-ifolds of M . Also, for any m "MH , the tangent space to MH is givenby

TmMH = {vm " TmM | Tm&h · vm = vm, .h " H}= (TmM)H = TmMH . (1.3.1)


Proof See for example [GS84b, BL97, Pal61, Bre72]. !

Remark 1.3.1 The requirement on the restriction to the connectedcomponents is not vacuous; in fact, any of the three sets defined inProposition 1.3.2 may contain components of di!erent dimensions;see [SL91, Remark 1.3] for an example. "

Remark 1.3.2 If M is a vector space on which H acts linearly, MH

is called in physics the space of singlets or space of invariantvectors. See [AS81, AS83, AS83]. "

Lemma 1.3.3 If G acts on M , µ " g", Gµ is the coadjoint isotropysubgroup of µ, and Gm = H, we have

Tm(G · m) * (TmM)H = Tm(N(H) · m),

Tm(Gµ · m) * (TmM)H = Tm(NGµ(H) · m).

Proof We clearly have Tm(N(H) · m) ' Tm(G · m). If - is in theLie algebra of N(H) and h " H, by Lemma 1.3.2 we conclude thatAdh- $ - " h so that (Adh- $ -)M (m) = 0 and hence

-M (m) = (Adh- $ -)M (m) + -M (m) = (Adh-)M (m)= (Adh-)M (h · m) = (&"

h"1-M )(h · m)= (Tm&h & -M & &h"1)(h · m) = Tm&h(-M (m))

which shows that -M (m) " TmM is fixed by all h " H, that is,-M (m) " (TmM)H . We have hence Tm(N(H) · m) ' (TmM)H andtherefore Tm(N(H) · m) ' Tm(G · m) * (TmM)H .

To prove the converse inclusion, note that if -M (m) " Tm(G ·m)*(TmM)H , that is, -M (m) is fixed by all h " H, then

-M (m) = Tm&h(-M (m)) = (Tm&h & -M & &h"1)(h · m)= (&"

h"1-M )(h · m) = (Adh-)M (m)

which states that Adh-$- " gm = h for all h " H. By Lemma 1.3.2 itfollows that - is in the Lie algebra of N(H) which proves the inclusionTm(G · m) * (TmM)H ' Tm(N(H) · m) and hence the lemma.

The second statement follows by replacing G by Gµ. !


Remark 1.3.3 If the G–action is proper, the equality in the thesisof Lemma 1.3.3 reads:

Tm(G · m) * TmMH = Tm(N(H) · m),

Tm(Gµ · m) * TmMH = Tm(NGµ(H) · m).

In this case, an alternative proof of this lemma can be given by usingidentity 1.3.1. Indeed,

Tm(G · m) * TmMH = Tm(G · m) * (TmM)H (by (1.3.1))

= (Tm(G · m))H

= Tm(G · m)H (again, by (1.3.1)).

However,

(G · m)H = {g · m | Gg·m = gHg%1 = H}= {g · m | g " N(H)}= N(H) · m.

So,

Tm(G · m) * TmMH = Tm(G · m)H = Tm(N(H) · m),

as required. "

Lemma 1.3.4 Let H be a compact subgroup of a Lie group G. Thenany of the inclusions gHg%1 ' H or H ' gHg%1 with g " G isequivalent to the identity gHg%1 = H.

Proof Clearly, it is enough to prove the result for just one of theinclusions. If gHg%1 ' H, we can construct a descending sequence

H / gHg%1 / g2Hg%2 / g3Hg%3 . . .

If this sequence stops let’s say at the nth step, that is, gn+1Hg%n%1 =gnHg%n, then the lemma follows. Actually this is so because of thecompactness of H; indeed, when we go from the ith to the (i + 1)thstep, since conjugation by g is a di!eomorphism, if the inclusion isstrict, gi+1Hg%i%1 is obtained from giHg%i just by dropping someof its connected components. Since H is compact it has only a finitenumber of components, so the strict inclusions have to stop after afinite number of steps. !


Proposition 1.3.3 The sets defined in Proposition 1.3.2 satisfy

(i) M(H) = G · MH .

(ii) MH = MH *M(H).

Proof (i) Let z "M(H) arbitrary. By definition, there exists an el-ement g " G, such that Gz = gHg%1. Hence Gg"1·z = g%1gHgg%1 =H, and therefore z = g · (g%1 · z) " G · MH . The converse inclusionfollows from MH 'M(H), and the G–invariance of M(H).(i) The only nontrivial inclusion is MH *M(H) ' MH . If m " MH

then H ' Gm. If also m " M(H), there is a g " G such that Gm =gHg%1, hence H ' gHg%1 which, by the previous lemma impliesthat Gm = H. !

If the proper action in question is the representation of a compactLie group G, that is, in this case our manifold is some vector spaceV on which G acts linearly, the array of tools available is immense.In particular, we quote for future reference two results that will beused in a variety of situations. Recall that a representation of G onthe vector space V is said to be irreducible if the only G–invariantsubspaces of V are {0} and V .

Proposition 1.3.4 Let G be a compact Lie group acting on V . LetW ' V be a G–invariant subspace of V . Then, there is a G–invariantcomplementary subspace Z ' V such that

V = W ( Z.

Proof See [GSS, Proposition 2.1]. !

Theorem 1.3.1 Let G be a compact Lie group acting on V . Then:

(i) Up to G–equivariant isomorphisms (G–isomorphisms), there area finite number of distinct (that is, not G–isomorphic) G–irreducible subspaces of V . Call these U1, . . . Ut.

(ii) Define Vk to be the sum of all G–irreducible subspaces W of Vsuch that W is G–isomorphic to Uk. Then

V = V1 ( · · ·( Vt.

§ 1.4. Hamiltonian Actions and Momentum Maps 19

We say that the above direct sum decomposition is the G–isotypic decomposition of V and that Vk is the isotypiccomponent of V of type Uk. By construction, this decomposi-tion is unique.

(iii) Let A : V % V be a G–equivariant linear mapping. ThenA(Vk) ' Vk for k = 1, . . . , t.

Proof See [GSS, Theorems 2.5 and 3.5]. !

1.4 Hamiltonian Actions and MomentumMaps

If the G–manifold M is also a Poisson (respectively symplectic) space,we will be interested in actions that respect this additional structure.

Definition 1.4.1 Let (M, {·, ·}) be a Poisson manifold (respectively(M, !) a symplectic manifold), let G be a Lie group, and let & :G )M % M be a smooth left action of G on M . We say that theaction & is canonical if & acts by canonical transformations; thatis, for any f, g " C!(M) and any g " G

&"g{f, g} = {&"

gf, &"gg} (resp. &"

g! = ! ).

As we announced in the introduction to the previous section inmany instances, the existence of canonical actions in a system, im-plies the presence of global conserved quantities for its associatedtime evolution. These integrals of motion are represented by the mo-mentum mapping associated to the action, whose existence takesplace whenever the infinitesimal generator vector fields -M are glob-ally Hamiltonian. In other words, for each - " g, there is a globallydefined function J# " C!(M), such that

-M = XJ! .

Definition 1.4.2 Let G be a Lie group acting canonically on thePoisson (respectively symplectic) manifold (M, {·, ·}) (respectively(M, !)). Suppose that for all - " g, the vector field -M is globallyHamiltonian, with Hamiltonian function J# " C!(M). The mapJ : M % g" defined by the relation

0J(z), -1 = J#(z),


for all - " g and z " M , is called the momentum mapping of theaction.

Notice that the momentum map is not uniquely determined; in-deed, J1 and J2 are momentum maps for the same canonical actioni! for any - " g

J#1 $ J#

2 " C(M).

Obviously, if M is symplectic and connected, then J is determinedup to a constant in g".

Example 1.4.1 Linear Momentum Let M be in this case thephase space of the N–particle system, that is T "R3N , and G, theadditive group R3 acting by spatial translation on every factor: v ·(qi, pi) = (qi + v, pi), with i = 1, . . . , N . This action is canonicaland has a momentum map associated that coincides with the classicallinear momentum, that is

J : T "R3N $% Lie(R3) 2 R3

(qi, pi) -$%%N

i=1 pi. "

Example 1.4.2 Angular Momentum Let SO(3) act on T "R3, bythe lifted action of SO(3) on R3, that is A · (q, p) = (Aq, Ap).This action is canonical and has a momentum map associated thatcoincides with the classical angular momentum, that is

J : T "R3 $% so(3) 2 R3

(q, p) -$% q ) p. "

Example 1.4.3 Symplectic Linear Actions Let (V, !) be a sym-plectic linear space and let G be a subgroup of the linear symplecticgroup, acting on V by matrix multiplication. By the choice of G thisaction is canonical and has a momentum map given by the expression:

0J(v), -1 =12!(-V (v), v),

for - " g and v " V arbitrary. "

The following celebrated theorem whose proof can be found forinstance in [MR94], shows the physical importance of the momentummap: it relates symmetries with conserved quantities.


Theorem 1.4.1 (Noether) Let G be a Lie group acting canonicallyon the Poisson (in particular symplectic) manifold M . We supposethat this action has an associated momentum map J : M % g". Ifh " C!(M) is a G–invariant Hamiltonian; that is h &&g = h for allg " G, then J is a constant of the motion for the Hamiltonian vectorfield associated to h; that is,

J & Ft = J,

for arbitrary time t. The same conclusion holds if the global G–invariance of h is relaxed to infinitesimal G–invariance, that is-M [h] = 0 for all - " g.

A very important class of canonical actions in a Poisson mani-fold endowed with momentum maps is constituted by the globallyHamiltonian actions. By construction, there is a well–defined ac-tion of G on both the domain and the range (the coadjoint actionof G on g") of J. A canonical action of G on M endowed with amomentum map is said to be globally Hamiltonian if J is equivariantwith respect to these actions, that is

Ad"g"1 & J = J & &g, (1.4.1)

for all g " G or, equivalently

JAdg#(g · z) = J#(z),

for all g " G, - " g, and z " M . It is very easy to verify that forglobally Hamiltonian actions, the mapping defined by

J : g $% C!(M)- -$% J#

is a Lie algebra homomorphism, that is

J[#, $] = {J#, J$}, (1.4.2)

for any -, . " g. The equivariance of the momentum map impliesthese commutation relations, however it is not necessary for themto hold; indeed J is an algebra homomorphism i! J is infinitesi-mally equivariant, that is, the equality coming from the derivativeof relation (1.4.1), with respect to g in the direction -, that is,

TzJ · -M (z) = $ad"#J(z),


is verified for all - " g. Actions admitting infinitesimally equiv-ariant momentum maps are called Hamiltonian actions. See forinstance [GS84b, MR94, Sou97], for the relation between equivari-ance and infinitesimal equivariance: if M and G are connected theyare equivalent.

One might ask about the circumstances in which a non equivariantmomentum map can be modified to make it be equivariant. The nextresult shows that if G is compact, this can always be done.

Theorem 1.4.2 Let G be a compact Lie group acting in a canonicalfashion on the Poisson manifold M and having a momentum mapJ : M % g". Then J can always be chosen to be equivariant.

Proof For each g " G, define

Jg(z) = Ad"g"1J(g%1 · z).

or, equivalently,

J#g = J(Adg"1-) & &g"1 .

Then, Jg is also a momentum map for the G–action on M . Indeed,if z " P , - " g, and F : M % R we have

{F,J#g}(z) = $dJ#

g(z) · XF (z)

= $dJAdg"1#(g%1 · z) · Tz&g"1 · XF (z)

= $dJAdg"1#(g%1 · z) · (&"gXF )(g%1 · z)

= $dJAdg"1#(g%1 · z) · X"#gF (g%1 · z)

= {&"gF,JAdg"1#}(g%1 · z)

= (Adg"1-)M [&"gF ](g%1 · z)

= (&"g-M )[&"

gF ](g%1 · z)

= dF (z) · -M (z)

= {F,J#}(z).

Therefore, {F,J#g$J#} = 0 for every F : M % R, that is, J#

g$J# is aCasimir function on M for every g " G and every - " g. Now define

0J1 =&

GJgdg


where dg denotes the Haar measure on G normalized such that thetotal volume of G is one. Equivalently, this definition states that

0J1# =&

GJ#

gdg

for every - " g. By the linearity of the Poisson bracket in each factor,it follows that

{F, 0J1#} =&

G{F,J#

g}dg =&

G{F,J#}dg = {F,J#}.

Thus 0J1 is also a momentum map for the G–action on M and 0J1#$J# is a Casimir function on M for every - " g, that is, 0J1 $ J "L(g, C(M)).

The momentum map 0J1 is equivariant. Indeed, noting that Jg(h·z) = Ad"

h"1Jg"1h(z) and using invariance of the Haar measure onG under translations and inversion, for any h " G, we have afterchanging variables g = hk%1 in the third equality below,

0J1(h · z) =&

GAd"

h"1Jg"1h(z)dg = Ad"h"1

&

GJg"1h(z)dg

= Ad"h"1

&

GJk(z)dk = Ad"

h"10J1(z). !

In our discussion we will come across canonical actions endowedwith a momentum map that cannot be chosen to be equivariant.In this case, one can modify the coadjoint action in g" so that themomentum map is equivariant with respect to this new G–structurein g". In the following discussion, we restrict to the situation in whichM is a symplectic manifold. For the Poisson case see [MR94, Chapter12].

Definition 1.4.3 Let G be a Lie group. A g"–valued one–cocyclein G is a map

0 : G $% g"

such that the cocycle identity

0(gh) = 0(g) + Ad"g"10(h)


holds. A cocycle ( is said to be a coboundary if there is a , " g"

such that

((g) = , $Ad"g"1,.

The set of on–cocycles of a group forms a vector space and thecoboundaries are a vector subspace. The quotient space is called thefirst g"–valued cohomology of G.

The proof of the following proposition can be found in [AM78,MR94].

Proposition 1.4.1 Let (M, !) be a symplectic manifold and G bea Lie group acting on M in a canonical fashion with an associatedmomentum map J : M % g". We define the non equivariancecocycle associated to J as the map

0 : G $% g"

g -$% J(&g(z))$Ad"g"1(J(z)).

Then,

(i) The definition of 0 does not depend on the choice of z "M .

(ii) The mapping 0 is a g"–valued one cocycle on G.

(iii) If J$ is another momentum map for the same canonical actionof G on M then its non equivariance cocycle 0$ is in the samecohomology class as 0; that is, 0 $ 0$ is a coboundary.

Using the non equivariance cocycle we can define a new action ofG on g", with respect to which the momentum map J is equivariant.

Definition 1.4.4 Let G be a Lie group acting canonically on thesymplectic manifold (M, !), with associated momentum map J :M % g". If 0 : G % g" is the non equivariance cocycle of J, wedefine the a!ne action of G on g" with cocycle 0, as the map

' : G) g" $% g"

(g, µ) -$% Ad"g"1µ + 0(g).

The proof of the following result is given in detail in [AM78].

Proposition 1.4.2 In the terms of the previous definition, the map' is an action of G on g". The momentum map J : M % g" isequivariant with respect to the symplectic action & in M , and thea"ne action ' in g".

§ 1.5. A First Approach to Reduction Theory 25

1.5 A First Approach to Reduction Theory

In this section we present a method to take advantage of the conservedquantities associated to a symmetric system to reduce or eliminatevariables in the search for the solutions of the problem. This pro-cedure, called reduction, is a modern formulation of the classicaltechniques of variables elimination, using the invariance properties ofthe problem system as, for instance in the reduction to the center ofmass frame in the two–body problem, using translational invariance,or the Jacobi’s elimination of the node, that allows one to eliminatefour variables in the n–body problem using rotational invariance.

The general setting for reduction goes back to E. Cartan [C22],Meyer [Mey73], Smale [S70], Souriau [Sou97], and Robbin [R73].We will follow the final formulation, due to Marsden and Wein-stein [MW74]. In all these works reduction is carried out in theabsence of singularities. As a first approach we will describe this sit-uation. The singular case is the subject of Chapter 2. Due to theneeds of subsequent discussions we make the presentation with thedegree of generality that will be needed there.

The simplest reduction strategy is the so called Poisson reduc-tion of a Poisson manifold with symmetry. Let (M, {·, ·}) be a Pois-son manifold and let G be a Lie group acting canonically on M . If weassume the action to be free and proper, Proposition 1.3.1 guaranteesthat the orbit space M/G, is a smooth manifold, and the canonicalprojection / : M % M/G is a smooth surjective submersion. More-over, the Poisson structure in M is projected naturally (or reduced)to the quotient, along with any dynamics generated on (M, {·, ·}) byany G–invariant Hamiltonian.

Theorem 1.5.1 (Poisson reduction) Let (M, {·, ·}) be a Poissonmanifold and let G be a Lie group acting canonically, freely and prop-erly on M . Then,

(i) The orbit space M/G is a Poisson manifold with the Poissonbracket {·, ·}M/G, characterized by

{f, g}M/G & / = {f & /, g & /},

where f, g : M/G % R are two arbitrary functions, and / :M %M/G is the canonical smooth surjective submersion.


(ii) Let h be a G–invariant function in M . The Hamiltonian flow Ft

of Xh commutes with the G–action, so it induces a flow FM/Gt

on M/G characterized by

/ & Ft = FM/Gt & /.

(iii) The flow FM/Gt is Hamiltonian in (M/G, {·, ·}M/G), with

Hamiltonian function [h] " C!(M/G) defined by

[h] & / = h.

We will call [h] the reduced Hamiltonian. The vector fieldsXh and X[h] are /–related.

Proof See Section 10.7 in [MR94]. !If the canonical action of G on M has an associated momen-

tum map, the previous result can be refined by taking advantage ofNoether’s Theorem, that is, the level sets of J are preserved by theHamiltonian dynamics induced by G–invariant Hamiltonians. By re-stricting the constructions of Theorem 1.5.1 to the level sets of themomentum map we will obtain new Poisson manifolds. When Mis symplectic, these Poisson manifolds are actually symplectic (theMarsden–Weinstein reduced spaces), and considered as subsetsof the Poisson reduced space M/G they will be its symplectic leaves.

Theorem 1.5.2 (Point reduction) Let (M, {·, ·}) be a Poissonmanifold, and let G be a Lie group acting canonically, freely, andproperly on M . Suppose that this action has an associated momen-tum map J : M % g", with non equivariance cocycle 0. Let µ " g"

be a regular value of J, and denote by Gµ the isotropy of µ under thea"ne action of G on g". Then,

(i) The set Mµ := J%1(µ)/Gµ is a Poisson manifold, with Poissonbracket {·, ·}Mµ, characterized by

{fµ, gµ}Mµ([m]µ) = {f, g}(m), (1.5.1)

for any fµ, gµ " C!(Mµ). The functions f, g " C!(M)G arearbitrary extensions of fµ &/µ, gµ &/µ " C!(J%1(µ))Gµ, where/µ : J%1(µ) % Mµ is the canonical surjective submersion. Wedenote [m]µ := /µ(m).


(ii) Let h " C!(M)G be a G–invariant Hamiltonian. The Hamilto-nian flow Ft of h leaves invariant the connected components ofJ%1(µ), and commutes with the G–action, so it induces a flowFµ

t on Mµ, uniquely determined by

/µ & Ft & iµ = Fµt & /µ, (1.5.2)

where iµ : J%1(µ) )%M is the canonical injection.

(iii) The flow Fµt is Hamiltonian in (Mµ, {·, ·}Mµ), with Hamiltonian

function hµ " C!(Mµ) defined by

hµ & /µ = h & iµ,

We will call hµ the reduced Hamiltonian. The vector fieldsXh and Xhµ are /µ–related.

(iv) Let k " C!(M)G another G–invariant function. Then, {h, k}is also G–invariant and {h, k}µ = {hµ, kµ}Mµ.

Remark 1.5.1 The hypothesis on the regularity of µ " g" in thestatement of the previous theorem, as well as in the coming resultswhere it also appears, can be relaxed to requiring µ to be just a cleanvalue of the momentum map J, that is, J%1(µ) is a submanifold ofM , whose tangent space at any z " J%1(µ) is such that Tz(J%1(µ)) =kerTzJ. This property holds for instance when J is a subimmersion,that is, a constant rank mapping (see [AMR, Theorem 3.5.17]). "

Before proving Theorem 1.5.2 we will introduce two lemmas thatwill be useful in a number of situations

Lemma 1.5.1 (Reduction Lemma) Let (M, {·, ·}) be a Poissonmanifold where G acts properly and canonically. Let J : M % g"

be a momentum map associated to this action, with non equivariancecocycle 0, and µ " g", m "M such that J(m) = µ. Then,

(i) J%1(G · µ) = G · J%1(µ) = {g · z | g " G and J(z) = µ}, wherethe dots indicate the canonical action of G on M and the a"neaction of G on g" with cocycle 0.

(ii) Gµ · m = (G · m) * J%1(µ), where Gµ denotes the stabilizer ofµ " g" under the a"ne action of G on g".


(iii) Tm(Gµ · m) = Tm(G · m) * kerTmJ. If µ " g" is a regular valueof J this expression amounts to saying that J%1(µ) and G · mintersect cleanly, that is,

Tm(Gµ · m) = Tm(G · m) * Tm(J%1(µ)).

(iv) Let µ " g" be a regular value of J. The map J is transversalto the orbit Oµ under the a"ne action by G; that is, for anyz " J%1(Oµ),

(TzJ)(TzM) + TJ(z)Oµ = g".

Consequently, J%1(Oµ) is a submanifold of M . Moreover,

Tz(J%1(Oµ)) = Tz(G · z) + ker(TzJ).

(v) If M is symplectic with symplectic form !, the sets kerTmJ andTm(G ·m) are !-orthogonal complements of each other, that is,

kerTmJ = (Tm(G · m))%.

Proof It is a straightforward generalization of the classical ReductionLemma that can be found for instance in [AM78, MR]. #

Lemma 1.5.2 Let (M, !) be a symplectic manifold and let G be aLie group acting freely, properly, and canonically on M with associ-ated momentum map J : M % g". Let µ " g" be a regular valueof J so that the level set J%1(µ) is a submanifold of M . Then,for any m " J%1(µ), every vector v " TmJ%1(µ) can be written asTmiµ(v) = Xf (m), where f " C!(M)G, and iµ : J%1(µ) )%M is thecanonical injection.

Proof It will be obtained as a corollary to the reconstruction equa-tions in Chapter 3 (see Lemma 3.4.2). #Proof of Theorem 1.5.2 (i) Since µ is a regular value of J and Jis G–equivariant, the set J%1(µ) is a closed Gµ–invariant submanifoldof M . Since Gµ is a closed subgroup of G the action by restrictionof the Lie group Gµ on J%1(µ) is proper and free which, by Propo-sition 1.3.1, implies that Mµ := J%1(µ)/Gµ is a well defined smoothmanifold such that /µ : J%1(µ)%Mµ is a surjective submersion.


We now show that the bracket {·, ·}Mµ in the statement is well–defined and induces a Poisson structure on Mµ. Firstly, the exten-sions f, g " C!(M)G of fµ & /µ, gµ & /µ " C!(J%1(µ))Gµ are alwaysavailable. Indeed, fµ &/µ can be extended smoothly to a G–invariantfunction f $ defined on J%1(Oµ) = G · J%1(µ) by

f $(g · z) = (fµ & /µ)(z), with g " G, z " J%1(µ).

Notice that f $ is well–defined: if g · z = g$ · z$, with g, g$ " G, andz, z$ " J%1(µ), then z = g%1g$ · z$. Applying J to both sides of thisequality we get µ = g%1g$ · µ, and hence g%1g$ " Gµ. Consequently,

f $(g · z) = (fµ & /µ)(z) = (fµ & /µ)(g%1g$ · z$)= (fµ & /µ)(z$) = f $(g$ · z$).

By construction, f $ is G–invariant. Its smoothness follows fromvisualizing G · J%1(µ) as G ) J%1(µ), which allows us to writef $ = fµ & /µ & p2, where p2 is the projection on the second factorin G ) J%1(µ). Now, by Proposition 1.3.1, there is a smooth G–invariant extension f of f $ to M .

It remains to be seen how the definition of {·, ·}Mµ in (1.5.1) is in-dependent of the extensions f, g " C!(M)G as well as of the choice ofm " Gµ ·m. First, let f, g " C!(M)G and f $, g$ " C!(M)G be twodi!erent extensions of fµ & /µ, gµ & /µ " C!(J%1(µ))Gµ respectively.We write

{fµ, gµ}Mµ([m]µ) = {f, g}(m) = Xg[f ](m)

=d

dt

$$$$t=0

f(F gt (m)) =

d

dt

$$$$t=0

f $(F gt (m));

where we used the fact that, by Noether’s Theorem, the flow F gt of

the Hamiltonian vector field Xg, satisfies F gt (m) " J%1(µ) for any

time t, and that f |J"1(µ)= f $ |J"1(µ). Since the same is true for gand g$, we continue

d

dt

$$$$t=0

f $(F gt (m)) = {f $, g}(m) = ${g, f $}(m)

= $Xf ! [g](m) = $ d

dt

$$$$t=0

g(F f !

t (m))

= $ d

dt

$$$$t=0

g$(F f !

t (m)) = {f $, g$}(m)

= {fµ, gµ}([m]µ),


as required. The choice of m " Gµ · m is equally irrelevant since iff, g " C!(M)G, then we also have {f, g} " C!(M)G; indeed, sincethe G–action is canonical, for any g " G

{f, g} & &g = &"g{f, g} = {&"

gf, &"gg} = {f, g}.

Hence, the bracket {·, ·}Mµ is well defined and it is a Poisson bracketsince it inherits all the necessary properties from the Poisson charac-ter of {·, ·}.(ii) By Noether’s Theorem Ft leaves invariant the connected compo-nents of J%1(µ). The G–invariance of h and the canonical characterof the action imply, as a corollary to Proposition 1.1.1, that for anyg " G,

Ft & &g = &g & Ft,

so Fµt is the flow on Mµ that makes the following diagram commuta-

tive:

J%1(µ)Ft#iµ$$$% J%1(µ)

&µ

""#""#&µ

MµF µ

t$$$% Mµ.

(iii) Due to the G–invariance of h, the function hµ " C!(Mµ) isuniquely determined by the identity hµ &/µ = h& iµ. Let Y " X(Mµ)be the vector field on Mµ whose flow is Fµ

t . By construction Y is /µ–related to Xh. Indeed, di!erentiating relation (1.5.2) with respect tothe time t, we obtain

T/µ &Xh & iµ = Y & /µ.

We now verify that Y is a Hamiltonian vector field with Hamiltonianfunction hµ, that is, Y = Xhµ . We will show that for any fµ "C!(Mµ)

Y [fµ] = {fµ, hµ}Mµ = Xhµ [fµ].

By construction, h is a G–invariant extension of hµ & /µ. Let f "


C!(M)G be a G–invariant extension of fµ&/µ. Then for any m "M ,

Xhµ [fµ]([m]µ) = {fµ, hµ}Mµ([m]µ) = {f, h}(m) = Xh[f ](m)

=d

dt

$$$$t=0

f(Ft(m)) =d

dt

$$$$t=0

(fµ & /µ & Ft & iµ)(m)

=d

dt

$$$$t=0

(fµ & Fµt & /µ)(m) = Y [fµ]([m]µ),

as required.(iv) The G–invariance of {h, k} was already shown in the proof ofpart (i). The function {h, k}µ " C!(Mµ) is the unique mapping forwhich {h, k}µ & /µ = {h, k} & iµ. By the definition of the Poissonstructure on Mµ, the function {hµ, kµ}Mµ satisfies this equality andtherefore coincides with {h, k}µ. !

We now show how, when M is symplectic, the point reducedspaces Mµ of Theorem 1.5.2 are symplectic and, understood as sub-sets of M/G, are its symplectic leaves. The manifold Mµ along withthe symplectic structure described in the following theorem is calledthe Marsden–Weinstein reduced space [MW74].

Theorem 1.5.3 (Symplectic point reduction) In the hypothe-ses of Theorem 1.5.2, let’s suppose that the Poisson bracket on Mcomes from a symplectic form !. Then the bracket {·, ·}Mµ in Mµ

comes from the symplectic form !µ, uniquely determined by the ex-pression

/"µ!µ = i"µ!. (1.5.3)

Let jµ : Mµ % M/G be the mapping defined by the commutativediagram

J%1(µ)iµ$$$% M

&µ

""#""#&

Mµjµ$$$% M/G.

The mapping jµ is a Poisson injective immersion. Moreover, thesymplectic manifolds (Mµ, !µ), considered as subsets of the Poissonmanifold (M/G, {·, ·}M/G) via jµ, are its symplectic leaves.


Proof We establish the symplectic character of Mµ by showing thatthe Poisson bracket {·, ·}Mµ can be written in terms of a nonde-generate symplectic form. Let v, w " TmJ%1(µ) arbitrary. ByLemma 1.5.2, v and w can be written as Tmiµ(v) = Xf (m) andTmiµ(w) = Xg(m), with f, g " C!(M)G. We define a two–form !µ

on Mµ by

!µ([m]µ)(Tm/µ(v), Tm/µ(w))= !µ([m]µ)(Tm/µ(Xf (m)), Tm/µ(Xg(m)))= !µ([m]µ)(Xfµ([m]µ), Xgµ([m]µ))= {fµ, gµ}Mµ([m]µ),

where we used that, by Theorem 1.5.2, the Hamiltonian vector fieldsXf , Xg, and Xfµ , Xgµ are /µ–related, respectively.

The Jacobi identity for the bracket {·, ·}Mµ and its antisymme-try guarantee that !µ is closed and antisymmetric. We now seethat !µ is nondegenerate. Indeed, if !µ([m]µ)(Tm/µ(v), Tm/µ(w)) =!µ([m]µ)(Xfµ([m]µ), Xgµ([m]µ)) = 0 for any w " TmJ%1(µ), then{fµ, gµ}Mµ([m]µ) = {f, g}(m) = df(m) · Xg(m) = df(m) · w = 0,and hence df |J"1(µ) (m) = d(f & iµ)(m) = dfµ([m]µ) & Tm/µ = 0.Since Tm/µ is surjective dfµ([m]µ) = 0 necessarily, and given thatXfµ([m]µ) (and thus Tm/µ(v) as well) depend on fµ only throughdfµ([m]µ), then Tm/µ(v) = 0 necessarily, as required. The pair(Mµ, !µ) is therefore a symplectic manifold.

We now see that !µ is determined by the Marsden–Weinsteinexpression (1.5.3). Let m be an arbitrary point in J%1(µ) and v, w "TmJ%1(µ). Using Lemma 1.5.2, we write again Tmiµ(v) = Xf (m)and Tmiµ(w) = Xg(m), with f, g " C!(M)G. The claim followsfrom the following chain of equalities in which we will use that, byTheorem 1.5.2, the Hamiltonian vector fields Xf , Xg, and Xfµ , Xgµ

are /µ–related, respectively:

/"µ!µ(m)(v, w) = !µ([m]µ)(Tm/µ(v), Tm/µ(w))

= !µ([m]µ)(Tm/µ(Xf (m)), Tm/µ(Xg(m)))= !µ([m]µ)(Xfµ([m]µ), Xgµ([m]µ))= {fµ, gµ}Mµ([m]µ) = {f, g}(m)= !(m)(Xf (m), Xg(m))= i"µ!(m)(v, w).


We now study the map jµ. A quick inspection of the commutativediagram that defines it shows that it is injective. In order to showthat it is an immersion let’s take [v]µ = Tm/µ(v) " T[m]µMµ such thatT[m]µjµ([v]µ) = 0. Since this is equivalent to Tm(jµ &/µ)(v) = Tm(/ &iµ)(v) = 0, we have that v " Tm(G · m) * TmJ%1(µ) = Tm(Gµ · m) =kerTm/µ, which implies that [v]µ = 0. Therefore, the set jµ(Mµ) isan immersed submanifold of M/G. It is also easy to see that jµ is aPoisson map. Indeed, for any f, g " C!(M/G):

j"µ{f, g}M/G([m]µ) = {f, g}M/G([m]) = {f & /, g & /}(m).

Now, using that f &/ and g&/ are G–invariant extensions of f &jµ&/µ

and g & jµ & /µ " C!(J%1(µ))Gµ to C!(M)G, we have that

{j"µf, j"µg}Mµ([m]µ) = {f & jµ, g & jµ}Mµ([m]µ)

= {f & /, g & /}(m) = j"µ{f, g}M/G([m]µ),

as required.Regarding the symplectic leaves of M/G, we first prove that for

any jµ([m]µ) " jµ(Mµ), the symplectic leaf in M/G that goes throughjµ([m]µ), is included in jµ(Mµ) because of Noether’s Theorem. In-deed, let FM/G

t be a Hamiltonian flow on M/G, coming from a G–equivariant Hamiltonian flow Ft on M . We show that for any time t,FM/G

t (jµ([m]µ)) stays in jµ(Mµ). By definition,

FM/Gt (jµ([m]µ)) = (FM/G

t & / & iµ)(m) = (/ & Ft & iµ)(m) = /(m$),

where, by Noether’s Theorem, m$ is some element in J%1(µ). Hence,

FM/Gt (jµ([m]µ)) = (/ & iµ)(m$) = jµ([m$]µ).

Since Mµ is symplectic, in order to complete the proof we just needto show that any function f " C!(jµ(Mµ)) admits an extension toC!(M/G), which follows easily from the proof to part (i) of The-orem 1.5.2, where we showed that any function on C!(J%1(µ))Gµ

admits a smooth G–invariant extension in C!(M)G. !As we saw in the Reduction Lemma, the G–equivariance of J

with respect to the a"ne action implies that J%1(Oµ) = G · J%1(µ).Since, by Noether’s Theorem, the set J%1(µ) is preserved by thedynamics induced by G–invariant Hamiltonians, J%1(Oµ) shares the


same property; moreover since this set is G–invariant, the quotientMOµ := J%1(Oµ)/G is a good candidate for a reduced space. In thefollowing results we see that this is the case and that, in addition,MOµ is canonically di!eomorphic to Mµ. We will call MOµ the orbitreduced space. This reduced structure was first characterized byMarle [Mar76] and by Kazhdan, Kostant, and Sternberg [KKS78].

Theorem 1.5.4 (Orbit reduction) Let (M, {·, ·}) be a Poissonmanifold, and let G be a Lie group acting canonically, freely, andproperly on M . Suppose that this action has a momentum map as-sociated J : M % g", with non equivariance cocycle 0. Let µ " g" bea regular value of J, and denote by Gµ the isotropy of µ under thea"ne action of G on g". Then,

(i) The set MOµ := J%1(Oµ)/G is a Poisson manifold, with Poissonbracket {·, ·}MOµ

, characterized by

{fOµ , gOµ}MOµ([m]Oµ) = {f, g}(m), (1.5.4)

for any fOµ , gOµ " C!(MOµ). The functions f, g "C!(M)G are arbitrary extensions of fOµ & /Oµ , gOµ & /Oµ "C!(J%1(Oµ))G, where /Oµ : J%1(Oµ) % MOµ is the canonicalsurjective submersion. We denote [m]Oµ := /Oµ(m).

(ii) Let h " C!(M)G be a G–invariant Hamiltonian. The Hamil-tonian flow Ft of h leaves invariant the connected componentsof J%1(Oµ), and commutes with the G–action, so it induces aflow F

Oµt on MOµ, uniquely determined by

/Oµ & Ft & iOµ = FOµt & /Oµ , (1.5.5)

where iOµ : J%1(Oµ) )%M is the canonical injection.

(iii) The flow FOµt is Hamiltonian in (MOµ , {·, ·}MOµ

), with Hamil-tonian function hOµ " C!(MOµ) defined by

hOµ & /Oµ = h & iOµ ,

We will call hOµ the reduced Hamiltonian. The vector fieldsXh and XhOµ

are /Oµ–related.

(iv) Let k " C!(M)G another G–invariant function. Then, {h, k}is also G–invariant and {h, k}Oµ = {hOµ , kOµ}MOµ

.


Proof It is identical to the proof of Theorem 1.5.2. In this case theextensions f, g " C!(M)G of fOµ &/Oµ , gOµ &/Oµ " C!(J%1(Oµ))G

are guaranteed directly by Proposition 1.3.1. !Analogous to the case of point reduction, if M is symplectic, MOµ

is symplectic and also, these orbit reduced spaces constitute the sym-plectic leaves of the Poisson manifold M/G; this circumstance givesus the first hint on the close relationship between the point and orbitreduction schemes. In order to sort out intrinsically the symplecticstructure of MOµ we discuss a little the symplectic character of thea"ne orbit Oµ. We first introduce the two–cocycle %, associatedto the one–cocycle 0, along with some of its properties. The proofof this result is a straightforward generalization of [AM78, Theorem4.2.8].

Theorem 1.5.5 Let G be a Lie group and 0 : G% g" a cocycle. Wedefine the two–cocycle % associated to 0 as

% : g) g $% R(-, .) -$% %(-, .) = d'0$(e) · -,

where '0$ : G% R is defined by '0$(g) = 0(g) · .. Then,

(i) % is a skew symmetric bilinear form on g satisfying Jacobi’s iden-tity.

(ii) If % is the two–cocycle associated to the non equivariance cocycleof a momentum map J : M % g" for the canonical action of Gon the Poisson manifold (M, {·, ·}), then for any -, . " g,

{J#, J$} = J[#, $] $ %(-, .).

This implies that for arbitrary z "M and . " g,

TzJ · .M (z) = $ad"$J(z) + %(., ·).

Since %(-, .) doesn’t depend on M , we also have that

X{J!,J"} = XJ[!, "] .

With the help of the two–cocycle we define a symplectic structurefor the orbits of the a"ne action of G on g". First, a straightforwardcomputation gives us the following


Lemma 1.5.3 Let G be a Lie group and 0 : G % g" a cocycle. Weconsider G acting on g" via the a"ne action. Let , be an arbitraryelement in the a"ne orbit Oµ of µ " g", then

T'Oµ = {-g#(,) = $ad"#, + %(-, ·) | - " g}.

The following theorem describes the symplectic nature of Oµ.The structure that we describe is associated to various names: Lie,Borel, Weil and, more recently Kirillov [K62, K76], Arnold [A66a],Kostant [Ko70], and Souriau [Sou97].

Theorem 1.5.6 In the terms of the previous lemma, the a"ne orbitOµ is a symplectic manifold with symplectic structures !±

Oµgiven by:

!±Oµ

(,)(-g#(,), .g#(,)) = ±0,, [-, .]1 3 %(-, .),

for arbitrary , " Oµ, and -, . " g. The symplectic structures !±Oµ

onOµ are usually called the ±–Kostant–Kirillov–Souriau (KKS)symplectic forms. The symbols O+

µ and O%µ will denote the pairs

(Oµ, !O+µ) and (Oµ, !O"

µ), respectively.

Proof It is a straightforward generalization of [MR94, Theorem14.4.1]. !.

We now show the orbit reduction analog to Theorem 1.5.3.

Theorem 1.5.7 (Symplectic orbit reduction) In the hypothesesof Theorem 1.5.4, let’s suppose that the Poisson bracket on M comesfrom a symplectic form !. Then, the bracket {·, ·}MOµ

in MOµ comesfrom the symplectic form !Oµ, uniquely determined by the expression

i"Oµ! = /"Oµ

!Oµ + J"Oµ!+Oµ

, (1.5.6)

where JOµ is the restriction of J to J%1(Oµ), and !+Oµ

is the +–symplectic structure for the a"ne orbit introduced in Theorem 1.5.6.Let jOµ : MOµ % M/G be the mapping defined by the commutativediagram

J%1(Oµ)iOµ$$$% M

&Oµ

""#""#&

MOµ

jOµ$$$% M/G.


The mapping jOµ is a Poisson injective immersion. Moreover, thesymplectic manifolds (MOµ , !Oµ), considered as subsets of the Pois-son manifold (M/G, {·, ·}M/G) via jOµ, are its symplectic leaves.

Proof We show the symplecticity of MOµ in a similar fashion to whatwe did for Mµ, in other words, we will see that the Poisson bracket{·, ·}MOµ

can be written in terms of a nondegenerate symplectic form.Let v, w " TmJ%1(Oµ) arbitrary. We assume, without loss of gener-ality, that m " J%1(µ). By Lemma 1.5.2, and the Reduction Lemma,v and w can be written as

v = -M (m) + Xf (m), with - " g, and f " C!(M)G

w = .M (m) + Xg(m), with . " g, and g " C!(M)G.

We define a two–form !Oµ on MOµ by

!Oµ([m]Oµ)(Tm/Oµ(v), Tm/Oµ(w))= !Oµ([m]Oµ)(Tm/Oµ(-M (m) + Xf (m)), Tm/Oµ(.M (m) + Xg(m)))= !Oµ([m]Oµ)(XfOµ

([m]Oµ), XgOµ([m]Oµ))

= {fOµ , gOµ}MOµ([m]Oµ),

where we used that, by Theorem 1.5.4, the Hamiltonian vector fieldsXf , Xg, and XfOµ

, XgOµare /Oµ–related, respectively.

The Jacobi identity for the bracket {·, ·}MOµand its antisymmetry

guarantee that !Oµ is closed and antisymmetric. We now see that !Oµ

is nondegenerate. Indeed, if !Oµ([m]Oµ)(Tm/Oµ(v), Tm/Oµ(w)) =!Oµ([m]Oµ)(XfOµ

([m]Oµ), XgOµ([m]Oµ)) = 0 for any w "

TmJ%1(Oµ), then {fOµ , gOµ}MOµ([m]Oµ) = {f, g}(m) = df(m) ·

Xg(m) = df(m) · (Xg(m) + .M (m)) = df(m) · w = 0, and hencedf |J"1(Oµ) (m) = d(f & iOµ)(m) = dfOµ([m]Oµ) & Tm/Oµ = 0.Since Tm/Oµ is surjective dfOµ([m]Oµ) = 0 necessarily, and giventhat XfOµ

([m]Oµ) (and thus Tm/Oµ(v) as well) depend on fOµ onlythrough dfOµ([m]Oµ), then Tm/Oµ(v) = 0 necessarily, as required.The pair (MOµ , !Oµ) is therefore a symplectic manifold.

We now show how !Oµ is determined by expression (1.5.6). Let mbe an arbitrary point that again we assume, without loss of generality,to be in J%1(µ) and v, w " TmJ%1(Oµ). We write v and w as v =Xf (m) + -M (m) and w = Xg(m) + .M (m), with f, g " C!(M)G

and -, . " g. The claim follows from the following chain of equalities


in which we will use that, by Theorem 1.5.4, the Hamiltonian vectorfields Xf , Xg, and XfOµ

, XgOµare /Oµ–related, respectively:

/"Oµ!Oµ(m)(v, w) = !Oµ([m]Oµ)(Tm/Oµ(v), Tm/Oµ(w))

= !Oµ([m]Oµ)(Tm/Oµ(Xf (m) + -M (m)),Tm/Oµ(Xg(m) + .M (m))

= !Oµ([m]Oµ)(Tm/Oµ(Xf (m)), Tm/Oµ(Xg(m))= !Oµ([m]Oµ)(XfOµ

([m]Oµ), XgOµ([m]Oµ))

= {fOµ , gOµ}MOµ([m]Oµ) = {f, g}(m)

= !(m)(Xf (m), Xg(m))= !(m)(v $ -M (m), w $ .M (m)).

Since both v $ -M (m) = Xf (m) and w $ .M (m) = Xg(m) belong toker TmJ = Tm(G · m)%, we can continue writing

!(m)(v $ -M (m), w $ .M (m)) = !(m)(v $ -M (m), w)= !(m)(v, w)$ !(m)(-M (m), w)= !(m)(v, w)$ !(m)(-M (m),

Xg(m) + .M (m))= !(m)(v, w)$ !(m)(-M (m), .M (m))= i"Oµ

!(m)(v, w)

$ !(m)(-M (m), .M (m)).

In order to conclude the required equality we just need to show that

!(m)(-M (m), .M (m)) = J"Oµ!+Oµ

(m)(v, w).

Hence, on one hand

J"Oµ!+Oµ

(m)(v, w) = !+Oµ

(µ)(TmJ(v), TmJ(w))

= !+Oµ

(µ)(TmJ(-M (m)), TmJ(.M (m)))

= !+Oµ

(µ)($ad"#µ + %(-, ·), $ad"

$µ + %(., ·)

= !+Oµ

(µ)(-g#(µ), .g#(µ)) = 0µ, [-, .]1 $ %(-, .),

where we used the expression for TmJ(-M (m)), given in Theo-rem 1.5.5. At the same time we can write for !(m)(-M (m), .M (m)):

!(m)(-M (m), .M (m)) = dJ#(m) · .M (m) = 0TmJ · .M (m), -1= 0µ, [-, .]1 $ %(-, .),


as required.Regarding the claim on the nature of the mapping jOµ and the

sets MOµ as the symplectic leaves of M/G, a computation basicallyidentical to what we did in Theorem 1.5.3 for the mapping jµ andthe sets Mµ completes the proof. !

As we already announced, the two approaches to reduction usingthe momentum map that we have studied namely, point and orbitreduction, are essentially identical. This idea is made more explicitin the following

Theorem 1.5.8 (The Reduction Diagram) Let (M, {·, ·}) be aPoisson manifold, and let G be a Lie group acting canonically, freely,and properly on M . Suppose that this action has a momentum mapassociated, with non equivariance cocycle 0. Let µ " g" be a regularvalue of J, and denote by Gµ the isotropy of µ under the a"ne actionof G on g". Let lµ : J%1(µ) )% J%1(Oµ) be the canonical injectionthen, the map Lµ : Mµ %MOµ defined by the commutative diagram

J%1(µ)lµ$$$% J%1(Oµ)

&µ

""#""#&Oµ

MµLµ$$$% MOµ .

is a Poisson di!eomorphism between (Mµ, !µ), and (MOµ , !Oµ).

Proof We will first show that Lµ is a bijective immersion, and there-fore a di!eomorphism [GHVI, Proposition IV, sec. 3.8]. First, itis one–to–one, because if [z]µ, [z$]µ " Mµ are such that [z]Oµ =Lµ([z]µ) = Lµ([z$]µ) = [z$]Oµ then, there exists an element g " Gsuch that z = g · z$. If we apply J to both sides of this equality weobtain that µ = g · µ, hence g " Gµ and, consequently [z]µ = [z$]µ.The surjectivity of Lµ is guaranteed by one of the implications of theReduction Lemma, that is, J%1(Oµ) = G · J%1(µ).

In order to check that Lµ is an immersion, we take v " Tm(J%1(µ))and [v]µ = Tm/µ · v such that T[m]µLµ([v]µ) = 0. Looking at thecommutative diagram that defines Lµ, this amounts to

Tm(Lµ & /µ)(v) = Tm(/Oµ & lµ)(v) = 0,

hence v " kerTm/Oµ = Tm(G ·m). Since v is also in Tm(J%1(µ)), bythe Reduction Lemma v " Tm(G · m) * Tm(J%1(µ)) = Tm(Gµ · m),hence [v]µ = 0, as required.


We verify that Lµ is Poisson. Let fOµ , gOµ " C!(MOµ), andf, g " C!(M)G, smooth G–invariant extensions of fOµ & /Oµ andgOµ & /Oµ " C!(J%1(Oµ))G respectively. Now, on one side

L"µ{fOµ , gOµ}MOµ

([m]µ) = {fOµ , gOµ}MOµ([m]Oµ) = {f, g}(m).

At the same time, since f, g are also extensions of fOµ & Lµ & /µ andgOµ & Lµ & /µ " C!(J%1(µ))Gµ respectively, we can write,

{L"µfOµ , L"

µgOµ}Mµ([m]µ) = {fOµ & Lµ, gOµ & Lµ}Mµ([m]µ) = {f, g}(m).

Since m "M is arbitrary we have that

L"µ{fOµ , gOµ}MOµ

= {L"µfOµ , L"

µgOµ}Mµ . !

The last Theorem, together with the previous ones on Poisson,point and orbit reduction, allows us to draw the following commuta-tive diagram (the reduction diagram)

J%1(µ)lµ$$$% J%1(Oµ)

&µ

""#""#&Oµ

MµLµ$$$% MOµ

!!

!"

##

#$

##

#%

!!

!&

M

M/G.

iµ iOµ

jµ jOµ

By the results just quoted, all the mappings involved are morphismswith respect to the corresponding categories in their domains andranges. The generalization of this diagram to the case in which sin-gularities are present will be treated in the following chapter.

Before we finish this section, let’s remark that the setup forreduction that we have shown is extremely general. The resultspresented can be adapted to a variety of relevant particular caseslike, for instance when the manifold M is a cotangent bundlewith the Poisson structure associated to the canonical Liouvillesymplectic form defined on it [AM78, Theorem 4.3.3]. The La-grangian side, that is, when M is a tangent bundle made sym-plectic via a Legendre transform, has been also extensively studied.See [MS93, MR94, HMR, CHMR, HMRa, MMPRa].

§ 1.6. Relative Critical Elements 41

1.6 Relative Critical Elements

In the previous section we have seen how, in the presence of a sym-metry, the quotient space forms, under certain restrictions, a newmanifold where the dynamics induced on M by G–equivariant vectorfields, projects naturally. We will take in this section the first stepstowards the study of the critical elements of these projected vectorfields, that we will call relative critical elements. More specifi-cally, we will try to visualize how these motions manifest themselvesin the original manifold and will device a definition of stability suit-able to their nature.

In the spirit of the previous section we will assume for the timebeing the absence of singularities; in other words, we will suppose thatthe symmetry is given by a Lie group G acting freely and properly onthe manifold M , in such a way that by Proposition 1.3.1, the quotientset M/G is a smooth manifold, and / : M % M/G is a surjectivesubmersion. Let X " X(M) be a G–equivariant vector field on M ;that is, for any g " G, X satisfies that T&g & X = X & &g. Such avector field induces on M/G a unique vector field, XM/G, /–relatedto X.

Definition 1.6.1 In the situation just described, we say that thepoint m " M is a relative equilibrium (respectively relative pe-riodic point (RPP) or relative periodic orbit (RPO)) withrespect to the G–symmetry of M , if the point [m] = /(m) is anequilibrium (respectively periodic point) of the projected vector fieldXM/G.

We now study how these relative critical elements look like in themanifold M . We start with the relative equilibria.

Theorem 1.6.1 Let m " M , and Ft be the flow associated to theG–equivariant vector field X. Then m is a relative equilibrium withrespect to the G–symmetry of M i! there exists an element - " g suchthat Ft(m) = exp t- · m or, equivalently, X(m) = -M (m). Moreover,

(i) If M is a Poisson manifold with Poisson bracket {·, ·}, the G–action is canonical, and X = Xh with h " C!(M)G a G–invariant Hamiltonian, then the relative equilibria of Xh co-incide with the elements m " M such that the points [m] =/(m) "M/G are equilibria of X[h].


(ii) If, in addition to the assumptions in (i), the canonical G–actionhas momentum map associated J : M % g", with non equiv-ariance cocycle 0 then, the relative equilibria m of Xh coincidewith the elements m " M such that [m]µ = /µ(m) " Mµ areequilibria of Xhµ, where J(m) = µ or, equivalently, the pointsm " M for which there exists an element - " gµ such thatFt(m) = exp t- · m. The element - " gµ is called a velocity ofthe relative equilibrium m.

(iii) If, in addition to the assumptions in (ii), the Poisson manifoldM is symplectic then, the relative equilibria m of Xh coincidewith the elements m " M such that [m]µ = /µ(m) " Mµ arecritical points of hµ, with µ = J(m) or, equivalently with thepoints m " M for which there exists an element - " gµ suchthat the Lyapunov function h$ J# has a critical point at m.

Proof By definition, if m " M is a relative equilibrium, thenXM/G([m]) = 0 and, therefore FM/G

t ([m]) = [m] for any time t.Since Ft and FM/G

t are related by / & Ft = FM/Gt , we have that

/(Ft(m)) = [m] hence, for each time t there is an element g(t) " Gsuch that Ft(m) = g(t) · m. Since F0 = id, and Ft+s = Ft & Fs,for arbitrary times t, s, the mapping t -% g(t) is necessarily a one–parameter subgroup of G. Consequently, there exists an element- " g such that Ft(m) = exp t- · m. Conversely, if m " M is suchthat Ft(m) = exp t- · m, for some - " g, then [m] is necessarilyan equilibrium of XM/G. (i) This is a direct consequence of Theo-rem 1.5.1 (iii).(ii) Let m be an arbitrary relative equilibrium of Xh, µ = J(m), and- " g such that Ft(m) = exp t- · m. By Noether’s Theorem and theG–equivariance of J with respect to the a"ne action with cocycle 0,J(Ft(m)) = J(m) for any time t and, consequently µ = exp t- · µ.Taking the derivative with respect to t at t = 0 in this equality, wesee that - " gµ necessarily. Using now the definition of Fµ

t given inTheorem 1.5.2:

Fµt ([m]µ) = (/µ & Ft & iµ)(m) = /µ(exp t- · m) = [m]µ,

since - " gµ and hence exp t- " Gµ for any time t. This guaranteesthat [m]µ is an equilibrium of Xhµ . The converse is straightforward.(iii) If M is symplectic, by Theorem 1.5.3 Mµ is symplectic so, bythe nondegeneracy of the Poisson structure, the equilibria of Xhµ


are given by the critical points of hµ. Analogously, if m is a rela-tive equilibrium such that J(m) = µ, we know from point (ii) thatXh(m) = -M (m), with - " gµ. Using the definition of J and takingv " TmM arbitrary, we write

!(m)(Xh(m), v) = !(m)(-M (m), v),

or, equivalently

dh(m) · v = dJ#(m) · v.

Since the vector v is arbitrary, d(h$ J#)(m) = 0, as required. !

An analogous result can be formulated for relative periodic orbits.

Theorem 1.6.2 Let m "M and let Ft be the flow associated to theG–equivariant vector field X. Then m is a relative periodic point withrespect to the G–symmetry of M i! there exists an element g " G (thephase shift), and a positive constant & > 0 (the relative period),such that

Ft+" (m) = g · Ft(m) for any t " R.

Moreover,

(i) If M is a Poisson manifold with Poisson bracket {·, ·}, the G–action is canonical, and X = Xh with h " C!(M)G a G–invariant Hamiltonian, then the relative periodic points of Xh

coincide with the elements m " M such that the points [m] =/(m) "M/G are periodic points of X[h].

(ii) If in addition to the assumptions in (i), the canonical G–actionhas associated momentum map J : M % g", with non equivari-ance cocycle 0 then, the relative periodic points m of Xh coin-cide with the elements m "M such that [m]µ = /µ(m) "Mµ isa periodic point of Xhµ, where J(m) = µ or, equivalently, thepoints m " M for which there exists an element g " Gµ suchthat

Ft+" (m) = g · Ft(m) for any t " R.


Proof If m " M is a RPP of X then, by definition, [m] " M/G isa periodic point of XM/G with period, say & > 0, such that for anytime t, FM/G

t+" ([m]) = FM/Gt ([m]). In terms of Ft, this implies that

/(Ft+" (m)) = /(Ft(m)) for any t.

In particular, for t = 0, (/µ & F" )(m) = /(m) and hence there existsan element g " G such that F" (m) = g · m. Thus, if t is arbitrary,

Ft+" (m) = (Ft & F" )(m) = Ft(g · m) = g · Ft(m),

as required. Conversely, if we apply / on both sides of the equalityFt+" (m) = g · Ft(m), we obtain that

/(Ft+" (m)) = /(g · Ft(m)) = /(Ft(m)),

or equivalently:

FM/Gt+" ([m]) = FM/G

t ([m]),

which shows that [m] is a periodic point of XM/G.The claim in (i) is a direct consequence of what we just did and

Theorem 1.5.1 (iii). The proof of (ii) is also straightforward anduses that in this case the phase shift is an element in Gµ. Indeed,taking J in both sides of

Ft+" (m) = g · Ft(m),

and using Noether’s Theorem, it follows that g " Gµ. !

Remark 1.6.1 The original definition of relative critical elements isdue to Poincare who introduced them not in terms of the reducedspaces, but using the theses of Theorems 1.6.1 and 1.6.2. "

As we pointed out in Section 1.2, a very important tool in thestudy of periodic orbits are the transversal sections and the Poincaremaps defined. This concept can be adapted to the treatment of RPOs.Proofs for the following results, and additional information can befound in [Fi91, Fi80].


Definition 1.6.2 Let X " X(M) be a G-equivariant vector field onthe G–manifold M . A G–invariant local transversal section ofX at m "M is a G-invariant submanifold S of codimension one withm " S such that for all z " S, X(z) is not contained in TzS.

If m "M is a RPP, with relative period & > 0, phase shift g " G,and S is a G–invariant local transversal section at m, then a G–equivariant Poincare map of the RPP m is a mapping $ : W0 %W1, where

(RPM1) W0, W1 ' S are open G–invariant neighborhoods of m " Sand $ is a G-equivariant di!eomorphism.

(RPM2) There is a continuous G–invariant function, called the pe-riod function, such that for all z " W0, (z, & $ ((z)) " DX ,and $(z) = F (z, & $ ((z)). The set DX is the domain of theflow F : DX 'M )R %M , and it is an open subset of M )R.

(RPM3) If t " (0, & $ ((z)), F (z, t) /"W0.

Theorem 1.6.3 (Existence and uniqueness of G–equivariantPoincare maps) Let m, M , and X be as in Definition 1.6.2.

(i) There exists a G–invariant local transversal section S, and a G–equivariant Poincare map $ : W0 %W1 for m.

(ii) If $ : W0 % W1 is a G–equivariant Poincare map for m inthe G–invariant local transversal section S, and $$ : W $

0 %W $

1 is the same for m$ := Ft0(m) in S$, then $ and $$ arelocally G–equivariant conjugate that is, there are G–invariantopen neighborhoods W2 of m " S, W $

2 of m$ " S$, and a G–equivariant di!eomorphism H : W2 % W $

2 such that W2 'W0 *W1, W $

2 'W $0 *W $

1, and the diagram

$%1(W2) *W2!$% W2 *$(W2)

H""#

""#H

W $2

!!$% S$

commutes.

We will dedicate much e!ort to study the stability of relativecritical elements. The first problem in this task is defining a sort of


stability suited to the additional structure present in our dynamicalsystem: the symmetry. In a symmetric system there are usually neu-tral directions associated to the invariance properties of the system.In these directions, the evolution of the system is basically ”free” (allthese ideas will be made more explicit in subsequent sections andillustrated in examples), which gives rise to drift phenomena mak-ing non–trivial the choice of a definition of stability, given that theobvious option, orbital stability, becomes too restrictive. The mostnatural and successful choice of stability for relative equilibria andRPOs consists by taking the definitions of stability for equilibria andperiodic orbits introduced in Section 1.2 and ”modding them out” bythe neutral directions of the system; this give rise to the concept ofstability relative to a subgroup, explicitly introduced in [Pat92],for relative equilibria, and in [OR3] for RPOs.

Definition 1.6.3 Let X " X(M) be a G-equivariant vector field onthe G–manifold M . Let G$ be a subgroup of G.

(i) A relative equilibrium m "M of X, is called G$–stable, or stablemodulo G$, if for any G$–invariant open neighborhood V of theorbit G$ · m, there is an open neighborhood U ' V of m, suchthat if Ft is the flow of the vector field X and u " U , thenFt(u) " V for all t 4 0.

(ii) A relative periodic point m of X is G$-stable, or stable moduloG$, if for any G$–invariant open neighborhood V of the set G$ ·{Ft(m)}t>0, there is an open neighborhood U ' V of m suchthat Ft(U) ' V , for any t > 0.

In the next chapters we will find specific criteria to establish thiskind of stability in particular systems (always Hamiltonian) and, inthe process, we will investigate the relation between stability relativeto a subgroup of a relative equilibrium (respectively RPO) and theLyapunov (respectively orbital) stability of the associated reducedequilibrium (respectively periodic orbit).

Chapter 2

Singular SymplecticReduction

Capıtulo XLII: Que trata de lo que mas sucedioen la venta y de otras muchas cosas dignas desaberse.Cervantes, Don Quijote de la Mancha

In the previous chapter we carried out reduction by a group ac-tion making very strong regularity assumptions. For instance, inmost cases, we assumed that the group action was free and, whena momentum map was involved, we only used its regular values.These hypotheses are very strong and omit many interesting phe-nomena like bifurcations, symmetry breaking, etc. The quotientspaces studied in Chapter 1 with the regularity assumptions elim-inated are called singular reduced spaces and the process bywhich they are obtained is called singular reduction. The structureof these Marsden–Weinstein reduced spaces [Mey73, MW74] in thesingular case was first investigated in Fischer, Marsden, and Mon-crief [FMM80, FMM80a], Arms, Marsden, and Moncrief [AMM81],Otto [Ot87], and Arms, Cushman, and Gotay [ACG91]. This was fol-lowed by the use of normal forms (see Chapter 3 for an extensive dis-cussion) to describe these reduced spaces as stratified spaces, firstintroduced by Sjamaar and Lerman [SL91], for compact Lie groupactions, and by Bates and Lerman [BL97], for proper Lie group ac-

48 Chapter 2. Singular Symplectic Reduction

tions. These and other reduction schemes are compared in [AGJ]. Inour exposition below we will take a di!erent, global, approach thatwill lead to an explicit description of all the structures involved inour constructions. This is not possible, in general, when using nor-mal forms, since the morphisms that link the reduced spaces with thenormal forms utilized to model them are usually implicitly (and nonconstructively) given.

Even though we will be mainly concerned with the symplecticcase, we will start by generalizing to the singular case [OR5] a theoremdue to Marsden and Ratiu [MR86], stated in the Poisson category.This theorem, regular in nature, contains as a particular case allthe Poisson reduction results stated in Chapter 1. We will see thatan appropriate generalization will do the same job in the singularsituation.

For future reference and as a first illustration of the interplaybetween points with symmetry and singularities of the momentummap we give a result, first stated in [AMM81].

Lemma 2.0.1 (Bifurcation Lemma) Let (M, !) be a symplecticmanifold and G be a group acting canonically on M with an associatedmomentum map J : M % g". For any m "M ,

range(TmJ) = (gm)#.

Where gm is the Lie algebra of the isotropy subgroup Gm, and(gm)# := {µ " g | µ|gm = 0} denotes the annihilator in g" of gm.

Proof Let - " gm and v " TmM be arbitrary. By the definition ofthe momentum map

0TmJ · v, -1 = dJ#(m) · v = !(m)(-M (m), v) = 0,

since -M (m) = 0, by the choice of - " gm. This implies thatrangeTmJ ' (gm)#. The equality follows from dimension counting:

rank(TmJ) = dimM $ dim(kerTmJ) = dimM $ dim(Tm(G · m))%

= dimG$ dimGm = dim(gm)#. !

2.1 Stratified Poisson Reduction by Folia-tions

The goal of this section is to show a generalization [OR5] of the mainresult in [MR86], that will be the basis of the presentation of singular

§ 2.1. Poisson Reduction by Foliations 49

Poisson reduction. We first review some concepts that will be usedin the exposition.

Definition 2.1.1 Let M be a di!erentiable manifold. A collectionof subspaces Dm ' TmM is called a smooth or di"erentiable dis-tribution if there are locally defined smooth vector fields {Xi}i'I inX(M), such that {Xi(m)}i'I spans Dm.

(i) D is called integrable if for any m " M , there is an injectivelyimmersed submanifold Sm 'M , such that TmSm = Dm.

(ii) D is called involutive if it is invariant under the (local) flowsassociated to vector fields with values in D.

Remark 2.1.1 Our definition of involutive distribution is more gen-eral than the traditional one, that is, D is involutive if [X, Y ] takesvalues in D whenever X and Y are vector fields with values in D. Thetwo concepts of involutivity are equivalent only when the dimensionof Dm is independent of m "M . "

Theorem 2.1.1 (Generalized Frobenius Theorem) A di!eren-tiable distribution D on a manifold M is integrable i! it is involutive.

Proof See [St74, Su73, MR94, LM87]. !

Definition 2.1.2 Let M be a di!erentiable manifold and S 'M bea subset of M . We say that S is a stratified subset of M withstrata {Si}i'I when

(S1) The subsets Si ' S, i " I, are submanifolds of M and form apartition of S.

(S2) The partition of S into the connected components {Sji }

j'Ji'I of

the subsets Si is locally finite.

(S3) If Sji * cl(Sj!

i! ) += 5 for (i, j) += (i$, j$), then Sji ' Sj!

i! anddim(Sj

i ) < dim(Sj!

i! ).

(S4) cl(Si)\Si is a disjoint union of strata of dimension strictly lessthan dim(Si).


We define the tangent bundle TS of the stratified subset S as

TS =(

i'I

TSi,

where TSi denote the ordinary tangent bundles of the manifolds Si.

Definition 2.1.3 Let M be a di!erentiable manifold and S 'M bea stratified subset of M with strata {Si}i'I . We say that D ' TM |Sis a smooth distribution on S adapted to the stratification{Si}i'I , if D * TSi is a smooth distribution on Si for all i " I. Thedistribution D is said to be integrable if D * TSi is integrable foreach i " I.

In the situation described by the previous definition, the integra-bility of the distributions D * TSi on Si allows us to partition eachSi into the maximal integral manifolds. Thus, there is an equiva-lence relation &i on Si whose equivalence classes are precisely thesemaximal integral manifolds. Doing this on each Si, we obtain anequivalence relation & on the whole set S by taking the union of thedi!erent equivalence classes corresponding to all the &i. We definethe quotient space S/& as

S/& :=(

i'I

Si/&i.

Notice that Definition 2.1.2 does not require the stratified subsetS to be a smooth manifold. In fact during part of our discussionwe will work with structures somewhat more general than manifolds,namely varieties.

Definition 2.1.4 A pair (X, C!(X)), where X is a topologicalspace and C!(X) ' C0(X) is a subset of continuous functions inX, is called a variety with smooth functions C!(X). If Y ' Xis a subset of X, the pair (Y, C!(Y )) is said to be a subvarietyof (X, C!(X)), if Y is a topological space endowed with the relativetopology defined by that of X and

C!(Y ) = {f " C0(Y ) | f = F |Y for some F " C!(X)}.

Sometimes C!(Y ) is called the set of Whitney smooth functionson Y with respect to X. A map " : X % Z between two varieties issaid to be smooth when it is continuous and ""C!(Z) ' C!(X).


In our discussion we will consider (S, C!(S)) as a subvarietyof (M, C!(M)). S/& is a variety whose set of smooth functions isdefined by the requirement that the canonical projection / : S % S/&is a smooth map, that is,

C!(S/&) : = {f " C0(S/&) | f & / " C!(S)}= {f " C0(S/&) | f & / = F |S for some C!(M)}.

In most applications of these concepts, the distribution D willbe induced by the tangent spaces to the orbits of a Lie group Gacting smoothly on M . By construction, D is integrable (the maximalintegral manifolds are the orbits). If this action is proper, some ofthe properties described in Proposition 1.3.1 are still true even whenS is not a manifold. More specifically we have the following:

Proposition 2.1.1 Let G be a Lie group acting properly on the ma-nifold M . Let (S, C!(S)) be a subvariety of (M, C!(M)) such thatS is a G–invariant subset of M . Then each G–invariant functionf " C!(S)G on S can be extended to M in a G–invariant fashion,that is, there is a F " C!(M)G such that F |S = f .

Proof See [ACG91, Proposition 2]. !

Definition 2.1.5 Let (M, {·, ·}) be a Poisson manifold and D 'TM be a smooth distribution on M . The distribution D is calledPoisson or canonical , if the condition df |D = dg|D = 0 for f, g "C!(M) implies that d{f, g}|D = 0

If the distribution D is defined by the tangent spaces to the orbitsof a Lie group G acting smoothly on M , the condition that D isPoisson can be expressed by the following condition: if f, g " C!(M)are such that -M [f ] = -M [g] = 0 for any - " g, then -M [{f, g}] = 0,for any - " g.

Definition 2.1.6 Let S 'M be a stratified subset, g " C!(S), andm " S. A local extension of g at m is a function G " C!(M)satisfying the following condition: there exists an open neighborhoodUm of m in M such that G|S&Um = g|S&Um.

Let D be an integrable distribution adapted to the stratified subsetS ' M . We say that D has the extension property if for any


f " C!(S/&) and any m " S the map f & / " C!(S) admits a localextension F " C!(M) at m such that dF |D = 0 (at all points ofM).

Remark 2.1.2 Note that if S is just a submanifold of M and D hasconstant dimension, that is, D is a usual smooth integrable subbundleof TM , the extension property is satisfied automatically: it su"cesto take a submanifold chart of S relative to M which is also a foliatedchart of S with respect to the distribution D|S . Also, if D is givenby the tangent spaces to the orbits of a proper G–action on M andS is a G–invariant subset of S, Proposition 2.1.1 guarantees that thetriplet (M, S, D) has the extension property. In general, note thatgiven two di!erent points m, m$ " S, the local extensions at m andat m$ need not coincide. "

Definition 2.1.7 Let (M, {·, ·}) be a Poisson manifold, S be a strat-ified subset of M with strata {Si}i'I , and D ' TM |S be a Poissonintegrable distribution adapted to S such that (M, S, D) has the ex-tension property. We say that (M, S, D) is Poisson reducible ifthe pair (C!(S/&), {·, ·}S/") is a well–defined Poisson algebra, wherethe bracket {·, ·}S/" is given by

{f, g}S/"(/(m)) = {F, G}(m), (2.1.1)

for every m " S, where i : S )% M is the natural inclusion, andF, H " C!(M) are smooth local extensions of f & /, g & / " C!(S)at m satisfying dF |D = dG|D = 0.

We now give a necessary and su"cient condition for (M, S, D)to be Poisson reducible. This result naturally generalizes the resultof Marsden and Ratiu [MR86] to the singular case.

Theorem 2.1.2 Let (M, {·, ·}) be a Poisson manifold with Poissontensor B : T "M % TM , S be a stratified subset of M with strata{Si}i'I , and D ' TM |S be a Poisson integrable distribution adaptedto S such that (M, S, D) has the extension property. Then (M, S, D)is Poisson reducible if and only if for any m " S we have

B((m) ' TmS + [(Sm]#, (2.1.2)

where

(m := {dF (m) | F " C!(M),dF |D = 0},


and

(Sm := {dF (m) " (m | F |Um&S is constant, for Um an open

neighborhood of m in M}.

Proof Notice first that an alternative way to write the condition inthe statement is

{XF (m) | F " C!(M),dF |D = 0} ' TmS + {dF (m) | F " C!(M),dF |D = 0, F |Um&S is constant, for Um an openneighborhood of m in M}#.

The proof of the theorem follows the strategy of [MR86]. First,we suppose that (M, S, D) is Poisson reducible. Let F " C!(M)satisfy dF |D = 0 and let $m " [TmS + [(S

m]#]# = [TmS]# * (Sm.

Thus $m = dK(m) for some K " C!(M) satisfying dK|D = 0, K isconstant on Um * S, where Um is an open neighborhood of m in M .Therefore, the functions F and K induce functions f, k " C!(S/&)by f&/ = F &i, k&/ = K&i and k is constant in an open neighborhoodof /(m) in S/&. Thus, by Poisson reducibility (2.1.1),

0$m, XF (m)1 = {K, F}(m) = {k, f}S/"(/(m)) = 0,

since k is a constant in a neighborhood of /(m). Since $m " [TmS +[(S

m]#]# is arbitrary, it follows that XF (m) " TmS + [(Sm]#.

Conversely, if B((m) ' TS + [(m]#, let f, g " C!(S/&) andF, G " C!(M) be smooth local extensions of f &/, g &/ " C!(S) atm such that dF |D = dG|D = 0. Since D is a Poisson distribution, itfollows that d{F, G}|D = 0, which implies that {F, G} is constant onthe equivalence classes of & and therefore induces a function, whichwe shall call {f, g}S/" " C!(S/&), satisfying the condition (2.1.1).If we show that this function does not depend on the extensionsinvolved, this defines the reduced bracket {f, g}S/" on S/&. Indeed,let G$ " C!(M) be another local extension of g & / " C!(S) atm such that dG$|D = 0. Since (G $ G$)|S&Um = 0, where Um is theneighborhood of m in M given by the hypothesis of local extendabilityof pull backs of functions from the quotient. Thus, d(G $ G$)(m)vanishes on TmS. It also vanishes on [(S

m]# by definition. Now usingthe hypothesis, for any m " S

0d(G$G$)(m), B(m)(dF (m))1 = 0, hence {F, G}(m) = {F, G$}(m),


which proves the independence on how g & / is extended. By anti-symmetry of {·, ·} it is also independent of the extension of f & /,therefore {f, g}S/" is well–defined and uniquely determined by theexpression (2.1.1). With this bracket (C!(S/&), {·, ·}S/") is a Pois-son algebra since the bracket {·, ·}S/" inherits all the properties of aPoisson bracket from those of {·, ·}. !

Remark 2.1.3 In the regular case considered in [MR86], S is asubmanifold and D is a smooth subbundle of TM . We have alreadyseen that in this situation the extension property is automaticallysatisfied. The condition of Poisson reducibility is stated as

B(D#) ' TS + D. (2.1.3)

Since the distribution D is adapted to the submanifold S, workingin a chart on M around a given point m " S, any $m " D#

m can bewritten as dF (m) for some smooth function F defined in this chartand constant on the local leaves of the foliation given by D. Nowchoose in every chart some function that is constant on the leavesof the foliation and construct a smooth function on M by adding allthese functions by means of a partition of unity. The resulting smoothfunction, also called F , is constant on the leaves of the foliation (sincedF |D = 0 by construction) and has the same di!erential at m, thatis, dF (m) = $m. This shows that in the regular case (m = D#

m.Let us now show that in the regular case, TmS + Dm = TmS +

[(Sm]#. Since (S

m ' (m, it follows that Dm = [(m]# ' [(Sm]# and

hence TmS+Dm ' TmS+[(Sm]#. To prove the converse, it su"ces to

show that [(Sm]# ' TmS + Dm = [TmS]## + [(m]# = [(TmS)# *(m]#,

or, equivalently, that (TmS)# * (m ' (Sm which is shown in the

following way. If $m " (TmS)# * (m, then $m = dF (m) for F "C!(M) satisfying dF |D = 0 and dF (m)|TmS = 0. One can replaceF by a smooth function vanishing on the distribution D and at thesame time being constant in U * S, for U an open neighborhood ofm in M . (To do this, replace in a chart at m the function F |U by theconstant function equal to $m, which is possible since $m vanisheson Dm and on TmS, and then patch this function with the restrictionof F to an open set V such that U 6 V = M by means of a partitionof unity. The resulting function is smooth, satisfies dF |D = 0 and isconstant on U * S.) This function is an element of (S

m which provesthe desired inclusion.


It should be also noted that the condition (2.1.3) is only su"cientin the singular case for the Poisson reducibility of (M,S,D), evenin the case when D is given by a group action. For example, con-sider the case of the S1 action by positive rotations on the complexline C. We take the integral manifolds of the distribution D to bethe concentric circles and the origin. This action has an equivariantmomentum map given by J(z) = |z|2/2 so that S := J%1(0) = 0.Then D0 = {0} so that D#

0 = C and hence, B(D#0) = C, since the

Poisson structure on C is induced by the standard symplectic struc-ture which is nondegenerate. On the other hand, T0S = 0, so thatT0S + D0 = {0}. which contradicts (2.1.3). However, (0 = {0} andthus this is consistent with (2.1.2). "

We now study the functorality property of Poisson reduction, thatwill be used to reduce the dynamics.

Proposition 2.1.2 Let (Mj , Sj , Dj) j = 1, 2, be Poisson reducible.We denote the Poisson bracket of Mj by {·, ·}j. Let " : M1 %M2 bea Poisson map such that "(S1) ' S2, and T"(D1) ' D2 (therefore "maps the equivalence classes of &1 into the equivalence classes of &2).Then " induces a unique smooth Poisson map '" : S1/&1 % S2/&2,characterized by /2 & " & i1 = '" & /1, where ij : Sj )% Mj are theinclusions and /j : Sj % Sj/&j are the projections. We call '" thereduction of ".

Proof By the hypotheses on ", the map '" exists, is smooth, and isunique. We show that it is Poisson. Let f, g " C!(S2/&2), m " S1,and F, G " C!(M2) be local extensions at "(m) " S2 of f & /2, g &/2 " C!(S2) respectively, such that dF |D2 = dG|D2 = 0. Then

'""{f, g}S2/"2(/1(m)) = {f, g}S2/"2

(('" & /1)(m))

= {f, g}S2/"2((/2 & ")(m))

= {F, G}2("(m)). (2.1.4)

Note that F & ", G & " " C!(M1) are smooth local extensions atm " S1 of f & '" & /1, g & '" & /1 " C!(S1) respectively, which satisfyd(F & ")|D1 = d(G & ")|D1 = 0 by the chain rule and the hypothesisT"(D1) ' D2. Therefore,

{'""f, '""g}S1/"1(/1(z)) = {F & ", G & "}1(z) = {F, G}2("(z)),


which coincides with (2.1.4) thereby proving the proposition. !

In Chapter 1 we saw how, in the framework of Poisson manifolds,the natural identification between derivations on the ring of smoothreal–valued functions and vector fields, allowed us to associate toeach function on the manifold a Hamiltonian vector field. In the caseof Poisson varieties like (S/&, C!(S/&)) we need something moregeneral to introduce dynamics since, in general, S/& is not a smoothmanifold and therefore, defining vector fields is not always possible.

Definition 2.1.8 Let (M, S, D) be a Poisson reducible system. Leth " C!(S/&). We define the Hamiltonian flow associated to h asthe smooth map FS/"

t : S/&% S/& such that for any f " C!(S/&)and any z " S/&, we have

d

dtf(FS/"

t (z)) = {f, h}S/"(FS/"t (z)).

Note that in this framework there is no standard Existence andUniqueness Theorem, as is the case for flows associated to Hamilto-nian vector fields on smooth manifolds. In fact, these two issues needto be addressed separately. The following result shows that existenceis always guaranteed.

Theorem 2.1.3 (Reduction of the dynamics) Let (M, S, D) bea Poisson reducible system and let h " C!(M) be a function suchthat dh|D = 0 and whose Hamiltonian flow Ft preserves the subsetS, that is, for any time t, Ft(S) ' S. Suppose also that for any t,TFt(D) ' D. Then there is a function hS/" " C!(S/&) uniquelydefined by the relation

hS/" & / = h & i,

called the reduced Hamiltonian , for which the reduction FS/"t of

Ft is a Hamiltonian flow induced by hS/". In addition, FS/"t is a

poisson map.

Proof The condition dh|D = 0 guarantees that h is constant onthe equivalence classes of & and therefore the relation hS/" & / =h & i defines hS/" uniquely. Proposition 2.1.2 ensures the existence


of FS/"t : S/& % S/& as the unique Poisson mapping satisfying the

equality

/ & Ft & i = FS/"t & /.

We verify that FS/"t is a Hamiltonian flow for hS/". Notice that,

by construction, h is a smooth extension hS/" & /. Thus, if fS/" "C!(S/&) is arbitrary, let f " C!(M) be a smooth local extensionat Ft0(m) " S of fS/" & /. By the flow property, for small |t$ t0|, fis also a smooth local extension at Ft(m) of fS/" & /. Thus we getfor such t

d

dtfS/"(FS/"

t (/(m))) =d

dtfS/"((/ & Ft & i)(m)) =

d

dtf(Ft(m))

= {f, h}(Ft(m)) = {fS/", hS/"}S/"(/(Ft(m)))

= {fS/", hS/"}S/"(FS/"t (/(m))),

which proves the claim. !

Let us remark again that FS/"t may not be the unique Hamil-

tonian flow associated to hS/". The following proposition, due toSjamaar and Lerman [SL91, BL97], describes a situation in whichthe uniqueness of the reduced flow is guaranteed.

Proposition 2.1.3 Let (M, S, D) be a Poisson reducible system. Ifthe functions in C!(S/&) separate points, then each HamiltonianhS/" " C!(S/&) has a unique associated Hamiltonian flow.

Proof The existence is guaranteed by the previous theorem since thereduction FS/"

t of the Hamiltonian flow Ft associated to any smoothlocal extension h " C!(M) of hS/" & / at an arbitrary point, suchthat dh|D = 0, does the job. Suppose now that GS/"

t is anotherHamiltonian flow for hS/". Since by hypothesis, the functions inC!(S/&) separate points, it is enough to show that for any fS/" "C!(S/&), /(m) " S/&, and any time t,

fS/"(GS/"t (FS/"

%t (/(m)))) = fS/"(/(m)).

This identity holds as a consequence of the following computation, inwhich we use the chain rule and the fact that FS/"

%t is a Hamiltonian


flow for $hS/":

d

dtfS/"(GS/"

t (FS/"%t (/(m)))) = {fS/", hS/"}S/"(GS/"

t (FS/"%t (/(m))))

+ {fS/" &GS/"t , $hS/" &GS/"

t }S/"(FS/"%t (/(m))) = 0.

since the flow GS/"t is Poisson by Theorem 2.1.3. !

This result is particularly relevant when the distribution D isgiven by the proper action of a Lie group, since in this case, Proposi-tion 1.3.1 guarantees that the hypothesis on the separation of pointsalways holds.

2.2 Singular Poisson, Point, and Orbit Re-duction

We will now use the results just proved as the main tool to gener-alize the Poisson reduction theorems introduced in the first part ofSection 1.5. As we already announced, many of these reduction pro-cedures will be particular cases of the reduction by foliations strategy,when the distributions are given by proper Lie group actions. Thesimplest case is given in the following Theorem.

Theorem 2.2.1 (Singular Poisson reduction) Let (M, {·, ·}) bea Poisson manifold and let G be a Lie group acting canonically andproperly on M . Then the following hold:

(i) The pair (C!(M/G), {·, ·}M/G) is a Poisson algebra, where thePoisson bracket {·, ·}M/G is characterized by

{f, g}M/G & / = {f & /, g & /},

for any f, g " C!(M/G); / : M %M/G denotes the canonicalsmooth projection.

(ii) Let h be a G–invariant function in M . The Hamiltonian flow Ft

of Xh commutes with the G–action, so it induces a flow FM/Gt

on M/G which is Poisson and is characterized by

/ & Ft = FM/Gt & /.

§ 2.2. Poisson, Point, and Orbit Reduction 59

(iii) The flow FM/Gt is the unique Hamiltonian flow of the function

[h] " C!(M/G) defined by

[h] & / = h.

We will call [h] the reduced Hamiltonian.

Proof (i) This part can be obtained as a corollary to Theorem 2.1.2by taking M = S, and D ' TM the distribution given by Dm = g·m.This distribution is smooth since for every m " M , if {-1, . . . , -n}is a basis of the Lie algebra g, the evaluation of the vector fields-1M , . . . , -nM at m, spans Dm. The distribution D is also triviallyintegrable since, by construction, the orbit G · m is a submanifold ofM such that Dm = Tm(G · m) = g · m, for arbitrary m " M . Thecanonical character of the G–action guarantees that D is Poisson inthe sense of Definition 2.1.5. Indeed, let f, g " C!(M) such thatdf |D = dg|D = 0, that is, for any m "M and any - " g

df(z) · -M (z) = dg(z) · -M (z) = 0.

Then,

d{f, g}(m) · -M (m) =d

dt

$$$$t=0

{f, g}(exp t- · m)

=d

dt

$$$$t=0

{f & &exp t#, g & &exp t#}(m)

=d

dt

$$$$t=0

Xg#"exp t! [f & &exp t#](m). (2.2.1)

As we said in previous arguments, the value of the Hamiltonianvector field Xg#"exp t!(m) at the point m, depends on g & &exp t#

only through d(g & &exp t#)(m). We now show that for any time t,d(g & &exp t#)(m) = dg(m) and therefore Xg#"exp t!(m) = Xg(m). In-deed, for v = d

ds

$$s=0

c(s) " TmM :

d

dtd(g & &exp t#)(m) · v =

d

dt

d

ds

$$$$s=0

(g & &exp t#)(c(s))

=d

ds

$$$$s=0

d

dtg(exp t- · c(s)) = 0,

since ddt(exp t- ·c(s)) " Tc(s)(G·c(s)) ' D, and, by hypothesis, dg|D =

0. This clearly implies that Xg#"exp t!(m) = Xg(m), as required. Since


the same can be done with the function f , expression (2.2.1) can bewritten as

d{f, g}(m) · -M (z) =d

dt

$$$$t=0

Xg[f & &exp t#](m)

=d

dt

$$$$t=0

{f & &exp t#, g}(m)

= $ d

dt

$$$$t=0

Xf#"exp t![g](m) =

= $ d

dt

$$$$t=0

Xf [g](m) = 0,

as required. Remark that the distribution D satisfies trivially theextension property, as well as the hypothesis of Theorem 2.1.2since B((m) ' TmM ' TmM + [(S

M ]#. This guarantees that(C!(M/G), {·, ·}M/G) is a Poisson algebra.(ii) Since the Lie group G acts canonically on M and the Hamil-tonian h is G–invariant, by Proposition 1.1.1, the Hamiltonian flowassociated to h satisfies that &g & Ft = Ft & &g for any g " G andtherefore, for any - " g, any m "M , and any time t

TmFt · -M (m) =d

ds

$$$$s=0

Ft(exp s- · m)

=d

ds

$$$$s=0

exp s- · Ft(m) = -M (Ft(m)),

which implies that TFt(D) ' D. The claim follows from Proposi-tion 2.1.2.(iii) It is a corollary of Theorem 2.1.3. The uniqueness follows fromProposition 2.1.3, and the properness of the action (see Proposi-tion 1.3.1). !

At this point we will assume that M is not only Poisson, but alsosymplectic, and that the canonical action of G on M is proper andhas an associated globally equivariant momentum map J : M % g",that is, the action is globally Hamiltonian. The natural step to takein this situation is studying point and orbit reduction. Regarding theformer, recall that in the regular case, if M was a symplectic mani-fold, so was the point reduced space Mµ := J%1(µ)/Gµ. If we are in agenuinely singular situation, the space J%1(µ)/Gµ is not even a man-ifold, however, it can be shown that in the sense of Definition 2.1.7,


it is endowed with a Poisson structure. The construction of this Pois-son structure constitutes the Universal Reduction Procedure ofArms, Cushman and Gotay [ACG91] which is described in detail inthe following theorem.

Theorem 2.2.2 (Singular point reduction) Let (M, !) be asymplectic manifold and let G be a Lie group acting properly on Min a globally Hamiltonian fashion with associated equivariant momen-tum map J : M % g". Let µ " g" be a value of J and denote by Gµ

the isotropy of µ under the coadjoint action of G on g". The followinghold:

(i) The set Mµ := J%1(µ)/Gµ is such that the pair(C!(Mµ), {·, ·}Mµ) is a Poisson algebra, with Poissonbracket {·, ·}Mµ characterized by

{fµ, gµ}Mµ([m]µ) = {f, g}(m), (2.2.2)

for any fµ, gµ " C!(Mµ). The functions f, g " C!(M)G arearbitrary smooth local extensions at m " J%1(µ) of fµ &/µ, gµ &/µ " C!(J%1(µ))Gµ, where /µ : J%1(µ)%Mµ is the canonicalsmooth projection, and [m]µ := /µ(m) "Mµ.

(ii) Let h " C!(M)G be a G–invariant Hamiltonian. The Hamil-tonian flow Ft of h leaves the connected components of J%1(µ)invariant and commutes with the G–action, so it induces a Pois-son flow Fµ

t on Mµ, uniquely determined by

/µ & Ft & iµ = Fµt & /µ, (2.2.3)

where iµ : J%1(µ) )%M is the canonical injection.

(iii) The flow Fµt is the unique Hamiltonian flow in (Mµ, {·, ·}Mµ),

with Hamiltonian function hµ " C!(Mµ) defined by

hµ & /µ = h & iµ,

We will call hµ the reduced Hamiltonian.

(iv) Let k " C!(M)G be another G–invariant function. Then,{h, k} is also G–invariant and {h, k}µ = {hµ, kµ}Mµ.


Proof Once more, we will obtain this result as a corollary to Theo-rem 2.1.2 taking M as the Poisson manifold, J%1(µ) as the stratifiedsubset S, and D as the distribution given by the tangent spaces tothe G–orbits in J%1(µ), that is, for any m " J%1(µ), Dm = g ·m. Weverify that J%1(µ) is a stratified subset in the sense of Definition 2.1.2and that D is a smooth, integrable, Poisson distribution, adapted tothe stratification of J%1(µ), for which the extension property holds.

Firstly, the equivariance of J with respect to the G–action impliesthat there is a well–defined continuous Gµ–action on the topologicalspace J%1(µ). Since the subset J%1(µ) and the subgroup Gµ areclosed in M and G respectively, the Gµ action on J%1(µ) is proper andtherefore a standard result (see, for instance [Bre72, BAR75, Fs70,B75]) guarantees that J%1(µ) can be stratified using the orbit typemanifolds associated to the Gµ–action, that is, J%1(µ) is a stratifiedsubset of M with strata the submanifolds of M

(J%1(µ))Gµ

(H) := J%1(µ) *MGµ

(H),

for any isotropy subgroup H ' Gµ. The submanifold character of thestrata is provided by the Bifurcation Lemma which proves that J|

MGµ(H)

is a constant rank map and hence, by the Subimmersion Theorem(see [AMR, Theorem 3.5.17]), (J|

MGµ(H)

)%1(µ) = J%1(µ) * MGµ

(H) =

(J%1(µ))Gµ

(H) is a submanifold of MGµ

(H) and therefore of M .Secondly, the distribution D is smooth since it is induced by a

smooth group action. We now verify that it is adapted to the strati-fication of J%1(µ) by Gµ–orbit types. Recall that the SubimmersionTheorem states that for any m " (J%1(µ))Gµ

(H),

Tm)(J%1(µ))Gµ

(H)

*= Tm((J|

MGµ(H)

)%1(µ)) = kerTmJ|M

Gµ(H)

= kerTmJ * TmMGµ

(H),

and therefore, using the Reduction Lemma and the Gµ–invariance ofM

Gµ

(H),

Dm * Tm)(J%1(µ))Gµ

(H)

*= kerTmJ * TmM

Gµ

(H) * g · m

= gµ · m * TmMGµ

(H) = gµ · m.

This implies that D coincides, stratum by stratum, with the smoothintegrable distribution induced by the Gµ–action, which guarantees


that D is integrable and adapted to the stratified subset J%1(µ). As inTheorem 2.2.1, the canonical character of the G–action implies thatthe distribution D is Poisson. The extension property of D followsfrom a result, Proposition 3.4.3, that will be easily proved later on us-ing normal forms. The conclusion of this result is that, under the hy-potheses of the current theorem, every function in C!(J%1(µ))Gµ ad-mits a smooth local G–invariant extension at any point to C!(M)G.

Finally, in order to show that Mµ is Poisson, we use Theorem 2.1.2to prove that the triplet (M, J%1(µ), D) is Poisson reducible, that is,we will verify for arbitrary m " J%1(µ) that

B((m) ' Tm(J%1(µ)) +)(J"1(µ)

m

*#.

Indeed, if F " C!(M)Gµ and H = Gm, we will show that

XF (m) " Tm)(J%1(µ))Gµ

(H)

*+)(J"1(µ)

m

*#

=)ker TmJ * TmM

Gµ

(H)

*+)(J"1(µ)

m

*#.

To see this, let

$m "+)

ker TmJ * TmMGµ

(H)

*+)(J"1(µ)

m

*#,#

=)ker TmJ * TmM

Gµ

(H)

*# *(J"1(µ)m ,

so that $m = dK(m) for some K " C!(M)G, constant on Um *J%1(µ), where Um is an open neighborhood of m in M . Then,

0$m, XF (m)1 = {K, F}(m) = XF [K](m).

However, by Noether’s Theorem, the Hamiltonian flow Ft of XF pre-serves the level sets of J, in particular J%1(µ). Therefore,

XF [K](m) =d

dt

$$$$t=0

K(Ft(m)) = 0,

since K |J"1(µ)= 0. This proves the required condition on XF (m) andhence implies that (Mµ, C!(Mµ)) is a Poisson algebra with bracket{·, ·}Mµ defined by

{fµ, gµ}Mµ([m]µ) = {f, g}(m),

for any fµ, gµ " C!(Mµ), and f, g " C!(M)G arbitrary smoothlocal G–invariant extensions at m of fµ&/µ, gµ&/µ " C!(J%1(µ))Gµ ,whose existence is again guaranteed by Proposition 3.4.3.


The remaining points are a straightforward consequences ofNoether’s Theorem, Proposition 2.1.2, and Theorem 2.1.3. Theuniqueness of the flow for the reduced Hamiltonian follows fromProposition 2.1.3, and the properness of the action (see Proposi-tion 1.3.1). !

A theorem completely identical can be stated for the singular orbitreduced space MOµ := J%1(Oµ)/G. The only di!erence in the proof,with respect to the one corresponding to Theorem 2.2.2, is that in thiscase J%1(Oµ) will play the role of S, which will be stratified by meansof the orbit types corresponding to the G–action defined on it. Theextension property follows directly in this case from Proposition 2.1.1,due to the G–invariance of J%1(Oµ).

Theorem 2.2.3 (Singular orbit reduction) Let (M, !) be asymplectic manifold and let G be a Lie group acting properly on Min a globally Hamiltonian fashion with associated equivariant momen-tum map J : M % g". Let µ " g" be a value of J, and denote byOµ the orbit of µ under the coadjoint action of G on g". Then thefollowing hold:

(i) The set MOµ := J%1(Oµ)/G is such that the pair(C!(MOµ), {·, ·}MOµ

) is a Poisson algebra, with Poissonbracket {·, ·}MOµ

, characterized by

{fOµ , gOµ}MOµ([m]Oµ) = {f, g}(m), (2.2.4)

for any fOµ , gOµ " C!(MOµ). The functions f, g " C!(M)G

are arbitrary local extensions at m of fOµ & /Oµ , gOµ & /Oµ "C!(J%1(Oµ))G, where /Oµ : J%1(µ) % MOµ is the canonicalsmooth projection and [m]Oµ := /Oµ(m) "MOµ.

(ii) Let h " C!(M)G be a G–invariant Hamiltonian. The Hamilto-nian flow Ft of h leaves the connected components of J%1(Oµ)invariant and commutes with the G–action, so it induces a flowF

Oµt on MOµ, uniquely determined by

/Oµ & Ft & iOµ = FOµt & /Oµ , (2.2.5)

where iOµ : J%1(Oµ) )%M is the canonical injection.


(iii) The flow FOµt is the unique Hamiltonian flow in

(MOµ , {·, ·}MOµ), with Hamiltonian function hOµ " C!(MOµ)

defined by

hOµ & /Oµ = h & iOµ .

We will call hOµ the reduced Hamiltonian.

(iv) Let k " C!(M)G be another G–invariant function. Then,{h, k} is also G–invariant and {h, k}Oµ = {hOµ , kOµ}MOµ

.

Remark 2.2.1 The Reduction Diagram As could be expectedfrom the regular case, the point and orbit reduction schemes areequivalent. In fact, we will now see, that using the same maps lµand Lµ that were considered in the regular case, the Poisson alge-bras (Mµ, C!(Mµ)) and (MOµ , C!(MOµ)) are Poisson isomorphic,giving as a result a reduction diagram identical to the one we hadin the regular case, but this time, in the category of Poisson vari-eties. Indeed, using Proposition 2.1.2, it is easy to see that Lµ is aPoisson smooth bijective map between Mµ and MOµ . We now verifythat the inverse of Lµ is smooth, which shows that Lµ is a Poissonisomorphism. It is easy to verify that L%1

µ : MOµ % Mµ is given byL%1

µ ([m]Oµ) = [g%1 · m]µ, where g " G is such that J(z) = Ad"g"1µ.

In order to prove that L%1µ is smooth we have to show, according

to Definition 2.1.4, that (L%1µ )"C!(Mµ) ' C!(MOµ). Thus, let

fµ " C!(Mµ) be arbitrary, fix m " J%1(Oµ) such that J(m) =Ad"

g"1µ, and let f " C!(M)G be a local G–invariant extension offµ & /µ " C!(J%1(µ))Gµ at m, whose existence is ensured by Propo-sition 3.4.3. Let fOµ " C!(MOµ) be the smooth function uniquelydefined by the relation

fOµ & /Oµ = f & iOµ .

We check now that (L%1µ )"fµ = fOµ , which proves the inclusion

needed. Indeed,

(L%1µ )"fµ([m]Oµ) = fµ([g%1 · m]µ) = (fµ & /µ)(g%1 · m)

= f(g%1 · m) = f(m) = (fOµ & /Oµ)(m)= fOµ([m]Oµ),

as required. "


Remark 2.2.2 Reduced Spaces and Invariant Theory A prob-lem that we have not addressed in our discussion is the explicit con-struction of the quotients that constitute the reduced spaces. Inthe case in which G is compact, the most powerful tool availableis the Theory of Invariants. The main idea behind this strategy isthat if G is a compact Lie group acting linearly on R2n (which is noloss of generality since, according to a theorem of Gotay and Tuyn-man [GT89], every canonical action of a compact Lie group on a sym-plectic manifold can be reduced to this case), then the algebra of G–invariant polynomials is finitely generated (see, for instance [Ke87]).If {01, . . . ,0k} is a set of generators, we can define the associatedHilbert map by

0 : R2n $% Rk

m -$% (01(m), . . . ,0k(m)),

whose image can be identified with R2n/G (see [Po76, AS83])in such a way that by Mather’s refinement of Schwarz’s Theo-rem [Ma77, Sch74], the space of smooth functions in R2n/G is iso-morphic to the space of Whitney smooth functions C!(0(R2n)),considering the set 0(R2n) as a semialgebraic subset of Rk. Noticethat since 0 is an polynomial map, the Tarski–Seidenberg theoremguarantees that R2n/G can be considered as a semialgebraic sub-set of Rk [Co74, AS83]. The point and orbit reduced spaces can beconstructed in this fashion by taking MOµ (and, therefore Mµ) asMOµ = 0(J%1(Oµ)). The reader is encouraged to check with [AS83]for further information, as well as with [CB97, ACG91] for specificexamples on the computation of the reduced spaces along with theirassociated reduced Poisson structures. "

2.3 Free Actions in a Singular World

To certain extent, the results presented in the previous section arenot completely satisfactory in the sense that, by reduction of a sym-plectic manifold, we obtained a semialgebraic Poisson variety. Thischange in categories is exclusively due to the presence of singularities.If, by reduction, we want to obtain again symplectic manifolds if westart with symplectic manifolds, we will have to further cut or stratifythe varieties Mµ and MOµ , in such a way, that the resulting pieceswill be certain regular Marsden–Weinstein reduced spaces that will

§ 2.3. Free Actions in a Singular World 67

naturally generalize Mµ and MOµ . We dedicate the next two sectionsto the definition and study of these smooth objects. The introduc-tion of these smooth symplectic reduced spaces is due to Sjamaarand Lerman [SL91], when the group is compact, and to Bates andLerman [BL97], when the group action is proper. In both instances,the treatment is based on the use of normal forms that give a localpicture of the quotients involved. In our presentation, we will keepall our statements and proofs completely global, which will allow usto identify explicitely all the structures associated to the spaces in-volved.

The main idea behind the discussion in the present section isthat, even when a G–action on a manifold M is not free, one cantake submanifolds of M and construct groups out of G, such thatthe new action on the new submanifold is free. Ordinary Marsden–Weinstein reduction in this setup will provide global models for theabove mentioned reduced spaces that generalize Mµ and MOµ ; thesewill be introduced in the next section.

Proposition 2.3.1 Let (M, !) be a symplectic manifold, and let Gbe a Lie group acting properly on M in a globally Hamiltonian fash-ion with associated equivariant momentum map J : M % g". Letµ " g" be a value of J and m " M such that J(m) = µ. We denoteH := Gm. Then the Lie group L := N(H)/H acts freely, properly,and canonically on the symplectic manifold MH . Moreover, this ac-tion has an associated momentum map JL : MH % l", given by theexpression

JL(z) = #"(J |MH (z)$ µ), (2.3.1)

where #" is a natural L-equivariant isomorphism

#" : (h#)H $% l",

between the H–fixed point set of vectors in the annihilator of h in g"

and the dual of the Lie–algebra of L = N(H)/H.

Proof Since H is closed and normal in N(H), the quotient N(H)/His a well defined Lie group, and the map

N(H)/H )MH $% MH

(nH, m) -$% n · m,


defines, by the closedness of N(H) on G, a free and proper action ofL on MH .

In order to introduce the isomorphism #", we first prove the fol-lowing

Lemma 2.3.1 For any linear representation of the compact Liegroup H on a vector space V , there is a canonical isomorphism

(V H)" 7= (V ")H . (2.3.2)

Proof We can choose a H–invariant inner product, 0·, ·1 on V (alwaysavailable because of the possibility of averaging over H). This allowsus to orthogonally decompose V as

V = V H (W,

for some W . Therefore,

V " = W # ( (V H)#,

where we naturally identify (V H)" with W # and W " with (V H)#. Ifv " V , we denote by v" " V " the element v" = 0v, ·1. Note that Hacts on V " via the dual action

h · v" = 0h%1 · v, ·1 for any h " H.

We now prove the isomorphism (V H)" 7= (V ")H . If v" " (V ")H , wehave for every h " H,

0h%1 · v, ·1 = h · v" = v" = 0v, ·1

which is equivalent to v " V H . Thus, the restriction v"|V H definesan element of (V H)". Conversely, given an element of (V H)" = W #,its extension by zero to W defines, by the same argument as above,an element of (V ")H . #

Since the G–action is proper, its isotropy subgroups are compact.In particular, H is a compact Lie group. It is easy to see that

L = N(H)/H = (G/H)H = (G/H)H .

Therefore, by Proposition 1.3.2,

l = T[e](N(H)/H) = (T[e](G/H))H .


Hence,

l" = ((T[e](G/H))H)" 7= ((g/h)H)" 7= ((g/h)")H 7= (h#)H (2.3.3)

This chain of isomorphisms defines #". In what follows we make itmore explicit and prove some of its properties.

Using again the compactness of H, we endow g with an AdH–invariant inner product, 0·, ·1. Let p be the orthocomplement of h inn := Lie(N(H)) according to 0·, ·1, that is,

n = h( p. (2.3.4)

Since the adjoint H–action on g leaves both n and h invariant, itfollows that p is also H–invariant.

Let / : N(H) % N(H)/H be the canonical projection. Recallthat if we consider N(H)/H endowed with the homogeneous smoothstructure (the smooth structure that makes it into a Lie group), theexistence of local sections implies that / is a surjective submersionand, therefore

ker Te/ = (Te/)%1([e]) = TeH = h.

Using the decomposition in (2.3.4), we can write each , " n as , =. + 1 with . " h = kerTe/ and 1 " p and hence each + " l as+ = Te/(+), with + " p. With this notation we define

# : l $% ((h#)H)" ' g+ = Te/(+) -$% +

.

We first check that # is well-defined. Indeed, if + = Te/(+1) =Te/(+2) then Te/(+1 $ +2) = 0 and hence +1 $ +2 " kerTe/ = h.However, both +1 and +2 lie in p so that +1 $ +2 " h * p = {0}.

We now verify that # actually maps into ((h#)H)". This is cer-tainly so i! for each + " p, 0+, ·1 " (h#)H (see (2.3.2)). Since pis orthogonal to h, clearly 0+, ·1 " h#. We still have to prove thath · + = + for all + " p and h " H. By definition,

h · + = Adh+ =d

dt

$$$$t=0

h(exp t+) h%1 =d

dt

$$$$t=0

(exp t+) h$(t)h%1,

(2.3.5)

where h$(t) is some element in H such that h exp t+ = exp t+h$(t),whose existence is guaranteed by the fact that exp t+ " N(H). By


construction, h$(t)h%1 is a curve in H through the identity. Hence,there is a 0 " h such that

d

dt

$$$$t=0

h$(t)h%1 =d

dt

$$$$t=0

exp t0 = 0.

Using the Leibniz rule on (2.3.5):

h · + =d

dt

$$$$t=0

(exp t+) h$(t)h%1 = ++ 0.

Since + " p, the AdH–invariance of the splitting (2.3.4) implies thath · + " p. The above identity implies then that 0 " h*p = {0}, henceh · + = +.

By (2.3.3), we have dim l = dim((h#)H)". Since # : l% ((h#)H)" islinear, it will be an isomorphism i! it is injective. Let + = Te/(+) " lbe such that #(+) = 0 with + " p. Then + = 0 and therefore + = 0.Note that this also proves that ((h#)H)" = p.

The mapping #" in the statement is the dual of #.We now study the equivariance properties of # and #" with re-

spect to the L–actions that will be defined below. Firstly, it is clearthat L acts on l and l" by means of the adjoint and coadjoint actionsrespectively. Regarding (h#)H ' g", we will see that the map

" : L) (h#)H $% (h#)H

(l = nH, .) -$% Ad"n"1.

is a well-defined L–action. Indeed, if nH = n$H, there is an elementh in H such that n$ = nh and therefore

Ad"(n!)"1. = Ad"

(nh)"1. = Ad"h"1n"1. = Ad"

n"1Ad"h"1. = Ad"

n"1..

We now verify that " really maps into (h#)H . If n " N(H) and. " (h#)H , the element Ad"

n"1. clearly belongs to h#, because for any, " h, if we denote 0·, ·1 the natural pairing between g" and g,

-0Ad"

n"1., ,.

= 0., Adn"1,1 = 0,

since . " h#, and Adn"1, " h. We now show that Ad"n"1. is also

H–fixed. For h " H arbitrary, we have

h · Ad"n"1. = Ad"

h"1Ad"n"1. = Ad"

(hn)"1. = Ad"(nh!)"1.

= Ad"n"1Ad"

(h!)"1. = Ad"n"1.,


as required. The element h$ " H is such that nhn%1 = h$.The contragradient map corresponding to ", namely "", defines

an L-action on ((h#)H)":

"" : L) ((h#)H)" $% ((h#)H)"

(l = nH, 1) -$% Adn1.

We now prove the equivariance of # with respect to these actions.If l = nH " L and + = Te/(+) " l are arbitrary (with + " p), we willshow that

#(l · +) = l · #(+).

The right hand side is l · #(+) = l · + = Adn+. To compute the lefthand side, note that

l · + = Adl(Te/(+)) =d

dt

$$$$t=0

l(expL tTe/(+))l%1

=d

dt

$$$$t=0

l((exp t+)H)l%1 =d

dt

$$$$t=0

(n(exp t+)n%1)H

= Te/(Adn+).

Since Adn+ " ((h#)H)", we conclude that the right hand side equals

#(l · +) = Adn+ = l · #(+),

as required.The equivariance of #" is a direct consequence of that of #. In-

deed, if l = nH is as before, 1" " p" = (h#)H , and + " l are arbitrary,then,

0#"(l · 1"),+1 = 0l · 1",#(+)1 = 01", l%1 · #(+)1= 01",#(l%1 · +)1 = 0#"(1"), l%1 · +1= 0l · #"(1"),+1.

Since + is arbitrary, this is equivalent to

#"(l · 1") = l · #"(1").

We next show that the mapping

JL(m) := #"(J|MH (m)$ µ)


is a well-defined momentum map for the L–action on MH . Recallthat by Proposition 1.3.2, the submanifold MH is symplectic in itsown right, with symplectic form !H := i"!, where i : MH )% M isthe inclusion. Notice that with respect to this symplectic form, theaction (nH, m) -% n · m introduced earlier is clearly canonical. Also,TmJ|MH (TmMH) ' h# for each m "MH , because if vm " TmMH and- " h are arbitrary, by the definition of the momentum map

0TmJ|MH · vm, -1 = !(m)(-M (m), vm) = 0,

since -M (m) = 0 for any m " MH . Without loss of generality, weassume that MH is connected. Let c(t) be a curve in MH joining mand z (for instance c(0) = m and c(1) = z) and denote

v(t) =d

dtJ|MH (c(t)) " h#.

Then,& 1

0v(t)dt =

& 1

0

d

dtJ|MH (c(t))dt = J(z)$ J(m) = J(z)$ µ

is an element of h#. Therefore, J(z) " µ + h# for any z " MH .The G–equivariance of the momentum map J implies that H ' Gµ.Hence

J(z) " µ + (h#)H

since z "MH . This justifies the right hand side of equality (2.3.1) asa well–defined expression.

We now show that JL is the momentum map seeked. If so, for+ " l arbitrary, the mapping JL needs to satisfy

dJ(L(m) · vm = !(m)(+MH (m), vm), (2.3.6)

for each vm = ddt

$$t=0

c(t) " TmMH . Using expression (2.3.1), the lefthand side can be written as

dJ(L(m) · vm =

d

dt

$$$$t=0

0#"(J|MH (c(t))$ µ), +1

= 0#"(TmJ|MH · vm), +1= 0TmJ|MH · vm, #(+)1 = dJ#(()(m) · vm

= !(m)(#(+)M (m), vm). (2.3.7)


However, if + " p is such that + = Te/(+), then

+MH (m) =d

dt

$$$$t=0

expL t+ · m =d

dt

$$$$t=0

expL Te/(t+) · m

=d

dt

$$$$t=0

exp t+ · m =d

dt

$$$$t=0

exp t#(+) · m = #(+)M (m).

This together with (2.3.7) implies expression (2.3.6). !

Remark 2.3.1 Note that JL is not equivariant in general. In fact,it is very easy to compute the associated non equivariance cocycle. Ifl = nH " L, then by equivariance of J and #", we have

0(l) = JL(l · m)$ l · JL(m)= #"(J|MH (l · m)$ µ)$ l · #"(J|MH (m)$ µ)= #"(J|MH (l · m)$ µ$ l · J|MH (m)$ l · µ)= #"(l · µ$ µ)= #"(n · µ$ µ). "

Note that the appearance of non-equivariance is an e!ect peculiarto the case µ += 0, since otherwise the non equivariance cocycle istrivial. Moreover, according to Theorem 1.4.2, if the group L iscompact, the momentum map can be chosen to be equivariant. Inwhat follows we shall often work under this hypothesis. We start bymaking a simple remark that will simplify some of our computations.

Lemma 2.3.2 In the notations of Theorem 2.3.1, the group L =N(H)/H is compact if and only if N(H) is compact.

Proof Since the canonical projection / : N(H)% N(H)/H is a con-tinuous surjective submersion, if N(H) is compact, N(H)/H is alsocompact. Conversely, suppose that L is compact. Since L, N(H),and H are di!erentiable manifolds and therefore metrizable topo-logical spaces, compactness is equivalent to sequential compactness,that is, every sequence has a convergent subsequence. Let {gn} bean arbitrary sequence in N(H). Since {/(gn)} = {[gn]} ' L is asequence in L, which, by hypothesis, is compact, it has a conver-gent subsequence [gnk ] % [g]. The submersivity of / implies theexistence of local sections 0, such that / & 0 is the identity func-tion on the domain of 0 (see, for instance [BC70, Proposition 6.1.4]).


Let 0 be a local section around the element [g]. As 0 is continuous,0([gnk ])% 0([g]). Let {hnk} be a sequence in H constructed in such away that 0([gnk ])hnk = gnk . The compactness of H (it is the isotropysubgroup of a proper action) implies the existence of a convergentsubsequence hnkj

% h. The sequence gnkj= 0([gnkj

])hnkj% 0([g])h

is the convergent subsequence of {gn} that we need in order to ensurethe compactness of N(H). !

By the previous lemma, we can assume the compactness of N(H)or of L interchangeably. Now, as promised, we take the subgroupN(H) (and therefore L) to be compact and we construct an equiv-ariant momentum map for the L–action on MH described in The-orem 2.3.1. First of all, the compactness of N(H) guarantees theexistence of an AdN(H)–invariant inner product on g that allows usto orthogonally split

g = h( s, g" = h" ( s".

According to this decomposition, h# = s" and s# = h". If . = .h#+.s#

is the decomposition of an element . in g" in terms of its h" and s"

components, we define the projection

1$ : g" $% h#

. = .h# + .s# -$% .s# .

The AdN(H)–invariance of the splitting guarantees that if . = .h# +.s# " (g")H , then .s# " (h#)H = (s")H and, therefore 1$ restricts to

1 : (g")H $% (h#)H .

Lemma 2.3.3 The projection 1 : (g")H % (h#)H is L–equivariant.

Proof We first make explicit the natural L–action on (g")H ; ifl = nH " L and . = .h# + .s# " (g")H , the map 2l(.) = Ad"

n"1.is well-defined, and 2l(.) " (g")H because for any h " H, there is ah$ " H such that

h · 2l(.) = hn · . = nh$ · . = n · . = 2l(.).

With this remark, 2 defines an action of L on (g")H . Moreover, theAdN(H)–invariance of the splitting utilized, guarantees that 2l(.) =n · . splits orthogonally as n · . = n · .h# +n · .s# " h"( s", and hence

1(l · .) = n · .s# = n · 1(.) = l · 1(.),


as required. !Note that since m "MH 'M , the equivariance of J implies that

µ = J(m) " (g")H , therefore, using the AdN(H)–invariant splitting ofg" previously introduced, it admits a decomposition of the form

µ = µh# + µs# = µh# + 1(µ).

Recall again that, by the AdN(H)–invariance of the splitting, for anyl = nH " L, l·µ splits orthogonally as l·µ = Ad"

n"1µ = n·µh#+n·µs# ,with n · µh# " h" and n · µs# " s". With this in mind, we prove thefollowing:

Proposition 2.3.2 If N(H) is compact, the map

KL := #" & 1 & J|MH : MH $% l",

is an equivariant momentum map for the canonical action of L =N(H)/H on MH .

Proof Since J(MH) ' µ + (h#)H , for any m " MH , 1(J(m) $ µ) =J(m)$ µ, and hence

KL(m) = (#" & 1)(J(m)) = (#" & 1)(J(z)$ µ + µ)= #"(J(z)$ µ) + (#" & 1)(µ) = JL(z) + #"(µs#).

Since KL and JL di!er by a constant and JL is a momentum map forthe L–action on MH , KL is too. The equivariance of KL is guaranteedby the L–equivariance of J|MH , 1 and #". !

For future reference we introduce at this point the following

Corollary 2.3.1 Let µ = J(m) " g" be as above and µ = µh# +µs# "h" ( s" be its orthogonal decomposition. Then, for any n " N(H),n · µh# = µh#.

Proof Since JL and KL are two momentum mappings for the sameaction, by Proposition 1.4.1, their non equivariance cocycles need tobe in the same cohomology class. As KL is L–equivariant, its nonequivariance cocycle is trivial. From Remark 2.3.1, we know that thenon equivariance cocycle of JL, which should be a coboundary, has


the form 0(l) = #"(l · µ $ µ). Hence there is some , = #"(.), with. " (h#)H , such that

#"(n · µ$ µ) = l · , $ , = l · #"(.)$ #"(.) = #"(n · . $ .),

where l = nH " L. Hence,

n · . $ . = n · µ$ µ = (n · µh# $ µh#) + (n · µs# $ µs#).

Since both n · µs# $ µs# and n · .$ . are in s", and n · µh# $ µh# is inh", it follows that n · µh# $ µh# = 0. !

Before we carry out symplectic reduction of MH with respect tothe L–action, we show that the momentum maps that will be involvedhave constant rank.

Lemma 2.3.4 The momentum maps JL and KL introduced inpropositions 2.3.1 and 2.3.2 respectively, are constant rank mappings.

Proof Since JL and KL di!er by a constant as we saw in the proofof Proposition 2.3.2, it is enough to prove the lemma for, say JL. Us-ing its expression in terms of J, that is, JL(z) = #"(J|MH (z) $ µ),and recalling that #" is a linear isomorphism, we conclude that forany m " MH , its derivative satisfies TmJL = #" & TmJ|MH . Hencerank (TmJL) = rank (TmJ|MH ). The Bifurcation Lemma finishes theproof since, for any m "MH , rank (TmJ|MH ) = dim(h)# and is there-fore a constant. !

The previous lemma guarantees that 0 " l" (respectively +# =#"(µs#) " l") is a clean value of JL (respectively of KL). Since theaction of the Lie group L on the symplectic manifold (MH , !H), with!H := i"!, is free, proper, and canonical, theorems 1.5.3 and 1.5.7 onregular point and orbit symplectic reduction apply. More specifically,we have the following result.

Theorem 2.3.1 Let (M, !) be a symplectic manifold and let G bea Lie group acting properly on M in a globally Hamiltonian fashionwith associated equivariant momentum map J : M % g". Let µ " g"

be a value of J and m "M be such that J(m) = µ. We denote H :=Gm. Then, in terms of the notations introduced in Propositions 2.3.1and 2.3.2, we have that:


(i) The space (MH)0 := J%1L (0)/L0 is a symplectic manifold, with a

symplectic form !0 uniquely determined by the equality:

/"0!0 = i"0!H ,

where i0 : J%1L (0) )% MH is the natural inclusion, and /0 :

J%1L (0)% (MH)0 is the canonical surjective submersion. More-

over, (MH)0 = (J%1(µ) *MH)/(NGµ(H)/H).

(ii) If the group L = N(H)/H is compact (i! N(H) is compact), thespace (MH)($ := K%1

L (+#)/L($, with +# = #"(µs#) = KL(m),is a symplectic manifold, with a symplectic form !($ uniquelydetermined by the equality

/"($!($ = i"($!H ,

where i($ : K%1L (+#) )% MH is the natural inclusion, and

/($ : K%1L (+#) % (MH)($ is the canonical surjective submer-

sion. Moreover, (MH)($ = (J%1(µ) *MH)/(NGµ(H)/H).

(iii) If the group L = N(H)/H is compact, the space (MH)O#$ :=K%1

L (O($)/L, with O($, the coadjoint orbit of the element+# " l", is a symplectic manifold, with a symplectic form !O#$uniquely determined by the equality

i"O#$!H = /"O#$

!O#$ + (KL)"O#$!+O#$

, (2.3.8)

where (KL)O#$ is the restriction of KL to K%1L (O($), !

+O#$

isthe (+) symplectic structure for the coadjoint orbit O($, intro-duced in Theorem 1.5.6, iO#$ : K%1

L (O($) )%MH is the naturalinclusion,and /O#$ : K%1

L (O($) % (MH)O#$ is the canonicalsurjective submersion. Moreover, (MH)O#$ = (J%1(ON(H)

µ ) *MH)/(N(H)/H), with ON(H)

µ the N(H)–orbit of µ " g", by thecoadjoint action.

Proof The well definiteness and symplecticity of the spaces intro-duced is a straightforward consequence of theorems 1.5.3 and 1.5.7,and the remarks that we made on the L–action and the momentummappings JL and KL.

We now show that (MH)0 = (J%1(µ) *MH)/(NGµ(H)/H) andthat (MH)($ = (J%1(µ) *MH)/(NGµ(H)/H). Firstly, by the defini-tion of JL, the element m " J%1

L (0) i! #"(J|MH (m)$µ) = 0. Since #"


is an isomorphism, this amounts to J|MH (m)$µ = 0, hence J%1L (0) =

J%1(µ)*MH . Analogously one shows that K%1L (+#) = J%1(µ)*MH .

As to the subgroups L0 and L($ , the subgroup L0 consists of theelements l " L that fix 0 " l" by the a"ne action induced by the nonequivariance cocycle of JL. In other words

L0 = {l " L | l · 0 = 0} = {l " L | 0(l) = 0}= {l " L | l · µ = µ} = NGµ(H)/H.

Analogously,

L($ = {l " L | l · +# = +#} = {l " L | l · #"(µs#) = #"(µs#)},

where now the dot means ordinary coadjoint action. Recall that, byCorollary 2.3.1, for any n " N(H), we have n · µh# = µh# . Nowlet l = nH " L be such that l · #"(µs#) = #"(µs#). We can writeequivalently that n · µs# = µs# i! n · (µs# + µh#) = µs# + µh# i!n · µ = µ, which guarantees that L($ = NGµ(H)/H.

Regarding the equality

(MH)O#$ = (J%1(ON(H)µ ) *MH)/(N(H)/H),

we just need to show that K%1L (O($) = (J%1(ON(H)

µ )*MH). Indeed,m " K%1

L (O($) i! there is an element l = nH " L, with n " N(H),such that l ·+# = l ·#"(µs#) = KL(m) = #"(J|MH (m)$µ)+#"(µs#).Since #" is an equivariant isomorphism this is equivalent to l · µs# =J|MH (m) $ µ + µs# = J|MH (m) $ µh# . Since by Corollary 2.3.1,l ·µh# = µh# , our condition is equivalent to n ·µs# = J|MH (m)$n ·µh#

i! J|MH (m) = n · µ i! m " (J%1(ON(H)µ ) *MH), as required. !

2.4 Singular Symplectic Point and Orbit Re-duction

We now make use of the spaces introduced in the previous sectionto construct the reduced structures that naturally generalize Mµ andMOµ in the singular situation.

Theorem 2.4.1 (Singular symplectic point reduction) Let(M, !) be a symplectic manifold and let G be a Lie group actingproperly on M in a globally Hamiltonian fashion with associated

§ 2.4. Symplectic Point and Orbit Reduction 79

equivariant momentum map J : M % g". Let µ " g" be a value of J,and m " M be such that J(m) = µ. We denote by Gµ the isotropyof µ under the coadjoint action of G on g" and H := Gm. Then, thefollowing hold:

(i) The set J%1(µ) *MGµ

(H) is a submanifold of MGµ

(H), and thereforeof M .

(ii) The set M (H)µ := (J%1(µ) * M

Gµ

(H))/Gµ has a unique quotientdi!erentiable structure such that the canonical projection

/(H)µ : J%1(µ) *M

Gµ

(H) $%M (H)µ

is a surjective submersion.

(iii) There is a unique symplectic structure !(H)µ on M (H)

µ charac-terized by

i(H) "µ ! = /(H) "

µ !(H)µ ,

where i(H)µ : J%1(µ) *M

Gµ

(H) )%M is the natural inclusion.

(iv) The symplectic manifolds (M (H)µ , !(H)

µ ), considered as subsetsof (Mµ, {·, ·}Mµ) and (M/G, {·, ·}M/G), are their symplecticleaves.

(v) Let h " C!(M)G be a G–invariant Hamiltonian. Then the flowFt of Xh leaves the connected components of J%1(µ) * M

Gµ

(H)invariant and commutes with the Gµ–action, so it induces aflow Fµ

t on M (H)µ that is characterized by

/(H)µ & Ft & i(H)

µ = Fµt & /(H)

µ .

(vi) The flow Fµt is Hamiltonian on M (H)

µ , with Hamiltonian func-tion h(H)

µ : M (H)µ % R defined by

h(H)µ & /(H)

µ = h & i(H)µ .

The vector fields Xh and Xh(H)

µare /(H)

µ –related. We will call

h(H)µ the reduced Hamiltonian.


(vii) Let k : M % R be another G–invariant function. Then {h, k}is also G–invariant and

{h, k}(H)µ = {h(H)

µ , k(H)µ }

M(H)µ

where { , }M

(H)µ

denotes the Poisson bracket induced by the

symplectic structure in M (H)µ .

Proof (i) The Bifurcation Lemma implies that J|M

(H)µ

is a constantrank mapping. The claim follows from the Subimmersion Theorem([AMR], theorem 3.5.17).

Since all the isotropy subgroups corresponding to elements ofJ%1(µ) * M

Gµ

(H) are conjugate to H, the claim (ii) is a direct con-sequence of Proposition 1.3.1.(iii) We will study the symplectic nature of M (H)

µ by showing thatthis space is di!eomorphic to the symplectic manifold ((MH)0, !0)introduced in Theorem 2.3.1 via a certain mapping F (H)

µ : (MH)0 %M (H)

µ and that the symplectic form defined by

!(H)µ := ((F (H)

µ )%1)"!0

satisfies

i(H) "µ ! = /(H) "

µ !(H)µ .

Indeed, let F (H)µ be the projection onto the quotients of the canonical

equivariant injection f (H)µ : J%1(µ) *MH )% J%1(µ) *M

Gµ

(H), that is,the mapping that makes the following diagram commutative:

J%1(µ) *MHf (H)

µ$$$% J%1(µ) *MGµ

(H)

&0

""#""#&(H)

µ

(MH)0F (H)

µ$$$% M (H)µ .

It is easy to verify that F (H)µ is a bijection. Moreover, we now show

that it is an immersion. Indeed, let [v]0 " T[m]0(MH)0 be such that[v]0 = Tm/0·v, with v = d

dt

$$t=0

c(t) " Tm(J%1(µ)*MH). If [v]0 is suchthat T[m]0F

(H)µ · [v]0 = 0, or equivalently, d

dt

$$t=0

(F (H)µ &/0)(c(t)) = 0,


by using the definition of the map F (H)µ , we have that d

dt

$$t=0

(/(H)µ &

f (H)µ )(c(t)) = 0, and hence T

[m](H)µ/(H)

µ (Tmf (H)µ · v) = 0. Therefore,

there exists a - " gµ such that v = Tmf (H)µ ·v = -M (m) " Tm(Gµ ·m).

By construction, we also have that v " TmMH , hence Lemma 1.3.3implies that v " Tm(NGµ(H) · m) = Tm(NGµ(H)/H · m) = Tm(L0 ·m) = kerT[m]0/0, and consequently, [v]0 = Tm/0 · v = 0, as required.We then have that F (H)

µ is a bijective immersion, and therefore adi!eomorphism [GHVI, Proposition IV, sec. 3.8].

We now check that the symplectic form !(H)µ := ((F (H)

µ )%1)"!0

satisfies the defining condition in the statement. Without loss ofgenerality, we take an arbitrary element m " J%1(µ)*MH ' J%1(µ)*M

Gµ

(H), and v, w " Tm(J%1(µ) * MGµ

(H)) two arbitrary vectors. ByProposition 3.4.5, there are two Lie algebra elements -, . " gµ andv$, w$ " Tm(J%1(µ) *MH), such that

v = -M (m) + v$ and w = .M (m) + w$.

Then,

/(H) "µ !(H)

µ (m)(v, w) = !(H)µ ([m](H)

µ )(Tm/(H)µ (-M (m) + v$),

Tm/(H)µ (.M (m) + w$))

= ((F (H)µ )%1)"!0([m](H)

µ )(Tm/(H)µ (v$), Tm/

(H)µ (w$))

= !0([m]0)(T/0 · v$, T/0 · w$)= !H(m)(v$, w$)= !(m)(v$, w$).

However, we also have,

i(H) "µ !(m)(v, w) = !(m)(-M (m) + v$, .M (m) + w$)

= !(m)(-M (m), .M (m)) + !(m)(-M (m), w$)+ !(m)(v$, .M (m)) + !(v$, w$)

= !(m)(v$, w$),

where we used that !(m)(-M (m), .M (m)) = 0, since -M (m) "Tm(Gµ · m) ' kerTmJ, and .M (m) " Tm(Gµ · m) ' Tm(G · m) =(kerTmJ)%. Analogously !(m)(-M (m), w$) = !(m)(v$, .M (m)) = 0,since v$, w$ " Tm(J%1(µ) *MH) ' kerTmJ, and -M (m), .M (m) "


Tm(G ·m) = (kerTmJ)%. Since m, v and w are arbitrary we concludethat

i(H) "µ ! = /(H) "

µ !(H)µ ,

as required.We skip (iv) momentarily and prove first (v), (vi) and (vii). In

order to establish (v), notice that the G–invariance of h, the canonicalcharacter of the G–action, Noether’s Theorem, and Proposition 1.1.1imply that

J & Ft = J for any time t and (2.4.1)&g & Ft = Ft & &g for any g " G. (2.4.2)

Let’s first note that if m " J%1(µ) *MGµ

(H), then Ft(m) " J%1(µ) *M

Gµ

(H) since, by (2.4.1), Ft(m) " J%1(µ), and by (2.4.2), GFt(m) = Gm.The identity (2.4.2) implies also the existence of a well defined flowFµ

t on M (H)µ that makes the following diagram commutative

J%1(µ) *MGµ

(H)

Ft#i(H)µ$$$$% J%1(µ) *M

Gµ

(H)

&(H)µ

""#""#&(H)

µ

M (H)µ

F µt$$$% M (H)

µ .

(vi) The relation h(H)µ & /(H)

µ = h & i(H)µ plus the G–invariance of h

defines h(H)µ : M (H)

µ % R uniquely.Let Y be the vector field whose flow is Fµ

t . By construction, Y is/(H)

µ –related to Xh, that is,

T/(H)µ &Xh & i(H)

µ = Y & /(H)µ .

We now verify that Y is the Hamiltonian vector field whose Hamil-tonian function is h(H)

µ .Let m be an arbitrary element in J%1(µ) * M

Gµ

(H). We denote

[m](H)µ = /(H)

µ (m). Analogously, if v " Tm(J%1(µ) *MGµ

(H)), we write


[v](H)µ = Tm/

(H)µ · v. Now,

dh(H)µ ([m](H)

µ ) · [v](H)µ = dh(H)

µ (/(H)µ (m)) · (Tm/

(H)µ · v)

= (/(H) "µ (dh(H)

µ ))(m)(v)

= d(h(H)µ & /(H)

µ ))(m)(v)

= d(h & i(H)µ )(m)(v)

= (i(H) "µ (dh))(m)(v)

= (i(H) "µ (iXh!))(m)(v). (2.4.3)

Since Tmi(H)µ ·Xh(m) = (Xh&i

(H)µ )(m) we can write expression (2.4.3)

as

(iXh(/(H) "µ !(H)

µ ))(m) · (v) = (/(H) "µ !(H)

µ )(m)(Xh(m), v)

= !(H)µ ([m](H)

µ )(Tm/(H)µ · Xh(m), Tm/

(H)µ · v)

= !(H)µ ([m](H)

µ )(Y [m](H)µ , [v](H)

µ ),

as required.(vii) The G–invariance of {h, k} comes from the fact that G actscanonically. Therefore, by part (vi), {h, k}(H)

µ is defined by

{h, k}(H)µ [m](H)

µ = ({h, k} & i(H)µ )(m)

= (i(H) "µ !)(m)(Xh(m), Xk(m))

= (/(H) "µ !(H)

µ )(m)(Xh(m), Xk(m))

= !(H)µ ([m](H)

µ )(Xh(H)

µ[m](H)

µ , Xk(H)

µ[m](H)

µ )

= {h(H)µ , k(H)

µ }M(H)

µ([m](H)

µ ).

(iv) The mappings that allow us to see M (H)µ as a subset of Mµ

and M/G are the injections that one obtains by projecting onto thequotients, namely i : J%1(µ)*M

Gµ

(H) )% J%1(µ) and iµ : J%1(µ) )%M .

In other words, the injections on the quotients are the maps j(H)µ and

jµ that make the following diagram commutative

J%1(µ) *MGµ

(H)i$$$% J%1(µ)

iµ$$$% M

&(H)µ

""#""#&µ

""#&

M (H)µ

j(H)µ$$$% Mµ

jµ$$$% M/G.


We first prove that the connected components of the symplectic man-ifolds (M (H)

µ , !(H)µ ) are the symplectic leaves of (Mµ, {·, ·}Mµ). As

a first step we show that the mapping j(H)µ is Poisson. Indeed, let

fµ, gµ " C!(Mµ) be arbitrary and let f, g " C!(M)G be smoothlocal G–invariant extensions at m " J%1(µ) of fµ & /µ, gµ & /µ "C!(J%1(µ))Gµ , always available by Proposition 3.4.3. Then, for any[m](H)

µ "M (H)µ , we have that

(j(H) "µ {fµ, gµ}Mµ)([m](H)

µ ) = {fµ, gµ}Mµ([m]µ) = {f, g}(m).

Let now f (H)µ , g(H)

µ " C!(M (H)µ ) be the reductions of f, g "

C!(M)G. Clearly, fµ & j(H)µ = f (H)

µ and gµ & j(H)µ = g(H)

µ hence,using part (vii), we can write

{j(H) "µ fµ, j(H) "

µ gµ}M(H)µ

([m](H)µ ) = {f (H)

µ , g(H)µ }

M(H)µ

([m](H)µ )

= {f, g}(H)µ ([m](H)

µ )

= {f, g}(m).

Since the point m " J%1(µ) is arbitrary, we have that

j(H) "µ {fµ, gµ}Mµ = {j(H) "

µ fµ, j(H) "µ gµ}M

(H)µ

,

as required.Note that by the dynamical part of Theorem 2.2.2, the Hamil-

tonian flows of functions in C!(Mµ) leave invariant the connectedcomponents of the sets M (H)

µ , which implies that the symplectic leavesof Mµ lie in the connected components of the sets M (H)

µ . Sincethe inclusion j(H)

µ is Poisson, that is, the restriction of {·, ·}Mµ toM (H)

µ is given by the symplectic form !(H)µ , and every function in

C!(M (H)µ ) can be extended to C!(Mµ) by Proposition 3.4.3, the

connected components of the sets M (H)µ are necessarily the symplectic

leaves of (Mµ, {·, ·}Mµ). The same thing can be proven identically for(M/G, {·, ·}M/G) by showing in a similar fashion that the inclusionjµ &j(H)

µ is also Poisson. In this case, the existence of extensions fromC!(M (H)

µ ) to C!(M/G) is also guaranteed by Proposition 3.4.3. !As we shall see in the next theorem, the spaces of the previous

section can also be used to construct symplectic orbit reduced spacesthat generalize MOµ to the singular case.


Theorem 2.4.2 (Singular symplectic orbit reduction) Let(M, !) be a symplectic manifold and let G be a Lie group actingproperly on M in a globally Hamiltonian fashion with associatedequivariant momentum map J : M % g". Let µ " g" be a value ofJ and let m " M be such that J(m) = µ. We denote by Oµ thecoadjoint orbit of µ under the coadjoint action of G on g" and letH := Gm. Then, the following hold:

(i) The set J%1(Oµ)*M(H) is a submanifold of J%1(Oµ)*M(H) andtherefore of M .

(ii) The set M (H)Oµ

:= J%1(Oµ) *M(H)/G has a unique quotient dif-ferentiable structure such that the canonical projection

/(H)Oµ

: J%1(Oµ) *M(H) $%M (H)Oµ

is a surjective submersion.

(iii) There is a unique symplectic structure !(H)Oµ

on M (H)Oµ

charac-terized by

i(H) "Oµ

! = /(H) "Oµ

!(H)Oµ

+ J(H) "Oµ

!+Oµ

,

where i(H)Oµ

: J%1(Oµ) * M(H) )% M is the natural inclusion.

The mapping J(H)Oµ

is defined by J(H)Oµ

:= J |J"1(Oµ)&M(H)and

!+Oµ

is the symplectic form on Oµ given in Theorem 1.5.6.

(iv) The symplectic manifolds (M (H)Oµ

, !(H)Oµ

), considered as subsetsof (MOµ , {·, ·}MOµ

) and (M/G, {·, ·}M/G), are their symplecticleaves.

(v) Let h " C!(M)G be a G–invariant Hamiltonian. Then the flowFt of Xh leaves the connected components of J%1(Oµ) *M(H)

invariant and commutes with the G–action, so it induces a flowF

Oµt on M (H)

Oµthat is characterized by

/(H)Oµ

& Ft & i(H)Oµ

= FOµt & /(H)

Oµ.

(vi) The flow FOµt is Hamiltonian on M (H)

Oµ, with Hamiltonian func-

tion h(H)Oµ

: M (H)Oµ

% R defined by

h(H)Oµ& /(H)

Oµ= h & i(H)

Oµ.


The vector fields Xh and Xh(H)Oµ

are /(H)Oµ

–related. We will call

h(H)Oµ

the reduced Hamiltonian.

(vii) Let k : M % R be another G–invariant function. Then {h, k}is also G–invariant and

{h, k}(H)Oµ

= {h(H)Oµ

, k(H)Oµ

}M

(H)Oµ

where { , }M(H)

Oµ

denotes the Poisson bracket induced by the

symplectic structure in M (H)Oµ

.

Proof The proof imitates that of Theorem 2.4.1. We just outlinethe changes that should be made with respect to that proof. Thesymplectic nature of M (H)

Oµis obtained by using the symplectic man-

ifold ((MH)0, !0). In this case, one sees that the map that resultsfrom projecting onto the quotient the canonical equivariant injectionf (H)Oµ

: J%1(µ) *MH )% J%1(Oµ) *M(H) is the di!eomorphism F (H)Oµ

that makes the following diagram commutative

J%1(µ) *MH

f (H)Oµ$$$% J%1(Oµ) *M(H)

&0

""#""#&(H)

Oµ

(MH)0F (H)Oµ$$$% M (H)

Oµ.

The symplectic form defined by

!(H)Oµ

:= ((F (H)Oµ

)%1)"!0

satisfies the defining condition of !(H)Oµ

in the statement. The claimsrelated to the reduction of the dynamics are proven in a fashion simi-lar to those of Theorem 2.4.1. Regarding the relation of (M (H)

Oµ, !(H)

Oµ)

with (MOµ , {·, ·}MOµ) and (M/G, {·, ·}M/G) we also follow the strat-

egy of the proof of Theorem 2.4.1, this time using the mappingsintroduced in the following commutative diagram

J%1(Oµ) *M(H)i!$$$% J%1(Oµ)

iOµ$$$% M

&(H)Oµ

""#""#&Oµ

""#&

M (H)Oµ

j(H)Oµ$$$% MOµ

jOµ$$$% M/G.


Notice that in this case, the existence of smooth G–invariant exten-sions of functions in C!(J%1(Oµ))G, is a straightforward corollary toProposition 2.1.1. !

Remark 2.4.1 The singular symplectic point and orbit reducedspaces introduced in theorems 2.4.1 and 2.4.2, clearly generalize theMarsden–Weinstein point and orbit reduced spaces since in the freecase, the isotropy groups H are always trivial which, by the Bifur-cation Lemma, implies that any value of the momentum mapping Jis always regular. In addition, M(H) = M

Gµ

(H) = M , which guar-

antees that (M (H)µ , !(H)

µ ) = (Mµ, !µ), and that (M (H)Oµ

, !(H)Oµ

) =(MOµ , !Oµ). "

Remark 2.4.2 In theorems 2.4.1 and 2.4.2, we have seen thatthe symplectic manifolds (M (H)

µ , !(H)µ ) and (M (H)

Oµ, !(H)

Oµ) constitute

the symplectic leaves of the Poisson varieties (Mµ, {·, ·}Mµ) and(MOµ , {·, ·}MOµ

), respectively. It can be seen that the propernessof the action guarantees, through the existence of slices (see Chap-ter 3), that the partition of these Poisson varieties in their symplecticleaves is locally finite [BL97, GS84b]. Moreover, Sjamaar and Ler-man [SL91] have shown, using normal forms, that these partitionsare actually stratifications in the sense of Definition 2.1.2. "

If the singular symplectic point and orbit reduced spaces gen-eralize the Marsden–Weinstein point and orbit reduced spaces, onecan expect the two procedures to be closely related. The followingtheorem shows that this is indeed the case.

Theorem 2.4.3 (The Singular Reduction Diagram) In thehypotheses of theorems 2.4.1 and 2.4.2, the mapping L(H)

µ defined asthe projection onto the quotients of the natural equivariant injectionl(H)µ : J%1(µ) *M

Gµ

(H) )% J%1(Oµ) *M(H), that is, the map makingthe diagram

J%1(µ) *MGµ

(H)

l(H)µ$$$% J%1(Oµ) *M(H)

&(H)µ

""#""#&(H)

Oµ

(M (H)µ , !(H)

µ )L(H)

µ$$$% (M (H)Oµ

, !(H)Oµ

),

commutative, is a symplectomorphism.


Proof It is a simple diagram chasing exercise. A straightforwardway to show this result is by recalling that both (M (H)

µ , !(H)µ ) and

(M (H)Oµ

, !(H)Oµ

) were introduced as symplectic manifolds by using nat-ural mappings that made them symplectomorphic to ((MH)0, !0).This implies that (M (H)

µ , !(H)µ ) and (M (H)

Oµ, !(H)

Oµ) are symplectomor-

phic and so it can be easily verified that L(H)µ is the specific symplec-

tomorphism in question. !As a conclusion to this chapter, we illustrate all the reduction

schemes that we have introduced, as well as the relation among them-selves, with the following diagram in which all the mappings involvedare morphisms with respect to the categories in which their domainsand ranges lie. Notice that the last line is only well defined when theLie group L is compact (see Theorem 2.3.1).

M&$$$% M/G

id$$$% M/G&8$$$ M

iµ

/"" jµ

/"" jOµ

/""/""iOµ

J%1(µ)&µ$$$% Mµ

Lµ$$$% MOµ

&Oµ8$$$ J%1(Oµ)

i

/"" j(H)µ

/"" j(H)Oµ

/""/""i!

J%1(µ) *MGµ

(H)

&(H)µ$$$% M (H)

µL

(H)µ$$$% M (H)

Oµ

&(H)Oµ8$$$ J%1(Oµ) *M(H)

f (H)µ

/"" F (H)µ

/"" F (H)Oµ

/""/""f (H)

Oµ

J%1L (0) &0$$$% (MH)0

id$$$% (MH)0&08$$$ J%1

L (0)

id

/"" id

/"" L#$

""#""#l#$

K%1L (+#) $$$%

&#$(MH)($ $$$%

L#$

(MH)O#$ 8$$$&#$K%1

L (O($)

Chapter 3

The Marle–Guillemin–Sternberg NormalForm

Capıtulo XXXI: Que trata de muchas y grandescosas.Cervantes, Don Quijote de la Mancha, II

Most of the good technical behavior that one encounters whendealing with proper actions of Lie groups comes from the existenceof slices and tubes associated to them. These structures providea privileged system of coordinates in which the group action takesan expression particularly simple and technically convenient. Theexistence of these charts adapted to the group action extends to thecase in which the action takes place on a symplectic manifold, in acanonical fashion. In this case, the tubular chart can be constructedin such a way that the expression of the symplectic form is verynatural and, moreover, if there is a momentum map associated tothis canonical action, this construction provides a normal form for it.This Symplectic Slice Theorem is also known as the Marle–Guillemin–Sternberg normal form. In this chapter we will describe in detail allthese local tools and we will give some preliminary examples on howto apply them, as a warm up for the following chapters where theseforms will play a crucial role.

90 Chapter 3. The MGS Normal Form

3.1 Proper Actions, Tubes, and Slices

The following definitions and results are standard in Lie theory(see [Bre72, Pal61]). They give a G–invariant local model for a man-ifold M on which there is a Lie group G acting properly.

Definition 3.1.1 Let G be a Lie group and H ' G be a compactsubgroup. Suppose that H acts on the left on the manifold A. Thetwist action of H on the product G)A is defined by

h · (g, a) = (gh, h%1 · a).

Note that this action is free and proper by the freeness of the actionon the G–factor and the compactness of H. The twisted productG)H A is defined as the orbit space corresponding to the twist action.The twisted product G)H A is naturally a G–space relative to the leftaction defined by g$ · [g, a] := [g$g, a].

Definition 3.1.2 Let M be a manifold and G be a Lie group actingproperly on M . Let m " M and denote Gm := H. A tube about theorbit G · m is a G–invariant di!eomorphism

" : G)H A $% U,

with U a G–invariant neighborhood of G ·m and A some manifold onwhich H acts. Note that the twisted product G )H A is well–definedsince by Proposition 1.3.1 the isotropy subgroup H is compact.

Definition 3.1.3 Let M be a manifold and G be a Lie group actingproperly on M . Let m " M and denote Gm := H. Let S be asubmanifold of M , such that m " S and H · S = S. We say that Sis a slice at m if the map

G)H S $% U[g, s] -$% g · s

is a tube about G · m, for some G–invariant open neighborhood ofG · m.

Theorem 3.1.1 Let M be a manifold and G be a Lie group actingproperly on M . Let m " M , denote Gm := H, and let S be asubmanifold of M such that m " S. Then the following statementsare equivalent:

§ 3.1. Proper Actions, Tubes, and Slices 91

(i) There is a tube " : G)H A $% U about G ·m such that "[e, A] =S. The space A can always be chosen to be an H–invariantneighborhood of 0 in the vector space TmM/Tm(G · m), whereH acts linearly by h · [v] = [h · v].

(ii) S is a slice at m.

(iii) G·S is an open neighborhood of G·m and there is an equivariantretraction

r : G · S $% G · m

such that r%1(m) = S.

Proof See for instance [Bre72, Pal61, ACG91, CB97]. !

Theorem 3.1.2 (Slice Theorem) Let M be a manifold and G bea Lie group acting properly on M . Let m "M and denote Gm := H.Then there is a slice for the G–action at m.

Proof See Proposition 2.2.2 and Remark 2.2.3 of [Pal61]. !

Remark 3.1.1 One way to construct the slice (see [Bre72]) consistsroughly (we omit here some technical details) of taking a H–invariantRiemannian metric g on M (always available by the compactness ofH) and letting A = Tm(G ·m)(, where 9 denotes orthogonality withrespect to g. The G–equivariant map

" : G)H A $% U[h, a] -$% h · expg a

is a tube around the orbit G · m. "

Remark 3.1.2 The Slice Theorem is due to J. L. Koszul [Kos53]in the case of compact group actions. It was generalized to propergroup actions by R. Palais [Pal61]. "

Proposition 3.1.1 Let G be a Lie group acting properly on thesmooth manifold M . Let m " M be a point with isotropy subgroupH := Gm. Then

((Tm(G · m))#)H = {df(m) | f " C!(M)G}.


Proof We first show that if f " C!(M)G, then df(m) " ((Tm(G ·m))#)H . It is clear that for any - " g,

0df(m), -M (m)1 =d

dt

$$$$t=0

f(exp t- · m) =d

dt

$$$$t=0

f(m) = 0.

Hence, df(m) " Tm(G · m)#. Now, df(m) is also H–fixed since forany h " H and any v = d

dt

$$t=0

m(t) " TmM with m(0) = m,

0h · df(m), v1 = 0df(m), h%1 · v1 =d

dt

$$$$t=0

f(h%1 · m(t))

=d

dt

$$$$t=0

f(m(t)) = 0df(m), v1.

Since the vector v is arbitrary, h · df(m) = df(m), as required.Since we are going to work locally, in order to prove the converse

inclusion, we do it in the tubular model provided by the Slice The-orem. Thus, the manifold M will be replaced by G )H V , whereV = TmM/Tm(G · m), and the point m " M is represented by[e, 0] " G)H V . It is easy to verify that

T[e, 0](G · [e, 0]) = {T(e, 0)/(-, 0) " T[e, 0](G)H V ) | - " g} 7= g/h) {0},

where / : G) V % G)H V is the canonical projection. Clearly,

(T[e, 0](G · [e, 0]))# 7= {0}) V " 7= V ",

and hence, using Lemma 2.3.1

((T[e, 0](G · [e, 0]))#)H 2 (V ")H 7= (V H)".

In the tubular model, the G–invariant functions f " C!(G )H

V )G are characterized by the condition f & / " C!(V )H . The claimthen follows if we show that

(V ")H = {dg(0) " V " | g " C!(V )H}.

Let g " C!(V )H and h " H be arbitrary. Then, for any v =ddt

$$t=0

c(t) " V with c(0) = 0, we have

0h · dg(0), v1 = 0dg(0), h%1 · v1 =d

dt

$$$$t=0

g(h%1 · c(t))

=d

dt

$$$$t=0

g(c(t)) = 0dg(0), v1.

§ 3.1. Proper Actions, Tubes, and Slices 93

Since v " V is arbitrary, it follows that h · dg(0) = dg(0).To prove the converse, we begin by decomposing V into its irre-

ducible H–components:

V = W1 ( . . .(Wk ( U1 ( . . .( Ur,

where dim W1 = . . . = dimWk = 1, and dimUi > 1 for i "{1, . . . , r}. Thus,

V H = W1 ( . . .(Wk. (3.1.1)

Let {w1, . . . , wk}, be a basis of V H adapted to the splitting (3.1.1).Define /1, . . . , /k " V " by

/i(wj) = (ij i, j " {1, . . . , k}/i|Up = 0 i " {1, . . . , k}, p " {1, . . . , r}.

By construction, the functionals /1, . . . , /k " V " are linear invariantsof the H–action on V . Moreover, they are the only ones. Indeed, since/1, . . . , /k is a basis of (V H)", there are no additional independentlinear invariants on V H . If $ : U1( . . .(Ur % R is another nontriviallinear invariant, there is some p " {1, . . . , r} such that $|Ur is not thezero functional. Therefore, ker($|Ur) += 0 is a nontrivial H–invariantsubspace of Up. Since this is impossible by the irreducibility of Up, itfollows that such an $ cannot exist.

We have thus shown that /1, . . . , /k " V ", or in general anybasis of (V H)", spans the set of all independent linear invariantsof the H–action on V . By the Hilbert Theorem, the ring of H–invariant polynomials on V is finitely generated. We complete theset {/1, . . . , /k} to a generating system {/1, . . . , /k, /k+1, . . . , /q}of the this ring. The Schwarz Theorem [Sch74, Ma77] guarantees thatevery H–invariant function f " C!(V )H can be locally written as

f = g(/1, . . . , /q),

with g " C!(Rq). Let now $ " (V ")H 7= (V H)" be arbitrary. Theform $ " (V H)" can be expanded as

$ = $1/1 + . . . + $k/k.

with $1, . . . , $k " R. Let g " C!(Rq) be such that

#g(0)#/i

= $i, i " {1, . . . , k}.


With this choice, the function f := g(/1, . . . , /q) belongs to C!(V )H

and satisfies

df(0) =#g(0)#/1

/1 + . . . +#g(0)#/k

/k

= $1/1 + . . . + $k/k = $,

where we used that d/j(0) = 0 for j " {k + 1, . . . , q} because theinvariants /j in this range of the indices are at least quadratic.. Since$ is arbitrary, the result follows. !

Remark 3.1.3 The properness condition in the statement of theprevious proposition is essential (and is not tied to the existenceof slices) since there are examples of non proper actions where thisresult doesn’t hold. Indeed, consider the irrational flow on the torus.Since the orbits of this action fill densely the torus, the only invariantfunctions in this particular case are the constant functions. Hencethe right hand side of the equality in Proposition 3.1.1 is trivial.However, if the torus in question is bigger than one dimensional, thevector space (Tm(G · m))# is non trivial. "

The Slice Theorem and the Tube Lemma in point set topology([Mun75], Lemma 5.8, page 169) imply the following corollary, thatwe state for future reference.

Corollary 3.1.1 Let G be a compact Lie group that acts on the man-ifold M and let m "M . Any open neighborhood V of the orbit G ·mcontains a G–invariant open neighborhood of G · m.

3.2 Hamiltonian Tubes: the Marle–Guillemin–Sternberg Normal Form

We have just constructed charts adapted to manifolds M on whichthere is a Lie group G acting properly, that is, the results just pre-sented locally model G–manifolds in such a way that the G–structureappears in a particularly simple fashion. Next, we consider the casein which the manifold M is symplectic, the G–action is canonical, andit has an associated momentum map J : M % g". We will see that,in the philosophy of the previous section, it is possible to constructa tube that incorporates this additional structure. This Hamiltonian

§ 3.2. Hamiltonian Tubes: the MGS Normal Form 95

tube is called the Marle-Guillemin-Sternberg (MGS) normalform, since it was introduced by Marle [Mar85], and by Guilleminand Sternberg [GS84a, GS84b], who considered compact groups intheir treatment. The generalization to proper group actions is due toBates and Lerman [BL97]. See also [CB97].

We first present the symplectic tube (Y, !Y ) that models oursymplectic G–space (M, !), following a generalization of the presen-tation made by Sjamaar and Lerman [SL91]. Once we have explicitelyconstructed (Y, !Y ) we will construct a G–equivariant local mappingthat puts into correspondence the model Y with the original manifoldM , preserving all the relevant structures involved.

Definition 3.2.1 Let (M, !) be a symplectic manifold and G be aLie group acting properly and canonically on M . Let m be a point inM . The vector space Vm := Tm(G ·m)%/(Tm(G ·m)% *Tm(G ·m)) iscalled the symplectic normal space at m.

The terminology chosen in the previous definition is justified by thefollowing lemma, whose proof is a straightforward exercise in sym-plectic algebra.

Lemma 3.2.1 Let (M, !) and m " M be as in Definition 3.2.1.The symplectic normal space Vm at m is a symplectic vector space,with the symplectic normal form !Vm defined by

!Vm([v], [w]) := !(m)(v, w),

for any [v] = /(v) and [w] = /(w) " Vm, and where / : Tm(G·m)% %Tm(G ·m)%/(Tm(G ·m)%*Tm(G ·m)) is the canonical projection. LetH := Gm be the isotropy subgroup of m. The mapping (h, [v]) -$%[h · v], with h " H and [v] " Vm, defines a canonical action of theLie group H on (Vm, !Vm), where g · u denotes the tangent lift of theG–action on TM , for g " G and u " TM .

Remark 3.2.1 The canonical H–action on Vm is linear by construc-tion. Example 1.4.3 shows that this action is necessarily globallyHamiltonian with Ad"–equivariant momentum map JVm : Vm % h"

given by

0JVm(v), .1 =12!Vm(.Vm(v), v),

for . " h and v " Vm arbitrary. Here, .Vm(v) = . · v is the inducedLie algebra representation of h on Vm. "


It can be shown that the space (Vm, !Vm) is isomorphic to a max-imal symplectic subspace of (Tm(G · m)%, !(m)). Notice that if thecanonical G–action has an associated momentum map J : M % g",and denoting J(m) = µ " g", the Reduction Lemma allows us towrite the symplectic normal space at m as

Vm = kerTmJ/Tm(Gµ · m),

where Gµ denotes the stabilizer of µ " g" under the a"ne action ofG on g". In some instances, we will realize the symplectic normalspace (Vm, !Vm) as a symplectic vector subspace of (TmM, !(m)) bychoosing in TmM a H–invariant inner product, always available bythe compactness of H. Using this inner product we define Vm as theorthogonal complement of Tm(G · m)% * Tm(G · m) in Tm(G · m)%,that is,

Tm(G · m)% = (Tm(G · m)% * Tm(G · m))( Vm,

where the symbol( denotes orthogonal direct sum. It is easy to verifythat (Vm, !(m)|Vm) is a symplectic vector space, symplectomorphicto (Tm(G · m)%/(Tm(G · m)% * Tm(G · m)), !Vm). Moreover, the H–invariance of the splitting used to construct Vm makes this spaceH–invariant under the lifted action of H to TM . This H–action onVm is canonical and the symplectomorphism between (Vm, !(m)|Vm)and (Tm(G · m)%/(Tm(G · m)% * Tm(G · m)), !Vm) is equivariant.

In what follows we assume that the canonical G–action is globallyHamiltonian, that is, it has an associated Ad"–equivariant momen-tum map J : M % g". Let J(m) = µ " g". We denote by gµ theLie algebra of the stabilizer of µ " g" under the coadjoint action ofG on g". Recall that by the equivariance of J, the isotropy subgroupH of m is a subgroup of Gµ and therefore h = Lie(H) ' gµ. Usingagain the compactness of H, we construct an inner product 0·, ·1 ong, invariant under the restriction to H of the adjoint action of G on g.Relative to this inner product we can write the following orthogonaldirect sum decompositions

g = gµ ( q, and gµ = h(m,

for some subspaces q ' g and m ' gµ. The inner product also allowsus to identify all these Lie algebras with their duals. In particular, wehave the dual orthogonal direct sums g" = g"µ ( q" and g"µ = h" (m"


which allow us to consider g"µ as a subspace of g" and, similarly, h"

and m" as subspaces of g"µ.The H–invariance of the inner product utilized to construct the

splittings gµ = h(m and g"µ = h"(m", implies that both m and m" areH–spaces using the restriction to them of the H–adjoint and coadjointactions, respectively. Indeed, if . " m; by definition, 0., -1 = 0, forany - " h. Therefore, for any h " H, 0Adh., -1 = 0., Adh"1-1 = 0,since Adh"1- " h. Since h " H is arbitrary, this proves that m is aH–space. Analogously, one proves the H–invariance of m".

We have now all the ingredients to present the symplectic tube(Yr, !Yr) that is going to model the symplectic G–manifold (M, !).

Proposition 3.2.1 Let (M, !) be a symplectic manifold and let G bea Lie group acting properly on M in a globally Hamiltonian fashion,with Ad"–equivariant momentum map J : M % g". Let m " M anddenote J(m) = µ " g". Let (Vm, !Vm) be the symplectic normal spaceat m " M . Relative to an H–invariant inner product on g considerthe inclusions m" ' g"µ ' g". Then there is a positive number r > 0such that, denoting by m"

r the open ball of radius r relative to theH–invariant inner product on m", the manifold

Yr := G)H (m"r ) Vm)

can be endowed with a symplectic structure !Yr with respect to whichthe left G–action g ·[h, ., v] = [gh, ., v] on Yr is globally Hamiltonianwith Ad"–equivariant momentum map JYr : Yr % g" given by

JYr([g, 1, v]) = Ad"g"1 · (µ + 1+ JVm(v)). (3.2.1)

Proof Since H is compact and there is a well defined free H–actionon Vm (by Lemma 3.2.1), on m" (by the H–invariance of the innerproduct used to construct it), and therefore on m")Vm (the productaction); the twisted product G)H(m")Vm) is therefore a well–definedsmooth manifold.

We now describe the symplectic form !Yr . Note that, since theabove described splittings allow us to see g"µ as a subset of g", themanifold Y1 := G) g"µ can be considered as a submanifold of G) g"

which is di!eomorphic to the cotangent bundle T "G (unless specifiedotherwise, we always use left trivializations –or body coordinates inthe language of continuum mechanics– for T "G and TG). Let !1 bethe pull–back of the canonical symplectic form from T "G to G ) g"


and then to G) g"µ. Denote by !µ the pull–back to G) g"µ of the +orbit symplectic structure !+

Oµon the coadjoint orbit Oµ through µ

(see Theorem 1.5.6) by the surjective submersion

/ : G) g"µ $% Oµ

(g, ,) -$% Ad"g"1µ,

and define the closed two–form ) on Y1 by ) := !1 + !µ.We will need below the formula of the pull back !B by left trans-

lation of the canonical symplectic form ! from T "G to G) g" givenin [AM78, Proposition 4.4.1]:

!B(g, ,)((TeLg-, 1), (TeLg.,0)) = $01, .1+ 00, -1+ 0,, [-, .]1(3.2.2)

for g " G, -, . " g, 1,0 " g".We now show that this closed two–form ) is non-degenerate at

each point of the form (g, 0) " Y1. Since non–degeneracy is an opencondition, this will guarantee that the form ) is non–degenerate in anopen neighborhood of the base of the trivial vector bundle Y1 % G.Thus, there is a positive number r > 0 such that G ) (h"r ( m"

r) issymplectic, where h"r and m"

r are the open balls of radius r relative tothe H–invariant inner products on h" and m" respectively. To provethis nondegeneracy of ) in a neighborhood of G){0}, let - = -1 +-2,. = .1 + .2 " g, with -1, .1 " gµ and -2, .2 " q. Let 1 = 01, ·1 and0 = 00, ·1 with 1, 0 " g"µ, 1, 0 " gµ and denote by 0·, ·1 the AdH–invariant inner product in g. Let’s suppose that

)(g, 0)00

d

dt

$$$$t=0

g exp t-, 1

1,

0d

dt

$$$$t=0

g exp t., 0

11= 0,

for any . " g and 0 " g"µ. Using (3.2.2), this implies that

)(g, 0)00

d

dt

$$$$t=0

g exp t(-1 + -2), 11

,

0d

dt

$$$$t=0

g exp t(.1 + .2),011

= 00, -1 + -21 $ 01, .1 + .21

+ !+Oµ

(g · µ)0

d

dt

$$$$t=0

g exp t- · µ,d

dt

$$$$t=0

g exp t. · µ1

= 00, -11 $ 01, .11

+ !+Oµ

(g · µ)0

d

dt

$$$$t=0

g exp t-2 · µ,d

dt

$$$$t=0

g exp t.2 · µ1

= 0,


for arbitrary .1 " gµ, .2 " q, and 0 " g"µ. This implies that - = 0and 1 = 0 and hence )(g, 0) is nondegenerate for all g " G.

Consider now the left action R of H on Y1 given by

Rh(g, ,) = (gh%1, Ad"h"1,)

and the orthogonal decomposition gµ = h(m. Using the definition of) and (3.2.2), it is straightforward to verify that this action is glob-ally Hamiltonian (on the presymplectic manifold Y1) with equivariantmomentum map JR : Y1 % h", given by

JR((g, (., 1))) = $., for any (., 1) " h" (m" = g"µ.

As we pointed out in Remark 3.2.1, the H–action on Vm is globallyHamiltonian with momentum map JVm : Vm % h" given by theformula:

0JVm(v), -1 =12!Vm(-Vm(v), v) for any - " h.

Putting together these two actions, we construct a product action ofH on the presymplectic manifold Y1 ) Vm (endowed with the sumpresymplectic form), which is Hamiltonian, with H–equivariant mo-mentum map ! : Y1 ) Vm

7= G ) m" ) h" ) Vm % h", given by thesum JR + JVm , that is,

! : G)m" ) h" ) V $% h"

(g, 1, ., v) -$% JVm(v)$ ..

The H–action on Y1 ) Vm is free and proper and 0 " h" is clearly aregular value of !. Therefore !%1(0)/H is a well–defined Marsden–Weinstein reduced presymplectic space which can be identified withY = G )H (m" ) Vm) by means of the quotient di!eomorphism L,induced by the H–equivariant di!eomorphism l:

l : G)m" ) Vm $% !%1(0) ' G)m" ) h" ) Vm

(g, 1, v) -$% (g, 1, JVm(v), v).

We define the presymplectic form !Y on Y as the pull back by L ofthe reduced presymplectic form )0 on !%1(0)/H. Thus, we have thefollowing commutative diagram with the lower arrow a presymplectic


di!eomorphism:

G)m" ) Vml$$$% !%1(0) ' G)m" ) h" ) Vm

&

""#""#&0

(G)H (m" ) Vm), !Y ) L$$$% (!%1(0)/H, )0).(3.2.3)

It is clear that the presymplectic manifolds constructed abovehave open symplectic submanifold by restricting m" to m"

r . Thisreplaces Y by Yr and !%1(0)/H by the reduction of G)m"

r)h"r)Vm atzero. Thus, any statements that can be made for Y as a presymplecticmanifold, can be made for Yr as a symplectic manifold and this is whywe shall work with Y in what follows.

We next show that Y is a presymplectic Hamiltonian G–spacewith the left G–action given in the statement. Let L be the leftG–action on Y1 defined by

L : G) (G) g"µ) $% G) g"µ(h, (g, .)) -$% (hg, .).

It is easy to verify that this action is canonical on (Y1, )). Moreover,it has an associated momentum map given by the expression

JL(g, .) = Ad"g"1(µ + .).

To show this, we verify that for any - " g we have the identity

i$-Y1 = dJ#L.

Notice that for any (g, ,) " Y1,

-Y1(g, ,) =0

d

dt

$$$$t=0

(exp t-)g, 01

=-g(Adg"1-), 0

2.

Then, for-

ddt

$$t=0

g exp t., 02" T(g, ')Y1 arbitrary, the definition of )


and (3.2.2) imply

)(g, ,)0-Y1(g, ,),

0d

dt

$$$$t=0

g exp t., 0

11

= 00, Adg"1-1+ 0,, [Adg"1-, .]1

+ !+Oµ

(g · µ)0

d

dt

$$$$t=0

(exp t-)g · µ,d

dt

$$$$t=0

g(exp t.) · µ1

= 00, Adg"1-1+ 0,, [Adg"1-, .]1+ 0g · µ, [-, Adg.]1,

(3.2.4)

where 0·, ·1 denotes the natural pairing between g" and g. On theother hand, from the expression of JL, we get

dJ#L(g, ,) ·

0d

dt

$$$$t=0

g exp t., 0

1

=d

dt

$$$$t=0

0Ad"exp(%t$) g"1(µ + , + t0), -1

= 0$Ad"g"1(ad"

$µ + ad"$,) + Ad"

g"10, -1= 00, Adg"1-1 $ 0µ, [., Adg"1-]1 $ 0,, [., Adg"1-]1= 00, Adg"1-1+ 0,, [Adg"1-, .]1+ 0g · µ, [-, Adg.]1,

which equals (3.2.4), as required.We now regard L as an action on Y1)Vm by letting G act trivially

on Vm, that is,

L : G) (Y1 ) Vm) $% Y1 ) Vm

(h, (g, ., v)) -$% (hg, ., v).

This action commutes with the H action on Y1 ) Vm and leaves thelevel sets of the H–momentum map ! invariant. This implies that theL–action restricts to !%1(0) and drops to a G–action on the reducedspace !%1(0)/H. Therefore, using the inverse of the presymplecticdi!eomorphism L on Y , we get an action ' on Y given by

'h([g, 1, v]) = [hg, 1, v], for any h " G and [g, 1, v] " Y.

It is straightforward to verify that the G–action on (!%1(0)/H,)0)is canonical with a momentum map J0 : !%1(0)/H % g" given bythe projection of JL, that is,

J0([g, 1, JVm(v), v]) = Ad"g"1(µ + 1+ JVm(v)).


This obviously implies that the G–action ' on (Y, !Y ) is canonicalwith momentum map

JY ([g, 1, v]) = Ad"g"1(µ + 1+ JVm(v)). !

The importance of the Hamiltonian tube Yr introduced in the pre-vious proposition is in the fact that it models the symplectic manifold(M, !) as a Hamiltonian G–space in a neighborhood of the orbit G·m.In order to prove this claim we reproduce here the approach of Batesand Lerman [BL97].

Definition 3.2.2 Let X be a submanifold of the symplectic manifold(M,!). For each m "M , we denote by Nm = TmX%/(TmX%*TmX)the symplectic normal space at m to X. If the dimension of Nm

is the same for each m " X, the resulting bundle over X is calledthe symplectic normal bundle of the embedding X )% M . Theword symplectic comes from the fact that, at each point m " X, thesymplectic form on M induces a natural symplectic structure !Nm onNm by

!Nm([v], [w]) := !(m)(v, w) for any v, w " TmX%.

Using this structure, the symplectic vector space (Nm, !Nm) is iso-morphic to a maximal symplectic subspace of (TmX%, !(m)).

Theorem 3.2.1 (Constant Rank Embedding Theorem) Let(M,!) and (M $,!$) be two symplectic manifolds. Suppose thati : N )% (M,!) and i$ : N )% (M $,!$) are two constant rankembeddings with isomorphic symplectic normal bundles, such thati"! = (i$)"!$. Then there exist neighborhoods U of i(N) in M andU $ of i$(N) in M $ and a di!eomorphism 2 : U % U $, such that2 & i = i$ and 2"!$ = !.

Furthermore, if G is a Lie group that acts properly on N , M ,and M $, preserves the forms ! and !$, and if the embeddings i and i$

are G–equivariant, the neighborhoods U and U $ can be chosen to beG–invariant and 2 to be G–equivariant.

Proof See [BL97]. !

We now introduce the main result of this section.


Theorem 3.2.2 (Marle-Guillemin-Sternberg Normal Form)Let (M,!) be a symplectic manifold and let G be a Lie group actingproperly on M in a globally Hamiltonian fashion, with associatedAd"–equivariant momentum map J : M % g". Let m " M anddenote J(m) = µ " g", H := Gm. Then the manifold

Yr := G)H (m"r ) Vm)

introduced in Proposition 3.2.1 is a Hamiltonian G–space and thereare G–invariant neighborhoods U of m in M , U $ of [e, 0, 0] in Y , andan equivariant symplectomorphism 2 : U % U $ satisfying 2(m) =[e, 0, 0] and JY & 2 = J.

Proof This is a consequence of the Constant Rank Embedding The-orem, with (M, !) and (Yr, !Yr) playing the role of the symplec-tic manifolds, and the orbit G · m 2 G )H ({0} ) {0}) playing therole of the submanifold N . We just need to show that the embed-dings iM : G · m )% M and iY : G )H ({0} ) {0}) )% Y , whichhave, by construction, constant rank, also have isomorphic symplec-tic normal bundles. The symplectic normal bundle V of the em-bedding iM : G · m )% M is constructed by taking at each pointz " G · m, the symplectic normal space at that point, that is,Vz = kerTzJ/Tz(Gµ · m). This is actually a bundle, since it is easyto see that for any g " G, the symplectic normal space at g · m isVg·m = (Tm&g(kerTmJ))/(Tm&g(Tm(Gµ · m))) 7= Vm, where &g de-notes the symplectomorphism defined by the group element g " Gacting on M .

Next, we deal with the embedding iY : G )H ({0} ){0}) )% Y . The symplectic normal space at [e, 0, 0] is V Y

[e,0,0] =kerT[e, 0, 0]JY /T[e, 0, 0](Gµ · [e, 0, 0]), which we shall now explicitlycompute. We begin by characterizing the subspace kerT[e,0,0]JY . Ifw = d

dt

$$t=0

[exp t0, t1, tv] " kerT[e,0,0]JY , then by (3.2.1) we have

d

dt

$$$$t=0

JY ([exp t0, t1, tv]) =d

dt

$$$$t=0

exp t0 · (µ + t1+ JVm(tv))

= $ad") · µ + 1+ T0JVm · v = 0.

Since JVm is homogeneous quadratic on Vm, it follows that T0JVm ·v =0 and hence the previous relation becomes $ad"

) ·µ+1 = 0. However,recalling that g = gµ ( q, noting that ad"

)µ " g#µ, the annihilator of


gµ in g", and 1 " m" ' g"µ7= q#, it follows that ad"

)µ = 0 and 1 = 0since g#µ * q# = {0} in g". Consequently,

kerT[e,0,0]JY7= gµ/h) Vm

7= Tm(Gµ · [e, 0, 0])) Vm.

Therefore, V Y[e, 0, 0] = kerT[e, 0, 0]JY /T[e, 0, 0](Gµ · [e, 0, 0]) 7= (Tm(Gµ ·

[e, 0, 0]))Vm)/Tm(Gµ ·m) 7= Vm. Thus, the symplectic normal spaceV Y

[e, 0, 0] is isomorphic to Vm. As before, the G–action guarantees thatall normal spaces at points [g.0.0] are isomorphic, which implies thatthe collection of these symplectic normal spaces forms a vector bundleV Y with base G)H ({0}) {0}). If we denote by L[g, 0, 0] : V Y

[g, 0, 0] %Vg·m the linear isomorphism between the fibers V Y

[g, 0, 0] and Vg·m, themap L defined by

L : V Y $% VvY[g, 0, 0] -$% L[g, 0, 0](vY

[g, 0, 0]),

is a vector bundle isomorphism between V Y and V . The ConstantRank Embedding Theorem guarantees the existence of U , U $, and2. The equality JY & 2 = J is a consequence of the following simplelemma. !

Lemma 3.2.2 Let (M1, !1) and (M2, !2) be two symplectic mani-folds and let G be a Lie group that acts in a globally Hamiltonianfashion on both M1 and M2. Let 2 : M1 % M2 be a symplectic G–equivariant di!eomorphism. If J1 : M1 % g" is a momentum map forthe G–action on M1, the map J2 : M2 % g" defined by J2 = J1 & 2%1

is a momentum map for the G–action on M2.

Proof The G–equivariance of 2 implies that the infinitesimal gener-ators of the G–actions on M1 and M2 are related by

-M2 = T2 & -M1 for any - " g.

We verify that J2 = J1&2%1 is a momentum map for the G–action onM2. Indeed, for any - " g, any point m2 "M2 such that m2 = 2(m1),

§ 3.3. The Reconstruction Equations 105

for some m1 "M1, and any v " Tm2M2, we have

dJ#2(m2) · v = 0Tm2J2(m2) · v, -1

= 0Tm2(J1 & 2%1)(m2) · v, -1= 0Tm1J1(Tm22

%1 · v), -1= !1(m1)(-M1(m1), Tm22

%1 · v)

= !1(2%1(m2))(Tm22%1 · -M2(m2), Tm22

%1 · v)= !2(m2)(-M2(m2), v), (since 2 is canonical)

as required. !

Remark 3.2.2 Note that since the Constant Rank Embedding The-orem only requires for its application isomorphism of symplectic nor-mal bundles, there is a freedom in the construction of the MGS nor-mal form at the time of choosing the symplectic normal space, that is,for any H–representation symplectic vector space V , H–equivariantlysymplectomorphic to (Vm, !Vm), one can construct a MGS normalform changing V by Vm, that shares the same properties that the oneintroduced in the previous theorem. "

Remark 3.2.3 The local picture provided by the MGS normal formfor a Hamiltonian G–space, should be thought of as the Darbouxcoordinates in this category. In fact, the MGS variables share withthe Darboux coordinates both its advantages and its disadvantages.On one hand, the appearance in this local model of the symplec-tic form, the G–action, and the expression of the momentum mapare extremely convenient. However, in practice, computing the G–invariant symplectomorphism 2 that links the symplectic G–tube Yr

with the manifold M on which we are working, is rarely possible.This drawback is a consequence of the non constructive nature of theproof of the Constant Rank Embedding Theorem, used to guaranteethe existence of this symplectomorphism (the arguments utilized aresimilar to the classical Moser proof of the Darboux Theorem). How-ever, even though constructing the MGS normal form in particularexamples is, in general, impossible, it is a priceless tool when provinggeneral results, as we will see in the following chapters.

There are some cases, like in bifurcation theory (see Chapter 7),where one only needs the value of the derivative of 2 at m. We willsee that it is possible to construct an isomorphism between TmM andT[e, 0, 0]Y that has the same normal properties as Tm2. "


3.3 The Reconstruction Equations

As we remarked in the previous section, the MGS normal form playsa role similar to the Darboux coordinates in the category of Hamil-tonian actions. It is in Darboux coordinates where the di!erentialequations defining a Hamiltonian vector field take the well–knowncanonical Hamiltonian form (see (1.1.1)). In this section we will de-termine the analog of Hamilton’s equations in MGS coordinates, thatis, we will take a G–invariant Hamiltonian and we will write down inMGS variables, the di!erential equations that determine its associ-ated Hamiltonian vector field.

As in the previous section, we will work with the presymplecticmanifold Y instead of the symplectic submanifold Yr, since we canalways restrict, if necessary, any statements to this open G–invariantsubmanifold. Let h " C!(Y )G be a G–invariant Hamiltonian on Y .Our aim is to compute the di!erential equations that determine theHamiltonian vector field Xh " X(Y ) associated to h and characterizedby

iXh!Y = dh. (3.3.1)

Since the projection / : G ) m" ) Vm % G )H (m" ) Vm) is a sur-jective submersion, there are always local sections available (see forinstance [BC70, Proposition 6.1.4]) that we can use to locally express

Xh = T/(XG, Xm# , XVm),

with XG, Xm# and XVm locally defined smooth maps defined on Yand having values in TG, Tm" and TVm respectively. Thus, for any[g, 1, v] " Y , one has XG([g, 1, v]) " TgG, Xm#([g, 1, v]) " T*m" =m", and XVm([g, 1, v]) " TvVm = Vm. Moreover, using the AdH–invariant decomposition of the Lie algebra g

g = h(m( q,

introduced in the previous section, the mapping XG can be written,for any [g, 1, v] " Y , as

XG([g, 1, v]) = TeLg-Xh([g, 1, v]) + Xm([g, 1, v]) + Xq([g, 1, v])

2,

with Xh, Xm, and Xq, locally defined smooth maps on Y with valuesin h, m, and q respectively. We will set up the di!erential equations


that define Xh, Xm, Xq, Xm# , and XVm . Once these functions havebeen computed, the value of Xh is uniquely determined by the rela-tion

Xh([g, 1, v]) = T(g, *, v)/-TeLg

-Xh([g, 1, v]) + Xm([g, 1, v])

+ Xq([g, 1, v])2, Xm#([g, 1, v]), XV ([g, 1, v])

2.

In order to carry this out, we will write down explicitely the expres-sion (3.3.1), using the characterization of !Y provided by the dia-gram (3.2.3). Indeed, let [g, 1, v] " Y and w = T(g, *, v)/((TeLg(vh +vm + vq), vm# , vVm) " T[g, *, v]Y arbitrary, with vh " h, vm " m, vq "q, vm# " m", and vVm " V . The value of Xh at [g, 1, v] is uniquelydetermined by the equality

!Y ([g, 1, v])(Xh([g, 1, v]), w) = dh([g, 1, v]) · w,

with w arbitrary or, equivalently,

!Y ([g, 1, v])(T(g, *, v)/(TeLg(Xh + Xm + Xq), Xm# , XVm),

T(g, *, v)/((TeLg(vh + vm + vq), vm# , vVm))

= dh([g, 1, v]) · w, (3.3.2)

where, for economy of notation, we suppressed the dependence ofXh, Xm, Xq, Xm# , and XVm on [g, 1, v]. Since, by construction (seediagram (3.2.3)), !Y = L")0, expression (3.3.2) can be rewritten as

)0((L & /)(g, 1, v))-T(g, $, v)(L & /)(TeLg(Xh + Xm + Xq), Xm# , XVm),

T(g, *, v)(L & /)(TeLg(vh + vm + vq), vm# , vVm)2

= )0((/0 & l)(g, 1, v))-T(g, *, v)(/0 & l)(TeLg(Xh + Xm + Xq),

Xm# , XVm), T(g, *, v)(/0 & l)(TeLg(vh + vm + vq), vm# , vVm)2

= )0(/0(g, 1, JVm(v), v))(T(g, *,JVm (v), v)/0(TeLg(Xh + Xm + Xq),

Xm# , TvJVm · XVm , XVm),T(g, *,JVm (v), v)/0(TeLg(vh + vm + vq), vm# , TvJVm · vVm , vVm))

= )(g, 1, JVm(v))((TeLg(Xh + Xm + Xq), Xm# , TvJVm · XVm),(TeLg(vh + vm + vq), vm# , TvJVm · vVm)) + !Vm(XVm , vVm)

= dh([g, 1, v]) · w. (3.3.3)

Note that since h " C!(G)H (m") Vm))G is G–invariant, the map-ping h &/ " C!(G)m")Vm)H can be understood as a H–invariant


function that depends only on the m" and Vm variables, that is,

h & / " C!(m" ) Vm)H .

This implies that dh([g, 1, v]) · w can be written as

dh([g, 1, v]) · w = d(h & /)(g, 1, v)((TeLg(vh + vm + vq), vm# , vVm)= Dm#(h & /)(g, 1, v) · vm# + DVm(h & /)(g, 1, v) · vVm ,

where Dm#(h &/)(g, 1, v) " (m")" = m and DVm(h &/)(g, 1, v) " V "m

are the partial derivatives of h & / with respect to the m" and Vm

variables, respectively. Using this remark and the expression for ) =!1 + !µ described in the previous section we can write (3.3.3) as

Dm#(h & /)(g, 1, v) · vm# + DVm(h & /)(g, 1, v) · vVm

= 0vm# , Xm1+ 0TvJVm · vVm , Xh1$ 0Xm# , vm1 $ 0TvJVm · XVm , vh1+ 01+ JVm(v), [Xh + Xm + Xq, vh + vm + vq]1+ 0Ad"

g"1µ, [Xh + Xm + Xq, vh + vm + vq]1+ !Vm(XVm , vVm). (3.3.4)

Using the explicit expression of JVm in terms of !Vm , we computeTvJVm · vVm . For any - " h,

0TvJVm · vVm , -1 =d

dt

$$$$t=0

0JVm(v + tvVm), -1

=d

dt

$$$$t=0

12!Vm(-Vm(v + tvVm), v + tvVm)

=12-!Vm(-Vm(vVm), v) + !Vm(-Vm(v), vVm)

2

= !V (-Vm(v), vVm),

since the h–representation on Vm is symplectic. In particular, if wetake Xh " h, we obtain

0TvJVm · vVm , Xh1 = !Vm((Xh)Vm(v), vVm),


which substituted into (3.3.4) yields

Dm#(h & /)(g, 1, v) · vm# + DVm(h & /)(g, 1, v) · vVm

= 0vm# , Xm1+ !Vm((Xh)Vm(v), vVm)$ 0Xm# , vm1 $ 0TvJVm · XVm , vh1+ 01+ JVm(v), [Xh + Xm + Xq, vh + vm + vq]1+ 0Ad"

g"1µ, [Xh + Xm + Xq, vh + vm + vq]1+ !Vm(XVm , vVm). (3.3.5)

Since this equality holds for arbitrary vh " h, vm " m, vq " q, vm# "m", and vVm " Vm, (3.3.5) separates into the following five di!erentialequations all evaluated at an arbitrary point [g, 1, v] for g " G, 1 " m",and v " Vm

Xm = Dm#(h & /) (3.3.6)iXVm+(Xh)Vm

!Vm = DVm(h & /) (3.3.7)

Pq#

3ad"

(Xm+Xh+Xq)(1+ JVm(v) + Ad"g"1µ)

4= 0 (3.3.8)

Pm#

3ad"


4= Xm# (3.3.9)

Ph#

3ad"


4= TvJVm · XVm ,

(3.3.10)

where Ph# , Pm# , and Pq# denote the projections in g" onto h", m",and q", respectively, according to the Ad"

H–invariant splitting

g" = h" (m" ( q".

These expressions are called the reconstruction equations at thepoint [g, 1, v]. They completely characterize, in MGS coordinates,the Hamiltonian vector field associated to the G–invariant Hamilto-nian h " C!(Y )G.

As a first illustration on how to handle these equations, and alsoas a self–consistency test on them, we verify, just by making somestraightforward calculations, that Noether’s Theorem holds at in-finitesimal level. Recall that this result implies that the Hamiltonianflow associated to a G–invariant Hamiltonian preserves the level setsof the momentum map, that is, in our case

JY & Ft = JY , for any time t,


where Ft is the flow of the vector field Xh. Di!erentiating withrespect to the time t and evaluating this expression at the pointm := [e, 0, 0] " Y , we obtain that

TmJY · Xh(m) = 0.

We verify that Xh, as provided by the reconstruction equations, sat-isfies this condition. Indeed, at the point m, equation (3.3.8) reducesto

Pq#

3ad"

(Xm(m)+Xh(m)+Xq(m))µ4

= 0,

which implies that ad"(Xm(m)+Xh(m)+Xq(m))µ " g"µ, and therefore, for

any - " gµ:

0ad"(Xm(m)+Xh(m)+Xq(m))µ, -1 = 0µ, ad(Xm(m)+Xh(m)+Xq(m))-1

= 0$ad"#µ, Xm(m) + Xh(m) + Xq(m)1

= 0.

Since - " gµ is arbitrary, it follows that ad"(Xm(m)+Xh(m)+Xq(m))µ = 0

and therefore Xm(m)+Xh(m)+Xq(m) " gµ. Consequently Xq(m) =0. By substitution of ad"

(Xm(m)+Xh(m)+Xq(m))µ = 0 in (3.3.9)and (3.3.10), one obtains that Xm#(m) = 0 and TvJVm ·XVm(m) = 0.Now,

TmJY · Xh(m) = TmJY (T(e, 0, 0)/(Xm(m) + Xh(m), 0, XVm(m)))

=d

dt

$$$$t=0

JY ([exp t(Xm(m) + Xh(m)), 0, tXVm(m)])

=d

dt

$$$$t=0

Ad"(exp t(Xm(m)+Xh(m)))"1(µ + JVm(tXVm(m)))

= $ad"Xm(m)+Xh(m)µ + TvJVm · XVm(m) = 0,

as required.

For future reference, we now describe how the reconstructionequations look like in the case in which the G–action is free. Sincein this situation the isotropy subgroup of any element is trivial, wehave that H = {e}, and therefore h = {0}. This implies that themomentum map JVm is not present and that m = gµ. The symplectictube Y reduces then to:

Y = G) g"µ ) Vm.

§ 3.4. The MGS Normal Form and Reduction 111

Consequently, the reconstruction equations at the point (g, 1, v) "G) g"µ ) Vm are, in this case:

Xgµ = Dg#µh (3.3.11)

iXVm!Vm = DVmh (3.3.12)

Pq#

3ad"

(Xgµ+Xq)(1+ Ad"g"1µ)

4= 0 (3.3.13)

Pg#µ

3ad"

(Xgµ+Xq)(1+ Ad"g"1µ)

4= Xg#µ . (3.3.14)

3.4 The MGS Normal Form and Reduction

The approach to singular reduction presented in Chapter 2 hasn’tmuch to do with how historically those concepts were developed. Thepoint of view that we took was based on the construction of someglobal models for the singular reduced spaces on which we studiedtheir structure. However, the way in which these singular spaceswere treated when they were first introduced by Sjamaar and Ler-man [SL91], in the case of compact Lie group actions, and by Batesand Lerman [BL97], in case of proper actions, was using the MGSnormal form. The use of normal forms produces a local picture ofthese spaces where it is very easy to visualize their structure. Forinstance, in Remark 2.4.2 we mentioned that the singular reducedspaces Mµ and MOµ have the structure of a stratified space; it is theMGS normal form that provides the necessary tools to prove thatstatement (see [SL91]). This section is the first illustration of thepower and the technical convenience of these normal forms. In fu-ture sections we will see how these advantages extend to the study ofstability, persistence, and bifurcation problems.

The first step to put singular reduction in the framework of normalforms is the following key result, due to Bates and Lerman [BL97].

Proposition 3.4.1 Let (Yr, !Yr) be the Hamiltonian G–tube de-scribed in Proposition 3.2.1. Then, for a small enough open neigh-borhood Y0 ' Yr of the orbit G · [e, 0, 0] one has:

(i) J%1Yr

(µ) * Y0 = {[g, 1, v] " Y0 | g " Gµ, 1 = 0 and JVm(v) = 0}.

(ii) If the coadjoint orbit Oµ is locally closed, then

J%1Yr

(Oµ) * Y0 = {[g, 1, v] " Y0 | 1 = 0 and JVm(v) = 0}.


Proof As usual, it is enough to work with the presymplectic manifoldY and the momentum map JY , making sure that at the end oneguarantees that the open neighborhood is shrunk so that it lies in Yr.

(i) We write the momentum map JY = 3 & b where

b : G)H (m" ) Vm) $% G)H g"µ[g, 1, v] -$% [g, 1+ JV (v)],

3 : G)H g"µ $% g"

[g, ,] -$% Ad"g"1(µ + ,).

We will show that the mapping 3 is a submersion at each point ofthe form [g, 0] " G)H g"µ. to do this, recall that g = gµ(q, the directsum being orthogonal relative to an H–invariant inner product on g.Applying Adg to both sides of the equation and recalling that gg·µ =Adg(gµ) we conclude that we have an orthogonal decomposition interms of annihilators g" = g#g·µ ( q#, where q = Adg(q), g#g·µ

7= q",and q# 7= g"g·µ. An arbitrary vector v[g, 0] " T[g, 0](G )H g"µ) can bewritten as v[g, 0] = d

dt

$$t=0

[(exp t-) g, t,], with - " g and , " q# 7= g"µ.Using the Leibniz rule we compute

T[g, 0]3 · v[g, 0] =d

dt

$$$$t=0

Ad"(exp t# g)"1(µ + t,) = $ad"

#(g · µ) + g · ,.

Since ad"#(g · µ) " Tg·µOµ = g#g·µ and g · , " g · q# = q# 7= g"g·µ, it is

clear that T[g, 0]3 maps onto g#g·µ( g"g·µ = g". Thus 3 is submersive atall points [g, 0].

Since submersivity is an open condition (see [AMR, Theorem3.5.2]), the mapping 3 is a submersion in an open neighborhood U ofthe base G · [e, 0] of the bundle G)H g"µ. The Submersion Theorem(see [AMR, Theorem 3.5.4]) guarantees that 3%1(µ)*U is a subman-ifold of G)H g"µ of dimension dimGµ$dimH. At the same time, thesubmanifold Gµ)H {0} of G)H g"µ is included in 3%1(µ)*U and hasalso dimension dimGµ $ dimH, therefore it is an open submanifoldin it, which allows us to choose an open subset V ' U in G )H g"µsuch that 3%1(µ)*V = Gµ)H {0}. We take Y0 := b%1(V ). With thischoice for the open set Y0, the claim of the Proposition is satisfied.Indeed:

J%1Y (µ) * Y0 = b%1(3%1(µ)) * b%1(V ) = b%1(Gµ )H {0})

= {[g, 1, v] " Y0 | g " Gµ, 1 = 0 and JVm(v) = 0}.


(ii) The proof is identical to part (i). The local closedness on thecoadjoint orbit Oµ is required so that 3%1(Oµ) is a submanifold ofG)H g"µ (see the proof in [BL97]). !

Remark 3.4.1 Notice that the neighborhood Y0 needs not be G–invariant, however, when the Lie group G is compact, Corollary 3.1.1implies that Y0 can always be chosen with that property. "

Proposition 3.4.2 Let Y0 be as in Proposition 3.4.1. The followingidentities hold:

(i) YH = N(H))H (m" ) Vm)H = N(H))H (m" ) Vm)H = Y H ,

(ii) Y(H) = G)H (m" ) Vm)H = G)H (m" ) Vm)H ,

(iii) J%1Y (µ) * YH / {[g, 0, v] | g " NGµ(H), v " V H

m }, andJ%1

Y (µ) * YH * Y0 = {[g, 0, v] | g " NGµ(H), v " V Hm }

(iv) J%1Y (µ) * Y

Gµ

(H) / {[g, 0, v] | g " Gµ, v " V Hm }, and

J%1Y (µ) * Y

Gµ

(H) * Y0 = {[g, 0, v] | g " Gµ, v " V Hm }

(v) If the coadjoint orbit Oµ is locally closed in g", thenJ%1

Y (Oµ) * Y(H) / {[g, 0, v] | g " G, v " V Hm } and

J%1Y (Oµ) * Y(H) * Y0 = {[g, 0, v] | g " G, v " V H

m }.

Proof We first prove the following lemma.

Lemma 3.4.1 Let n = [g, 1, v] " Y . Then, Gn = gH(*, v)g%1.

Proof Let k " Gn. By definition, k · n = n or, equivalently,[kg, 1, v] = [g, 1, v], which implies that there exists an elementh " H such that kg = gh and (1, v) = h%1 · (1, v). The last equal-ity guarantees that h " H(*, v) and, since k = ghg%1, we have thatk " gH(*, v)g

%1. Conversely, let ghg%1 " gH(*, v)g%1. By definition,

ghg%1 · n = ghg%1 · [g, 1, v] = [gh, 1, v]

= [g, h%1 · 1, h%1 · v]

= [g, 1, v] = n, since h%1 " H(*, v). #

We now proceed to the proof of the Proposition. Recall that sincethe action is proper and H is an isotropy subgroup, H is necessarilycompact.


(i) If n = [g, 1, v] " YH , by the previous lemma we have

H = Gn = gH(*, v)g%1.

Since H(*, v) ' H, we have that H = gH(*, v)g%1 ' gHg%1. The

compactness of H and Lemma 1.3.4 imply that H = gHg%1 =gH(*, v)g

%1, which guarantees that g " N(H) and that H = H(*, v).Hence (1, v) " (m" ) Vm)H ' (m" ) Vm)H , thereby showing thatYH ' N(H))H (m" ) Vm)H ' N(H))H (m" ) Vm)H .

Conversely, if n = [g, 1, v] " N(H) )H (m" ) V )H , then H 'H(*, v) and g " N(H). However, since we always have H(*, v) 'H, it follows that H(*, v) = H. Hence, by the above lemma, Gn =gH(*, v)g

%1 = gHg%1 = H. This shows that N(H))H (m" ) V )H 'YH , thus proving the equalities

YH = N(H))H (m" ) Vm)H = N(H))H (m" ) Vm)H .

We now prove that Y H = N(H) )H (m" ) V )H . One inclusionis clear since N(H) )H (m" ) V )H = YH ' Y H . Conversely, ifn = [g, 1, v] " Y H we have that for any h " H, h · n = n or[hg, 1, v] = [g, 1, v]. This implies the existence of some h$ " H suchthat hg = gh$, that is, g%1hg " H. Since h is arbitrary we havethat g%1Hg ' H and, by Lemma 1.3.4 we obtain that g%1Hg = Hand hence g " N(H). Let now h " H be arbitrary and considerthe element gh%1g%1 " H (since g " N(H)). Since, by hypothesis,gh%1g%1 · n = n we have that

[g, 1, v] = [gh%1g%1g, 1, v] = [gh%1, 1, v] = [g, h1, hv],

which implies that h · (1, v) = (1, v) for all h " H, that is, (1, v) "(m" ) V )H .(ii) follows from part (i), by taking into account that Y(H) = G ·YH .(iii) It is easy to show, just by using the definition of the H–momentum map JVm in terms of the symplectic form !Vm , thatJVm(v) = 0 for any v " V H

m . Let n = [g, 0, v] be such thatg " NGµ(H) and v " V H

m . Then JY (n) = g · (µ + JVm(v)) = µ, thatis, n " J%1

Y (µ). Now we show that also n " YH . Since the isotropysubgroup Hv is a subgroup of H, we have that Gn = gH(0,v)g

%1 =gHvg%1 ' gHg%1 = H, because g " N(H). Also, for any h " H, wehave g%1hg " H (again because g " N(H)). Consequently,

h · [g, 0, v] = [hg, 0, v] = [g(g%1hg), 0, v]

= [g, 0, (g%1hg)%1 · v] = [g, 0, v],


since. by hypothesis, v " V Hm . Therefore H ' Gn and so we have

Gn = H and hence n " YH . This proves the inclusion {[g, 0, v] | g "NGµ(H), v " V H

m } ' J%1Y (µ) * YH .

To prove the second equality, note that by Proposition 3.4.1{[g, 0, v] | g " NGµ(H), v " V H

m } ' J%1Y (µ)*Y0, which, together with

the inclusion just shown, proves {[g, 0, v] | g " NGµ(H), v " V Hm } '

J%1Y (µ) * YH * Y0. Conversely, let n = [g, 1, v] " J%1

Y (µ) * YH * Y0.By Proposition 3.4.1, we have g " Gµ, 1 = 0, and v " J%1

Vm(0). In

addition, by part (i), g " N(H) and v " V Hm , which shows that

J%1Y (µ) * YH * Y0 ' {[g, 0, v] | g " NGµ(H), v " V H

m }.(iv) If n = [g, 0, v] with g " Gµ and v " V H

m , then JY (n) = g ·(µ + JVm(v)) = µ, so n " J%1

Y (µ). In addition, by Lemma 3.4.1,Gn = gH(0,v)g

%1 = gHvg%1 ' gHg%1. Conversely, if h " H,

ghg%1 · n = ghg%1 · [g, 0, v] = [gh, 0, v]

= [g, 0, h%1 · v] = [g, 0, v] = n,

since v " V Hm . Thus, gHg%1 ' Gn and hence Gn = gHg%1 with

g " Gµ, by hypothesis. This shows that n " YGµ

(H) and the firstinclusion is proved.

By Proposition 3.4.1, {[g, 0, v] | g " Gµ, v " V Hm } ' J%1

Y (µ) * Y0,and hence, by the inclusion just proved, we have, {[g, 0, v] | g "Gµ, v " V H

m } ' J%1Y (µ) * Y

Gµ

(H) * Y0. Conversely, if n = [g, 1, v] "J%1

Y (µ) * YGµ

(H) * Y0, we have g " Gµ, 1 = 0, v " J%1Vm

(0), by Proposi-tion 3.4.1, and Gn = kHk%1 for some k " Gµ. For h " H arbitrary,we therefore have khk%1 · n = n, that is, [khk%1g, 0, v] = [g, 0, v].Thus, there is some h$ " H such that khk%1g(h$)%1 = g and h$ ·v = v.In particular, this determines h$ uniquely as h$ = (g%1k)h(g%1k)%1 "H. This shows that (g%1k)H(g%1k)%1 ' H, so that by Lemma 1.3.4,(g%1k)H(g%1k)%1 = H. Therefore, g%1k " N(H). Since h, h$ " Hmutually determine each other by the equality h$ = (g%1k)h(g%1k)%1,it follows that h$ " H is also arbitrary. Since h$ · v = v this meansthat v " V H

m which shows that n " {[g, 0, v] | g " Gµ, v " V Hm },

thereby proving the stated equality.(v) The proof of this is identical to that of (iv), with the only ex-ception that one uses the second identity in Proposition 3.4.1. !

Using the technical convenience of these local models we nowprove some results that were of much use in the previous chapters.


The following computations are the first illustration of the use of thenormal forms to prove general facts about Hamiltonian G–spaces.The procedure is always the same: given a point in a symplecticmanifold where there is a Hamiltonian G–action, according to The-orem 3.2.2, we can always substitute, in a neighborhood of the orbitof the given point, the original space by the symplectic tube providedby the MGS normal form. The nature of the constructions we useimpose two restrictions. First, the nature of the claims that one canmake is local, that is, only valid in a G–invariant neighborhood of agiven orbit. Second, according to Remark 3.4.1, if one uses the mod-els provided by Proposition 3.4.2, even the G–invariance needs to bequestioned. Finally, let us again emphasize that the implicit natureof the symplectomorphism that between the original space and theMGS model allows us to prove only purely abstract results.

Proposition 3.4.3 Let (M, !) be a symplectic manifold and let Gbe a Lie group acting properly on M in a globally Hamiltonian fashionwith associated equivariant momentum map J : M % g". Let m "Mand denote J(m) = µ, H := Gm. Then every f " C!(J%1(µ))Gµ

(respectively f " C!(J%1(µ) *MGµ

(H))Gµ) admits a local G–invariant

extension at m to C!(M)G.

Proof By Proposition 3.4.1 (respectively Proposition 3.4.2), the sub-set J%1(µ) can be locally modeled as J%1(µ) 2 Gµ)H ({0})J%1

Vm(0))

(respectively J%1(µ) *MGµ

(H) 2 Gµ )H ({0}) V Hm )). If we denote by

/ the canonical projection / : G ) m" ) Vm % G )H (m" ) Vm),the mapping f & /|Gµ){0})J"1

Vm(0) " C!(Gµ ) {0} ) J%1

Vm(0))H is

a H–invariant function that depends only on the J%1Vm

(0) factor,that is, F $ := f & /|Gµ){0})J"1

Vm(0) " C!(J%1

Vm(0))H (respectively

f & /|Gµ){0})V Hm" C!(Gµ ) {0} ) V H

m )H is a H–invariant functionthat depends only on the V H

m factor, that is, F” := f &/|Gµ){0})V Hm"

C!(V Hm )H = C!(V H

m )). By Proposition 1.3.1, F $ (respectively F”)can be locally extended to F " C!(Vm)H . The required local exten-sion for f at m is the function f " C!(M)G defined by

f([g, 1, v]) = F (v). !

Proposition 3.4.4 Let (M, !) be a symplectic manifold and let Gbe a Lie group acting properly on M in a globally Hamiltonian fash-ion with associated equivariant momentum map J : M % g". Let


m " M and denote J(m) = µ, H := Gm. Then every functionf " C!(MH)N(H) = C!(MH)L admits a local extension at m toC!(M)G.

Proof By Proposition 3.4.2, the submanifold MH can be locally mod-eled by

MH 2 N(H))H (m" ) Vm)H .

If we denote by / : N(H) ) (m" ) Vm)H % N(H) )H (m" )Vm)H the canonical projection, the mapping f " C!(MH)N(H) 2C!(N(H))H (m")Vm)H)N(H) is such that the composition f &/ isa function in C!((m" ) Vm)H). Since (m" ) Vm)H is a H–invariantvector subspace of m" ) Vm, Proposition 1.3.1 guarantees that thereis a H–invariant extension F " C!(m" ) Vm)H of f & / to m" ) Vm.The mapping F " C!(M)G defined by

F ([g, 1, v]) = F (1, v)

is the local extension required. !

Proposition 3.4.5 In the hypotheses of the previous proposition,let m " J%1(µ) * MH ' J%1(µ) * M

Gµ

(H). Then every vector

v " Tm-J%1(µ) *M

Gµ

(H)

2, can be written as

v = -M (m) + v$,

with -M (m) " Tm(Gµ · m) and v$ " Tm(J%1(µ) *MH).

Proof By Proposition 3.4.2, the submanifold J%1(µ) *MGµ

(H) ' M

can be locally modeled around the point m , [e, 0, 0] by

J%1(µ) *MGµ

(H) 2 Gµ )H ({0}) V Hm ) 2 Gµ )H V H

m .

Let / : Gµ ) V Hm % Gµ )H V H

m be the canonical projection. Since /is a submersion, the vector v can be written as

v = T(e, 0)/(-, v$) = T(e, 0)/(-, 0) + T(e, 0)/(0, v$)

, -M (m) + T(e, 0)/(0, v$),


for some - " gµ and v$ " V Hm . Since T(e, 0)/(0, v$) " T(e, 0)/({0} )

V Hm ) ' T(e, 0)/(Lie(NGµ(H)) ) V H

m ) 2 Tm(J%1(µ) *MH), the resultfollows. !

As an example of the combined power of the MGS normalform with the reconstruction equations we now give a proof ofLemma 1.5.2, whose statement was:

Lemma 3.4.2 Let (M, !) be a symplectic manifold and let G be aLie group acting freely, properly, and canonically on M with associ-ated equivariant momentum map J : M % g". Let µ " g" be a regularvalue of J so that the level set J%1(µ) is a submanifold of M . Then,for any m " J%1(µ), every vector v " TmJ%1(µ) can be written asTmiµ(v) = Xf (m), where f " C!(M)G, and iµ : J%1(µ) )%M is thecanonical injection. That is, if we omit iµ, we can write

TmJ%1(µ) = {Xf (m) | f " C!(M)G}.

Proof The inclusion {Xf (m) | f " C!(M)G} ' TmJ%1(µ) is astraightforward consequence of the Noether Theorem. Conversely,recall that in the free case the symplectic tube Y provided by theMGS normal form coincides with

Y = G) g"µ ) Vm.

By Proposition 3.4.1, the submanifold J%1(µ) can be locally modeledaround the point m , (e, 0, 0) by

J%1(µ) 2 Gµ ) {0}) Vm.

Therefore, a vector u " TmJ%1(µ) is represented in the normal formas u 7= (-, 0, v), with - " g"µ and v " Vm. We need to find a certainf " C!(G)g"µ)Vm)G such that u = Xf (m) , Xf (e, 0, 0) = (-, 0, v).Let f " C!(g"µ ) Vm) such that Dg#µf(0, 0) = - and DVmf(0, 0) =!Vm(v, ·). By construction, the function f considered as defined onG)g"µ)Vm is G–invariant and, in view of the reconstruction equationsin the free case (3.3.11)–(3.3.14),

Xf (e, 0, 0) = (-, Xg#µ(e, 0, 0), v).

We finish the proof by showing that Xg#µ(e, 0, 0) = 0. Indeed, thereconstruction equations (3.3.13) and (3.3.14) evaluated at (e, 0, 0)


yield

Pq#

3ad"

(Xgµ (e, 0, 0)+Xq(e, 0, 0))µ4

= 0

Pg#µ

3ad"

(Xgµ (e, 0, 0)+Xq(e, 0, 0))µ4

= Xg#µ(e, 0, 0).

The first equation implies that ad"(Xgµ (e, 0, 0)+Xq(e, 0, 0))µ " g"µ. There-

fore, the projection on the left hand side of the second equation isnot needed and we have for any - " gµ,

0Xg#µ(e, 0, 0), -1 = 0ad"(Xgµ (e, 0, 0)+Xq(e, 0, 0))µ, -1

= $0ad"#µ, Xgµ(e, 0, 0) + Xq(e, 0, 0)1 = 0

Since - " gµ is arbitrary, this implies that Xgµ((e, 0, 0) = 0. !

Remark 3.4.2 A reasonable question to ask is if the previous lemmageneralizes to the singular case, that is, given a point m in a globallyHamiltonian system (M, !, G, J : M % g") such that the isotropyH := Gm is nontrivial, can we write every vector in kerTmJ as theevaluation at m of the Hamiltonian vector field corresponding to aG–invariant Hamiltonian? After what we saw for the regular caseand the remarks made after the introduction of the reconstructionequations, where we verified that the converse was true, in agreementwith Noether’s Theorem, an a"rmative answer is tempting. However,the answer is negative. If we try a proof similar to the regular one, wewill end up needing a H–invariant function h " C!(m" ) V )H withprescribed values for its derivative at (0, 0), which may be impossibleto do. This was not a problem in the regular case because there wereno invariance requirements.

Indeed, a simple counterexample may be constructed. Considerthe canonical lifted action of the Lie group S1 on the symplectic man-ifold (T "R2, !). This action is globally Hamiltonian with equivariantmomentum map given by the well–known two–dimensional angularmomentum:

J : T "R2 $% (Lie(S1))" 2 R(q1, q2, p1, p2) -$% q1p2 $ q2p1.

It is easy to see that ker T(0, 0, 0, 0)J = T(0, 0, 0, 0)(T "R2). At the sametime, the invariant theory, in particular a theorem of Weyl [We46]


combined with a result of Schwarz [Sch74], states that every f "C!(T "R2)S1 is locally a function of the following four invariants

/1 = (q1)2 + (q2)2 /2 = p21 + p2

2

/3 = q1p1 + q2p2 /4 = q1p2 $ q2p1,

that is, any f " C!(T "R2)S1 is such that f , f(/1, /2, /3, /4). Thisimplies that

df(0, 0, 0, 0) =#f

#/1d/1(0) + . . . +

#f

#/4d/4(0) = (0, 0, 0, 0),

since d/1(0) = d/2(0) = d/3(0) = d/4(0) = (0, 0, 0, 0). Conse-quently, Xf (0, 0, 0, 0) = (0, 0, 0, 0), for any f " C!(T "R2)S1 . "

As we saw in the previous remark, in the singular case, the vec-tor space ker TmJ does not coincide in general with {Xf (m) | f "C!(M)G}. Nevertheless, the following proposition describes this lat-ter space explicitely.

Proposition 3.4.6 Let (M, !) be a symplectic manifold and let Gbe a Lie group acting properly on M in a globally Hamiltonian fashionwith associated equivariant momentum map J : M % g". Let m "Mand denote J(m) = µ, H := Gm. Then,

{Xf (m) | f " C!(M)G} = (Tm(L · m))%(m)|MH = kerTmJL.

Proof The last equality is a straightforward consequence of the Re-duction Lemma applied to the globally Hamiltonian action of L onMH . Let now f " C!(M)G. Noether’s Theorem guarantees thatXf (m) " kerTmJ. The flow Ft of the Hamiltonian vector field Xf

commutes with the G–action, which implies that Xf (m) " TmMH .Hence,

Xf (m) " ker TmJ * TmMH = Tm(J%1(µ) *MH)

= Tm(J%1L (0)), by the proof of Theorem 2.3.1

= (Tm(L · m))%(m)|MH , by the Reduction Lemma.

Conversely, since the action of L on MH is free, by Lemma 3.4.2 wehave that:

(Tm(L · m))%(m)|MH = Tm(J%1L (0)) = {Xg(m) | g " C!(MH)L}.


However, Proposition 3.4.4 guarantees that every g " C!(MH)L =C!(MH)N(H) admits a local G–invariant extension at m toC!(M)G. Therefore,

{Xf (m) | f " C!(M)G} = {Xg(m) | g " C!(MH)L},

which concludes the proof. !We now tackle the problem of expressing the various approaches

to singular reduction pointed out in the previous chapter in terms ofthe normal form. We start with Poisson reduction.

Theorem 3.4.1 Let (M, !) be a symplectic manifold, let m "M , and let G be a Lie group acting canonically and properly onM . Then, using the notation of Theorem 3.2.2, there is a neigh-borhood of [m] " M/G such that the restriction of the Pois-son algebra (C!(M/G), {·, ·}M/G) described in Theorem 2.2.1 tothis neighborhood is isomorphic to the Poisson algebra (C!((m" )V )/H), {·, ·}(m#)V )/H).

Suppose, additionally, that the G–action on M is globally Hamil-tonian, with associated equivariant momentum map J : M % g", andthat µ " g" is a value of J. The Poisson algebra (C!(Mµ), {·, ·}Mµ)introduced in Theorem 2.2.2 is locally Poisson isomorphic to the Pois-son algebra (C!((Vm)0), {·, ·}(Vm)0), with (Vm)0 = J%1

Vm(0)/H, the

reduced space at 0 " h" with respect to the momentum map associ-ated to the Hamiltonian H–action on Vm. Moreover, if the coadjointorbit Oµ is locally closed, the Poisson algebra (C!(MOµ), {·, ·}MOµ

)described in Theorem 2.2.3 is also locally Poisson isomorphic to thePoisson algebra (C!((Vm)0), {·, ·}(Vm)0).

Proof The claim on (C!(M/G), {·, ·}M/G) is a direct consequenceof the MGS Normal Form Theorem. Since locally (M, !) can beidentified with (Y, !Y ), the same thing holds for the quotients M/Gand Y/G along with the Poisson structures defined on them via The-orem 2.2.1. Since Y/G = (G)H (m" ) Vm))/G 2 (m" ) Vm)/H, theresult follows.

Regarding (C!(Mµ), {·, ·}Mµ), recall that by Proposition 3.4.1,the subset J%1(µ) can be identified with Gµ )H J%1

Vm(0) and, hence

J%1(µ)/Gµ 2 (Gµ)H J%1Vm

(0))/Gµ 2 J%1Vm

(0)/H = (Vm)0. Recall that(Vm)0 carries a Poisson bracket {·, ·}(Vm)0 defined via Theorem 2.2.2.If the coadjoint orbit Oµ is locally closed, we can use Proposition 3.4.1


and follow the same steps with respect to (C!(MOµ), {·, ·}MOµ). In

this case, J%1(Oµ) can be identified with G )H J%1V (0) and, hence

J%1(Oµ)/G 2 (G)H J%1Vm

(0))/G 2 J%1Vm

(0)/H = (Vm)0. !

Remark 3.4.3 Note that the previous theorem gives us for free thePoisson isomorphism Lµ that links point and orbit reduction, sincewe have shown that the two reductions yield (C!((Vm)0), {·, ·}(Vm)0)."

Remark 3.4.4 A consequence of the previous result with importantpractical implications is that reduction by Lie groups acting on man-ifolds is reduced to the case of reduction by linear actions of compactgroups on vector spaces (recall that the H–action on Vm is actuallya linear representation). This opens the door to the utilization ofinvariant theory (see Remark 2.2.2) in the actual computation of thereduced spaces. "

Proposition 3.4.7 In the hypotheses of Theorem 3.4.1, the level setJ%1(µ) and the reduced spaces Mµ and MOµ are locally path con-nected.

Proof In Proposition 3.4.1, we saw that J%1(µ) can be locallyidentified with Gµ )H J%1

Vm(0). The Lie group Gµ is a manifold

and therefore locally path connected. Also, JVm is by construc-tion a quadratic map, hence the level set J%1

Vm(0) is a cone and

therefore locally path connected, which makes the Cartesian prod-uct Gµ ) J%1

Vm(0) locally path connected. Since the canonical projec-

tion Gµ ) J%1Vm

(0) % Gµ )H J%1Vm

(0) is a continuous open map, thespace J%1(µ) 2 Gµ )H J%1

Vm(0) is locally path connected. Regarding

the reduced spaces, they can be identified with (Vm)0 = J%1Vm

(0)/H.Again, the cone J%1

Vm(0) is locally path connected. The result fol-

lows from the continuity and openness of the canonical projection/ : J%1

Vm(0)% J%1

Vm(0)/H. !

Remark 3.4.5 The previous result is related to the quadratic natureof the singularities of the momentum map associated to a Hamilto-nian proper action. This remarkable property was first noticed byArms, Marsden, and Moncrief in [AMM81], where they used an ap-proach not based on normal forms but on the so called Kuranishi


map, which presents the advantage of admitting an infinite dimen-sional generalization. "

We now express the symplectic reduced spaces in MGS normalform.


be a value of J, and let m " M be such that J(m) = µ. We denoteby Gµ the isotropy of µ under the coadjoint action of G on g" anddenote H := Gm. Then, if the coadjoint orbit Oµ is locally closed,the reduced spaces (M (H)

µ , !(H)µ ) and (M (H)

Oµ, !(H)

Oµ) introduced in the-

orems 2.4.1 and 2.4.2, respectively, are locally symplectomorphic to(V H , !Vm |V H

m), where Vm = kerTmJ/Tm(Gµ · m) is the symplectic

normal space at m.

Proof Recall that since Vm is naturally symplectic by Lemma 3.2.1,Proposition 1.3.2 guarantees that V H

m is too. The proof of the The-orem is provided by Proposition 3.4.2. For instance, in the case of(M (H)

µ , !(H)µ ), the set M (H)

µ = (J%1(µ) * MGµ

(H))/Gµ can be locallymodeled by

M (H)µ 2 (J%1(µ) *M

Gµ

(H))/Gµ 2 (Gµ )H V Hm )/Gµ 2 V H

m /H = V Hm ,

which proves the claim. In the case of (M (H)Oµ

, !(H)Oµ

) the proof isidentical. The local closedness of Oµ in the hypothesis guaranteesthat the hypotheses of Proposition 3.4.2 are satisfied and therefore alocal model is available. !

Remark 3.4.6 Similarly to what happened in Theorem 3.4.1 withthe Poisson case, the use of normal forms provides a very short proofof the equivalence of the point and orbit reduction methods (that is,the existence of the symplectomorphism L(H)

µ ), since, according to theprevious result, (M (H)

µ , !(H)µ ) and (M (H)

Oµ, !(H)

Oµ) are both locally sym-

plectomorphic to the same symplectic vector space (V Hm , !Vm |V H

m). "

Remark 3.4.7 Notice that a presentation of singular reductionbased on global models, as the one on Chapter 2, presents the advan-tage of not only allowing the possibility of explicitely writing down


all the structures involved, but of also being less demanding tech-nically. For instance, the local closedness of the coadjoint orbits isnever present in the global approach. "

Chapter 4

Persistence andSmoothness of RelativeCritical Elements

Capıtulo XXVIII: De cosas que dice Benengelique las sabra quien le leyere, si las lee con atencion.Cervantes, Don Quijote de la Mancha, II

In this chapter we present the first application of the results pre-viously introduced on singular reduction and normal forms.

The persistence problem of relative critical elements consists ofstudying under what conditions a given relative critical element ofa Hamiltonian system with symmetry is surrounded by, or persiststo, other relative critical elements in a nearby region of the phasespace. In our discussion we will focus on a series of results introducedby J. Montaldi [MO97, MO97a] and G. Patrick [Pat95] in which,provided that the relative critical element in question satisfies certainnon degeneracy conditions and has trivial symmetry, one can give alower bound for the number of persisted elements. Using singularreduction, these results have been extended [OR97] to the generalcase in which arbitrary isotropies are present. We will describe indetail all these results.

We will make a distinction between persistence and bifurcation.The di!erence between these two close terms is that in the former,

126 Chapter 4. Persistence of Relative Critical Elements

the surrounding or persisted relative critical elements that we willstudy, posses the same symmetry, that is, isotropy subgroup, as thesource of the persistence phenomenon, while in a bifurcation, thebifurcated elements have strictly smaller symmetry. In the jargon ofbifurcation of dynamical systems with symmetry, one says that thereis a symmetry breaking phenomenon. We postpone the study ofbifurcations to Chapter 7.

4.1 Singular Relative Critical Elements

In Section 1.6 we introduced the definition of relative critical elementsin the context of free actions and we saw how these solutions lookedlike both in the original manifold, as well as in the Marsden–Weinsteinreduced spaces, available in that situation. Since the treatment in thissection will include singularities, we will formulate these statementsin this new level of generality. As could be expected, the singularreduced spaces introduced in Chapter 2 will replace the Marsden–Weinstein ones.

Definition 4.1.1 Let (M, !) be a symplectic manifold and let G bea Lie group acting properly on M in a globally Hamiltonian fashionwith associated equivariant momentum map J : M % g". Let µ " g"

be a value of J and m "M be such that J(m) = µ. We denote by Gµ

the isotropy of µ under the coadjoint action of G on g" and H := Gm.Let h " C!(M)G be a G–invariant Hamiltonian. We say that thepoint m "M is a relative equilibrium (respectively relative peri-odic point (RPP) or relative periodic orbit (RPO)) of h withrespect to the G–symmetry of M , if the point [m](H)

µ = /(H)µ (m) is an

equilibrium (respectively periodic point) of the Hamiltonian dynamicalsystem (M (H)

µ , !(H)µ , h(H)

µ ).

As we did in the free case, we study the appearance of thesesolutions in the original manifold M .

Theorem 4.1.1 In the hypotheses of Definition 4.1.1, the followingstatements are equivalent:

(i) The point m " J%1(µ) *MGµ

(H) with H := Gm is a relative equi-librium.

§ 4.1. Singular Relative Critical Elements 127

(ii) There is a unique + " Lie(NGµ(H)/H ' l such that

Ft(m) = expL t+ · m for all t " R,

with Ft the flow of the Hamiltonian vector field Xh and expL :l % L the exponential map associated to the Lie group L :=N(H)/H. The element + " l is called the canonical velocityof the relative equilibrium m.

(iii) There is a - " Lie(NGµ(H)) such that

Ft(m) = exp t- · m for all t " R.

The element - " Lie(NGµ(H)) is called a velocity of the rela-tive equilibrium m. The set of all possible velocities of a givenrelative equilibrium coincides with the set of representatives ofthe canonical velocity in the Lie algebra of NGµ(H), that is,- " Lie(NGµ(H)) is a velocity of the relative equilibrium m ifand only if [-] = +, where + is given in (ii).

(iv) There is a - " Lie(NGµ(H)) such that

Xh(m) = -M (m).

(v) There is a - " Lie(NGµ(H)) such that the Lyapunov functionL# := h$ J# satisfies

dL#(m) = 0.

Proof (i)!(ii) By hypothesis, the point [m](H)µ " M (H)

µ is an equi-librium of X

h(H)µ

and hence

Fµt

3[m](H)

µ

4= [m](H)

µ for arbitrary time t " R.

This implies that /(H)µ (Ft(m)) = /(H)

µ (m), so that Ft(m) = g(t) · mfor some g(t) " Gµ. The G–equivariance of Ft implies that GFt(m) =Gm = H. Thus H = GFt(m) = g(t)Hg(t)%1, is equivalent to g(t) "NGµ(H) for all t. Since the action of NGµ(H)/H ' N(H)/H on MH

is free, let l(t) = g(t)H be the unique element of NGµ(H)/H suchthat Ft(m) = l(t) · m. The fact that Ft+s = Ft & Fs implies that l(t)is a one–parameter subgroup of NGµ(H)/H ' N(H)/H and hence


l(t) = expL t+, for some + in the Lie algebra of NGµ(H)/H. Thus wehave

Ft(m) = expL t+ · m with + " Lie(NGµ(H)/H) ' l.

Next, we show that + is unique. Indeed, if there was another +$ in theLie algebra of NGµ(H)/H such that Ft(m) = expL t+·m = expL t+$·mfor all t " R, then expL($t+) expL t+$ · m = m for all t " R, that is,expL($t+) expL t+$ = [e] for all t " R since Lm = {e}. Thus, takingt so small that both t+ and t+$ lie in the neighborhood where expL isa di!eomorphism, we conclude that + = +$.(ii)!(iii) We now show that we can find a - " Lie(NGµ(H)) suchthat Ft(m) = exp t- · m, for all time t. Let’s recall first that onLie(NGµ(H)/H), we are taking the Lie group homogeneous di!eren-tiable structure which guarantees that the projection / : NGµ(H)%NGµ(H)/H is a surjective submersion; in particular, Te/ is surjective.Let - " Lie(NGµ(H)) be such that Te/(-) = +. By construction, / is aLie group homomorphism and hence (exp t-)H = expNGµ (H)/H t+ =expL t+. Therefore, expL t+ · m = exp t- · m, since H = Gm and weconclude by (ii) that

Ft(m) = exp t- · m with - " Lie(NGµ(H)).

(iii)!(i) We show that the point [m](H)µ " M (H)

µ is an equilib-rium of the Hamiltonian dynamical system (M (H)

µ , !(H)µ , h(H)

µ ):

Fµt

3[m](H)

µ

4= /(H)

µ (Ft(m)) = /(H)µ (exp t- · m) = /(H)

µ (m) = [m](H)µ ,

since - " Lie(NGµ(H)) ' gµ. The point m is therefore a relativeequilibrium.

(iii)!(iv) The map Ft(m) = exp t- · m is the flow of the vectorfield -M , hence

-M (m) =d

dt

$$$$t=0

Ft(m) = Xh(m).

(iv)!(iii) It is obvious, by the uniqueness of the integral curvesof a vector field.

(iv)"(v) If - " Lie(NGµ(H)), the definition of the momentummap implies that -M (m) = XJ!(m). Thus, Xh(m) = -M (m) if andonly if XJ!(m) = Xh(m) which is equivalent to dh(m) = dJ#(m). !


There are a few specific cases in which the previous theorem guar-antees that a given relative equilibrium is actually a genuine equilib-rium.

Corollary 4.1.1 In the conditions of Theorem 4.1.1, suppose thatm "M is a relative equilibrium. Then the following hold:

(i) If the isotropy of m coincides with Gµ, that is, m " J%1(µ) *M

Gµ

(Gµ) = J%1(µ) *MGµ, then m is an equilibrium.

(ii) If m " J%1(µ) *MGµ

(H) and NGµ(H) = H, then m is an equilib-rium.

(iii) If NGµ(H) is finite, then m is an equilibrium.

Proof Follows trivially from Theorem 4.1.1 since in all of these casesthe Lie algebra of NGµ(H)/H is trivial. !

Theorem 4.1.2 (Souriau–Smale–Robbin) In the hypotheses ofDefinition 4.1.1, the point m " M is a relative equilibrium if andonly if m is a critical point of

h|MH ) JL : MH $% R) l",

with L = N(H)/H.

Proof Proposition 2.3.1 states that the action of L on MH is free,proper, and admits a momentum map JL : MH % l" which is, ingeneral, non–equivariant. The explicit construction of JL given thereguarantees that JL(m) = 0. Moreover, part (ii) of Theorem 4.1.1shows that m is a relative equilibrium of the G–space (M, !, h) i! mis a relative equilibrium of the L–space (MH , !|MH , h|MH ). Since theL–action is free, we now follow the traditional proof of the Souriau–Smale–Robbin Theorem (see for instance [AM78, Proposition 4.3.8]).By Theorem 4.1.1, the point m is a relative equilibrium of the L–space(MH , !|MH , h|MH ), if and only if [m]0 = /0(m) is an equilibrium inthe Marsden–Weinstein reduced space ((MH)0, (!|MH )0, (h|MH )0),that is d(h|MH )0([m]0) = 0. Since, by definition, (h|MH )0 & /0 =h|MH & i0, and /0 is a surjective submersion, the previous condition isequivalent to dh|MH (m)|ker TmJL = 0. There are two possible cases:


• dh|MH (m) = 0: In this case m is clearly an equilibrium on MH ,and hence a relative equilibrium on MH , and also a critical pointof h|MH ) JL.

• dh|MH (m) += 0: We have seen that m is a relative equilibriumif and only if dh|MH (m)|ker TmJL = 0. Note also that TmJL :TmMH % l" is surjective since the L–action is free on MH (seeLemma 2.0.1). The algebraic lemma below (see [AM78, Lemma4.3.9]) will show that this is equivalent to dh|MH (m) ) TmJL

not being surjective, which, by definition, means that m is acritical point of h|MH ) JL. !

Lemma 4.1.1 Let E and F be Banach spaces and let $ : E % Rand L : E % F be continuous linear maps. Suppose that ker L has aclosed complement in E and that L is surjective. Then $ is surjectiveon kerL if and only if $ ) L : E % R ) F , given by ($ ) L)(v) =($(v), L(v)), is surjective.

Proof By hypothesis, there is a closed complement E$ of kerL in E,that is, E = kerL( E$.

First, assume that $ is surjective on ker L and let (x, w) " R)Fbe arbitrary. By surjectivity of L, there is some v$ " E$ such thatL(v$) = w and by surjectivity of $ on kerL there is some u " kerLsuch that $(u + v$) = x. Thus, ($)L)(u + v$) = ($(u + v$), L(v$)) =(x, w) which implies that $) L : E % R) F is surjective.

Conversely, if $ ) L : E % R ) F is surjective, for an arbitraryx " R we can find some v " E such that (x, 0) = ($ ) L)(v) =($(v), L(v)), that is, there some v " ker L such that $(v) = x. Thus,$ is surjective on kerL. !

We introduce an analog of Theorem 4.1.1, for relative periodicorbits.


be a value of J and let m "M such that J(m) = µ. We denote by Gµ

the isotropy of µ under the coadjoint action of G on g" and H := Gm.Let h " C!(M)G be a G–invariant Hamiltonian. Then the followingstatements are equivalent:

(i) the point m is a RPP;


(ii) there is a constant & > 0 (the relative period) and g " NGµ(H)(the phase shift) such that

Ft+" (m) = g · Ft(m), for any time t " R,

where Ft is the flow of Xh;

(iii) there is a constant & > 0 and a unique element l " NGµ(H)/Hsuch that

Ft+" (m) = l · Ft(m), for any time t " R,

where Ft is the flow of Xh.

Proof (i)!(ii) By hypothesis, there is a & > 0 such thatFµ

t+" ([m](H)µ ) = Fµ

t ([m](H)µ ) for any time t. Thus,

/(H)µ (Ft+" (m)) = /(H)

µ (Ft(m)), for any time t.

In particular, for t = 0, (/(H)µ & F" )(m) = /(H)

µ (m). Hence, thereexists an element g " Gµ such that F" (m) = g · m. Thus, if t isarbitrary,

Ft+" (m) = (Ft & F" )(m) = Ft(g · m) = g · Ft(m).

At the same time, the G–invariance of the Hamiltonian h, implies theG–equivariance of the flow Ft, for any t, and hence Gm = GFt(m); inparticular, for t = & one has:

H := Gm = GF$ (m) = Gg·m = gGmg%1 = gHg%1,

which implies that g " N(H) and, hence g " NGµ(H), as required.(ii)"(iii) Take l = gH. The uniqueness of l is a consequence of

the freeness of the action of N(H)/H, and hence of NGµ(H)/H, onMH .

(ii)!(i) If we apply /(H)µ on both sides of the equality Ft+" (m) =

g · Ft(m) and recall that g " Gµ, we obtain that

/(H)µ (Ft+" (m)) = /(H)

µ (g · Ft(m)) = /(H)µ (Ft(m)),

or, equivalently,

Fµt+" ([m](H)

µ ) = Fµt ([m](H)

µ ),

where Fµt is the flow of the Hamiltonian vector field on M (H)

µ definedby the reduced Hamiltonian function h(H)

µ . This shows that [m](H)µ is

a periodic point of (M (H)µ , !(H)

µ , h(H)µ ) with period & > 0. !


4.2 Persistence of Singular Relative CriticalElements

We start by introducing some concepts and technical lemmas thatwill be needed in what follows..

Definition 4.2.1 In the conditions of Theorem 4.1.1, the pointm " M is said to be a non degenerate relative equilibrium ifthe corresponding reduced equilibrium [m](H)

µ " M (H)µ is such that

dh(H)µ ([m](H)

µ ) is a non degenerate quadratic form.

For the next result we need a technical lemma (see for in-stance [BrL75, page 125]).

Lemma 4.2.1 (Splitting Lemma) Let f " C!(V ) W ) with Vand W finite dimensional vector spaces and such that the mappingf |W , defined by f |W (w) := f(0, w), has a non–degenerate criticalpoint at 0. Then there is a local di!eomorphism defined around thepoint (0, 0), of the form 4(v, w) = (v, 41(v, w)), such that

(f & 4)(v, w) = f(v) + Q(w),

where Q is the non–degenerate quadratic form Q = 12d

2f |W (0), andf is a smooth function on V .

The following two results describe the basics of the symplecticslicing technique, due to Guillemin and Sternberg (see [GS84b,Theorem 26.7]), which will prove to be very useful.

Lemma 4.2.2 Let (M, !) be a symplectic manifold and let G be acompact Lie group acting on (M, !) in a globally Hamiltonian fash-ion, with associated equivariant momentum map J : M % g". Letm " M and denote J(m) = µ " g". Let : ·, · ; be a Ad–invariantinner product on g and consider the Ad–invariant splitting

g = gµ ( q,

where q = g(µ . Let

g" = q# ( g#µ

be the dual Ad"–invariant splitting, where q# 7= g"µ and g#µ7= q". Then

§ 4.2. Persistence of Relative Critical Elements 133

(i) TµOµ * q# = {0},

(ii) TµOµ + q# = g",

(iii) TµOµ ' TmJ(TmM) = (gm)#,

where Oµ is the coadjoint orbit through µ " g".

Proof (i) Let , " TµOµ*q#. There exist elements - " g and . " gµ

such that , = ad"#µ " TµOµ and , =: ., ·;. Hence, for any 1 " gµ:

: ., 1; = ,(1) = 0ad"#µ, 11

= $0µ, ad*-1 = $0ad"*µ, -1 = 0.

Therefore: ., 1;= 0 for 1 " gµ arbitrary, which implies that . = 0and hence , = 0.(ii) Since clearly TµOµ + q# ' g", a dimension count argument willconclude the proof. Indeed,

dim(TµOµ + q#) = dimTµOµ + dim q#, by part (i)= dimOµ + dim g"µ

= dimG$ dimGµ + dimGµ = dimG = dim g".

(iii) The equivariance of the momentum map implies that Gm ' Gµ,and hence gm ' gµ. Let , = ad"

#µ " TµOµ be arbitrary. Then forany . " gm ' gµ:

,(.) = 0ad"#µ, .1 = $0ad"

$µ, -1 = 0,

so, , " (gm)#, as required. !

Proposition 4.2.1 In the hypotheses and notations of Lemma 4.2.2,there is a local symplectic Gµ–invariant submanifold Q 'M contain-ing m, such that the Gµ–action on Q, obtained by restriction of theoriginal G–action, is globally Hamiltonian with equivariant momen-tum map JQ : Q% q# 7= g"µ, given by

JQ(z) = J(z)$ µ.

Proof By Lemma 4.2.2, we have

TmJ(TmM) + q# / TµOµ + q# = g",


hence, the mapping J is transversal to q# at m " M . By the LocalTransversality Theorem (see [AR67, Theorem 17.1]), there is an openneighborhood U of µ " g" such that Q = J%1(U * q#) is a regularsubmanifold of M . Let us prove that the point m belongs to Q,that is, that µ " q#. Using the Ad–invariant inner product on g,there is a unique - " g such that µ =: -, · ;. The element µbelongs to q# i! - " gµ. Indeed, µ|q = 0 i! : -, . ;= 0 for all. " q i! - " q( = gµ. Finally, - " gµ because for any 5 " g,we have 0ad"

#µ, 51 = 0µ, [-, 5]1 =: -, [-, 5] ;=: [-, -], 5 ;= 0, byAd–invariance of the inner product.

Since the isotropy subgroup Gµ is compact and the orbit Gµ ·µ = {µ}, Corollary 3.1.1 guarantees that U , and therefore Q, canbe chosen to be Gµ–invariant sets. Since symplecticity is an opencondition, in order to establish it for Q, we prove that the vectorspace (TmQ, !(m)|TmQ) is symplectic, and then we shrink Q further,if necessary, in a Gµ–invariant fashion to conclude that Q is a Gµ–invariant symplectic manifold. To prove the latter statement, we willshow that if W = TmQ = Tm(J%1(U * q#)), then W *W% = {0}.Indeed, since ker TmJ 'W , by the Reduction Lemma,

W% ' (kerTmJ)% = Tm(G · m).

So let 6M (m) "W *W%, with 6 " g, and decompose 6 = 6gµ +6q "gµ ( q. For any .gµ + .q " gµ ( q = g we have, by equivariance of J,

0TmJ · 6M (m), .gµ + .q1 = $0ad"+J(m), .gµ + .q1

= 0ad"$gµ

µ, 61 $ 0ad"+µ, .q1

= $0ad"+µ, .q1.

However, since 6M (m) " W = (TmJ)%1(q#), we also have $ad"+µ =

TmJ(6M (m)) " q# so that the above expression vanishes. This showsthat TmJ · -M (m) = 0. Therefore,

-M (m) " kerTmJ * Tm(G · m) = Tm(Gµ · m),

by the Reduction Lemma, and we conclude that - " gµ. Since -M (m)also belongs to W%, we have

0 = !(m)(-M (m), v) = dJ#(m) · v = 0TmJ(v), -1,

for any v "W = Tm(J%1(U * q#)) = (TmJ)%1(q#). Since TmJ(W ) =q#, this implies that - " (q#)# = q, whence - " q * gµ = {0}. Thus,W *W% = {0} and hence Q is symplectic.


Let !Q be the pull back of ! to Q by the inclusion of Q intoM . The previous arguments prove that (Q, !Q) is a local symplecticGµ–submanifold of (M, !) around m " M . A momentum map JQ :Q % g"µ for the globally Hamiltonian Gµ–action on (Q, !Q) is givenby JQ(z) = J(z); note that it is not necessary to restrict J(z) toq#, since, by construction, J(Q) ' U * q#. Since the group Gµ fixesµ " g" and we already proved that µ " q#, we can modify JQ to theequivariant momentum map JQ : Q% q# 7= g"µ given by

JQ(z) = J(z)$ µ. !

Remark 4.2.1 The results just introduced will allow us to reduceour study to the case µ = 0 (notice that by construction JQ(m) = 0).Another way to do this is using the so called shifting trick (see forinstance [MR]). The main di!erence between this procedure and thesymplectic slicing technique just described is that, in the shifting trickone works on a di!erent manifold with the same group G in order toachieve the reduction at µ = 0, while in the slicing technique we justdescribed, it is enough to use a certain symplectic submanifold of Mand changing to original symmetry group G to the subgroup Gµ. "

Before we state the main result in this section, we introduce somenotation. Let (M, !) be a symplectic manifold and let G be a Liegroup acting properly on M in a globally Hamiltonian fashion withassociated equivariant momentum map J : M % g". Let µ " g"

be a value of J and let m " M be such that J(m) = µ. Let Gµ

be the coadjoint isotropy subgroup of µ and denote H := Gm. IfNGµ(H)/H is compact, the Weyl group of NGµ(H)/H will be denotedby W (H)

µ and the cardinality of the orbit W (H)µ · . through . " (h#)H

by W(H)µ (.).

Theorem 4.2.1 Let (M, !, h) be a Hamiltonian G–space, with G aLie group acting properly on M , and let J : M % g" be the asso-ciated equivariant momentum map. Suppose that m " M is a non-degenerate relative equilibrium with Gm = H, such that N(H) iscompact and J(m) = µ. Then there is a (NGµ(H))–invariant neigh-borhood V ' (h#)H of the origin such that for each $ = µ+. " µ+V

there are at least 12 dim(NGµ(H) · .) + 1 relative equilibria in M (H)

,

or W(H)µ (.), if they are all non degenerate.


Proof We first prove the theorem in the case in which there are nosingularities, that is, H = {e}. This is the case presented by Montaldiin [MO97], whose proof we describe in what follows. Additionally, wesuppose that µ = 0. The result in full generality will be a corollary ofthis case, together with the results on singular reduction introducedin Chapter 2, and the symplectic slicing technique. Notice that ifH = {e} and µ = 0, the subgroups NGµ(H) and N(H), as well as thehypotheses that go with them, reduce to G, which consequently willnow be compact. In this simplified case we have to show that thereis a neighborhood V of 0 in g" such that for each $ " V , there are atleast 1

2 dim(O,)+1 relative equilibria in M,, or W0($) if they are allnon degenerate, where O, denotes the coadjoint orbit of $ " g", andW0($) is now the cardinality of the Weyl group orbit of G through$.

Since what we are trying to prove is a general statement aboutHamiltonian dynamical systems endowed with a compact symmetry,we will make a very strong use of the MGS normal form. Indeed, let

Y = G) g" ) Vm

be the symplectic tube around the orbit m " M provided by Theo-rem 3.2.2, in which the point m is represented by (e, 0, 0), Vm is thesymplectic normal space at m, and the momentum map J is given bythe expression

J(g, ., v) = Ad"g"1. := g · .. (4.2.1)

Lemma 4.2.3 If Gm = {e}, a neighborhood of [m] in the Marsden–Weinstein reduced space (M,, !,), with $ " g" close enough to theorigin, is symplectomorphic to O%

, )Vm, endowed with the sum sym-plectic structure.

Proof Using expression (4.2.1), it is easy to compute that, in theMGS normal form around m , (e, 0, 0),

J%1($) = {(g, g%1 · $, v) " Y | g " G and v " Vm}.

We define the mapping

2 : J%1($) $% O%, ) Vm

(g, g%1 · $, v) -$% (g%1 · $, v).


The mapping 2 is clearly di!erentiable and invariant with respect tothe action of the coadjoint isotropy subgroup G, on J%1($), so itdrops to a smooth map

& : J%1($)/G, $% O%, ) Vm

[g, g%1 · $, v] -$% (g%1 · $, v).

This map is clearly surjective. It is also injective because if &[g, g%1 ·$, v] = &[g$, (g$)%1 ·$, v$], we obviously have that g%1 ·$ = (g$)%1 ·$and v = v$, which implies that g$g%1 · $ = $ and therefore [g, g%1 ·$, v] = [g, g%1(g$g%1)%1 ·$, v$] = [g, (g$)%1 ·$, v$] = [(g$g%1)g, (g$)%1 ·$, v$]. In order to show that & is a di!eomorphism, it is enough toshow that it is a local di!eomorphism, that is, for any [g, g%1 ·$, v] "J%1($)/G,, the map T(g, g"1·,, v)2 is surjective with kernel equal to

T(g, g"1·,, v)(G, · (g, g%1 · $, v))

= {TeRg., 0, 0) " TgG) g" ) Vm | . " g,}.

Surjectivity is obvious. If v = (vG, vg# , vVm) " kerT(g, g"1·,, v)2 itimmediately follows that vg# = 0 and vVm = 0. Therefore, the vectorv can be written as v = d

dt

$$t=0

(g(t), g(t)%1 · $, v) with g(0) = gand g(t)%1 · $ = $ for any t. Thus g(t) " G, for all t and hencev = (g$(0), 0, 0) = (TeRg(TgRg"1g$(0)), 0, 0), where TgRg"1g$(0) "g,, thereby showing that kerT(g, g"1·,, v)2 ' T(g, g"1·,, v)(G, · (g, g%1 ·$, v)). The converse inclusion is obvious.

To conclude the proof, endow O%, ) Vm with the sum symplec-

tic structure !%O%

+ !Vm , where O, carries the ($) orbit symplecticstructure !%

O%and Vm its natural symplectic form !Vm . We verify

that & : J%1($)/G, % O%, ) Vm is a symplectic map. For this,

let /, : J%1($) % J%1($)/G, be the projection and i, : J%1($) )%G)g")Vm = Y be the inclusion. Recall that the symplectic form onY is !B+!Vm where !B is given by (3.2.2). If (g, g%1 ·$, u) " J%1($),-, . " g, and v, w " Vm, we have

-/",&

"(!%O%

+ !Vm)2(g, g%1 · $, u)

-(TeLg-, ad"

#(g%1 · $), v), (TeLg., ad"

$(g%1 · $), w)

2

=-2"(!%

O%+ !Vm)

2(g, g%1 · $, u)

-(TeLg-, ad"

#(g%1 · $), v), (TeLg., ad"

$(g%1 · $), w)

2

= !%O%

(g%1 · $)-ad"

#(g%1 · $), ad"

$(g%1 · $)

2+ !Vm(v, w)

= $0g%1 · $, [-, .]1+ !Vm(v, w).


On the other hand, by (3.2.2),

i",(!B + !Vm)(g, g%1 · $, u)-(TeLg-, ad"

#(g%1 · $), v), (TeLg., ad"

$(g%1 · $), w)

2

= !B(g, g%1 · $)-(TeLg-, ad"

#(g%1 · $)), (TeLg., ad"

$(g%1 · $))

2

+ !Vm(v, w)

= $0ad"#(g

%1 · $), .1+ 0ad"$(g

%1 · $), -1+ 0g%1 · $, [-, .]1+ !Vm(v, w)

= $0g%1 · $, [-, .]1+ !Vm(v, w).

Since the two expressions are identical , the characterization of thesymplectic form on J%1($)/G, implies that &"(!%

O%+ !Vm) equals

this reduced symplectic form and the lemma is proved. #We now implement the hypothesis of the theorem using the nor-

mal form Y . Recall that in these variables, the G–invariance of theHamiltonian h implies that it is a function only of the g" ) Vm coor-dinates, that is,

h : g" ) Vm $% R.

By definition, the non degeneracy of m as a relative equilibrium im-plies that [m]0 is a non degenerate equilibrium of the reduced system(M0, !0, h0), that is, the Hessian d2h0([m]0) is a non degeneratequadratic form. By Theorem 3.4.2, this means that in the MGSnormal form, the Hamiltonian dynamical system (Vm, !Vm , hVm) hasa non degenerate equilibrium at 0 " Vm, where hVm is defined byhVm(v) = h(0, v), for any v " Vm. With this in mind, we can useLemma 4.2.1 to find new variables g" ) V $

m, related to g" ) Vm via alocal di!eomorphism around the element (0, 0)

4 : g" ) Vm $% g" ) V $m

(., v) -$% (., 41(., v))

provided by Lemma 4.2.1, in which the Hamiltonian h is representedby the map h$ defined by:

h$(., v$) = (h & 4)(., v) = h(.) +12d2hVm(0)(v, v),

for a certain mapping h " C!(g") and any (., v$) = 4(., v).


For any given $ " g" close enough to zero, Lemma 4.2.3 guaran-tees that the relative equilibria of our system with momentum $ " g",are given by the critical points of the restriction h, of h to O%

, )Vm.By the nature of the splitting given in Lemma 4.2.1, the critical pointsof h, coincide with the critical points of h$

,, that is, the restriction ofh$ to O%

, )V $m. Additionally, the non degeneracy of d2hVm(0) implies

that a point (., v) " O%, ) V $

m is a critical point of h$, i! . " O%

, is acritical point of h|O% . We have thus reduced the problem of findingrelative equilibria of the original system to the problem of findingcritical points of the function

h|O% : O, % R.

Algebraic topological arguments give us the first estimate on thenumber of critical points of h|O% . As Montaldi [MO97] points out,the Ljusternik–Shnirelman category cat(K) of a compact manifold Kis at least 1 greater than its cup–length and, in the case of a compactsymplectic manifold K,

cup–length(K) 4 dim(K)/2

Since the group G is compact, O%, is also compact. Therfore, there

are at least 12 dim(O,)+ 1 critical points for h|O% , which gives us the

first estimate in the statement of the theorem for the case µ = 0,Gm = {e}.

We will provide a proof for the second estimate using Morse–theoretic arguments. If - " g, the natural pairing, defines a linearfunctional in g" which restricts to a function

-, : O%, $% R. -$% 0., -1.

If - is a regular element of the compact Lie algebra g, that is, itbelongs to only one Cartan subalgebra, this function has remarkableproperties, as we shall prove in the following proposition.

Proposition 4.2.2 Let - be a regular element of the Lie algebra gand $ " g". Then the function -, : O%

, % R is a perfect Morsefunction with precisely W0($) critical points.

Proof Relative to the (-) orbit symplectic structure on O%, , the

Hamiltonian vector field X#% , associated to the function -, "C!(O,), is given by the expression

X#%(,) = $ad"#,, for any , " O,.


Since in a Hamiltonian dynamical system on a symplectic manifoldthere is a bijective correspondence between critical points of theHamiltonian and the equilibria of the associated Hamiltonian vec-tor field, the critical points of -, are the points , " O, for whichad"

#, = 0. We now identify the Lie algebra g with its dual g" bymeans of a G–invariant inner product, which allows us to abuse thenotation in order to write the singular locus %(-,) of -, as the set

%(-,) := g# *O, = t *O, = W0 · $

where t is the unique Cartan subalgebra of g containing the givenregular element -.

We now show that -, is a perfect Morse function, that is, that allthe critical points W0 · $ are non–degenerate and have even Morseindex. Indeed, let T (-) := 0exp t-1 be the torus generated by theelement - " g. Using the action of g on O%

, , we can define an actionof the torus T (-) on the tangent space of O%

, at the critical pointsof -,, that is, if , " O%

, is an arbitrary critical point of -,, hencead"

#, = 0, we will show that T (-) acts linearly on T'(O,). The firstrequisite for this to happen is that T (-) should not move ,; this is sobecause

exp t- · , = Ad"exp(%t#), = e%t ad#

!, = ,.

Once we know this we can lift the T (-) action to T'(O%, ) by

g · ad"), = Ad"

g"1 (ad"),)

= (ad) &Adg"1)", = (Adg"1 & adAdg))",= ad"

Adg) Ad"g"1, = ad"

Adg),,

for any g " T (-). We therefore have a linear canonical action givenby

T (-)) T'(O%, ) $% T'(O%

, )(g, ad"

),) -$% ad"Adg),.

By Example 1.4.3, we known that any linear canonical action isHamiltonian. The general form of the momentum map for these ac-tions particularizes in our case to the mapping Jt(#) : T'(O,)% t(-)"

given by the expression

0Jt(#)(ad"),), 51 =

12!%O%

(,)(5T&(O%)(ad"),), ad"

),),


for any 5 " t(-) and where !%O%

is the negative orbit symplectic formon O%

, . Since 5T&(O%)(ad"),) = ad"

[-,)],, we have

0Jt(#)(ad"),), 51 =

12!%O%

(,)(ad"[-,)],, ad"

),)

= $120,, [[5, 0], 0]1 = $1

20ad"

) ad"),, 51.

Therefore, we can write

Jt(#)(ad")(,)) = $1

2ad"

) ad"),.

A straightforward computation shows that the Hessian of -, can bewritten as

d2-,(,)(ad")1,, ad"

)2,) = 0ad"

)1ad"

)2,, -1.

This allows us to conclude that

12d2-,(,)(ad"

),, ad"),) = $J#

t(#)(ad"),).

In our hypotheses, there are coordinates (x, y) in which J#t(#) has the

expression

J#t(#)(x, y) =

12

n!

i=1

(x2i + y2

i )$i(-),

(see [GS82, Proposition 4.6]) where the $i’s are the weights of thesymplectic representation of T (-) on T'(O,). In these coordinates,the Hessian matrix takes the form

d2-,(,) = $

5

6666666666667

$1(-) 0 · · · · · · · · · · · · 00 $1(-) · · · · · · · · · · · · 0...

... $2(-) · · · · · · · · · 0...

...... $2(-) · · · · · · 0

......

...... . . . · · · 0

......

......

... $n(-) 00 0 0 0 0 0 $n(-)

8

999999999999:

.

The regularity of the element - implies that $i += 0, for all i "{1, . . . , n}. Hence, the critical points of -, are non degenerate. The


condition on the even index follows trivially from the specific form ofthe Hessian matrix d2-,(,). The function -, is therefore a perfectMorse function. #

The result that we just proved implies that for -, the Morse in-equalities are actually equalities and hence, any smooth function onO, has at least as many critical points as -, has. This implies thath|O% has at least W0($) critical points, from which the result in thecase µ = 0 with trivial isotropy follows.

We now prove the result for the case µ += 0, as a corollary ofthe case µ = 0, using the symplectic slicing technique introduced inProposition 4.2.1. Suppose that J(m) = µ += 0. Let (Q, !Q, hQ) bethe local Hamiltonian Gµ–space through m " M , whose existenceis guaranteed by Proposition 4.2.1 and where the Hamiltonian hQ "C!(Q)Gµ is defined by hQ = h & iQ, with iQ : Q )% M the naturalinclusion. Notice that by the construction of Q and by the NoetherTheorem, the Hamiltonian flow Ft of h leaves Q invariant. Moreover,since iQ is a canonical map, by Proposition 1.1.1, the Hamiltonianflows Ft of h and FQ

t of hQ, are related by

Ft & iQ = iQ & FQt .

Since by hypothesis m " M is a relative equilibrium, Theorem 4.1.1implies that there is an element - " gµ such that

Ft(m) = exp t- = FQt (m),

which again, by Theorem 4.1.1, guarantees that m " Q is a relativeequilibrium of the Gµ–space (Q, !Q, hQ) such that JQ(m) = 0. Weare therefore in the conditions in which we worked before, hence, thereis an open neighborhood V of 0 in g"µ such that for any . " V , thereare at least 1

2 dim(Gµ · .) + 1 relative equilibria with Q–momentum., or Wµ(.), if they are all non degenerate. Notice that, by thedefinition of JQ, if z " Q is such that JQ(z) = ., then J(z) = µ + .,and the result for µ += 0 follows.

Assume now that, additionally, the isotropy H := Gm of therelative equilibrium m is non trivial. By Theorem 4.1.1 there is aunique + " Lie(NGµ(H)/H) ' l such that

Ft(m) = expL t+ · m.


This implies that m "MH is also a relative equilibrium of the Hamil-tonian L–space (MH , !|MH , h|MH ). Notice that by hypothesis, theLie group N(H) is compact which, by Proposition 2.3.2 allows usto choose an Ad"–equivariant momentum map KL for the L–actionon MH , such that KL(m) = +# " l". Moreover, since this action isfree, we can use our previous results and hence we know that there isa L($–invariant neighborhood U of 0 in Lie(L($) = Lie(NGµ(H)/H)such that for each % " U , there exist at least 1

2 dim(NGµ(H)/H ·%)+1relative equilibria in (MH)($+., or W(H)

µ (%), if they are all non-degenerate, with W(H)

µ (%) the cardinality of the Weyl group orbitof NGµ(H)/H through % " Lie(NGµ(H)/H). Using the definition ofKL, note that these are also relative equilibria for (M, !, h) with J–momentum equal to µ+(#")%1(%). If we take V such that U = #"(V )and . satisfying % = #"(.), the result follows. !

With the same setup used for relative equilibria, it is very easyto generalize an analogous persistence result, also due to Mon-taldi [MO97a], dealing with RPOs , originally proved for free actions.

Definition 4.2.2 Let (M, !) be a symplectic manifold and let G bea Lie group acting properly on M in a globally Hamiltonian fash-ion with associated equivariant momentum map J : M % g". Leth " C!(M)G be a G–invariant Hamiltonian. Let µ " g" be a valueof J and m " M be a RPP of the Hamiltonian dynamical system(M, !, h). Denote J(m) = µ. Let Gµ be the coadjoint isotropysubgroup at µ and denote H := Gm. We say that m " M is anon degenerate RPP if the corresponding reduced periodic point[m](H)

µ " M (H)µ is such that the occurrence of 1 as a characteristic

multiplier is minimal, that is, the multiplicity of 1 as an eigenvalueof the linearized Poincare map is 1.

Theorem 4.2.2 Let ' be a non degenerate relative periodic orbit ofthe Hamiltonian dynamical system described in Definition 4.2.2, ofenergy E0 and period T0. Assume that m " ', [m](H)

µ " M (H)µ , and

N(H) is connected. Then there is a NGµ(H)–invariant neighborhoodV ' (h#)H of the origin such that for each $ = µ+. " µ+V and eachE close to E0, there is at least one relative periodic orbit in M (H)

, ofenergy E and period close to T0 or W(H)

µ (.) relative periodic orbitsif they are all non-degenerate.


Proof Montaldi’s result ([MO97a, Theoreme 1]) proves the theo-rem in the case H = {e}, which can be generalized to the singularcase following similar steps to those in Theorem 4.2.1, that is, weidentify m " MH as a RPP of the Hamiltonian dynamical system(MH , !|MH , h|MH ), with respect to the free L–action defined on it.Choices for . and V identical to Theorem 4.2.1 complete the proof.!

Remark 4.2.2 Notice that Theorems 4.2.1 and 4.2.2 are genuinepersistence results, that is, the relative critical elements whose exis-tence is proved have the same symmetry as the one we start with.This somehow limits the range of applicability of these statementssince the systems where the phenomena described by them takesplace, need to be large enough so that one encounters non trivialorbit types. A scenario where the prediction capabilities of theseresults is somewhat limited is the one created by an Abelian symme-try. Since in this case the coadjoint action and the Weyl groups aretrivial, Theorems 4.2.1 and 4.2.2 predict just one new relative criti-cal element in each neighboring momentum level. In Chapter 7 wewill see how bifurcation techniques help to overcome some of theselimitations. "

4.3 Manifolds of Relative Equilibria

G. Patrick [Pat95, Theorem 17] was the first who realized that, if inTheorem 4.2.1 we make certain regularity assumptions on the mo-mentum value and on the velocity of the relative equilibrium, theresult in the previous section can be vastly improved. Indeed, onecan predict not only a finite number of relative equilibria around thegiven non degenerate one, but an entire manifold of them. The orig-inal result of Patrick is formulated for free actions of compact Liegroups and, in these terms it says that the set of relative equilibriaaround a given non degenerate relative equilibrium, with regular ve-locity - and momentum µ, is a local smooth symplectic manifold ofdimension rank(G) in M/G.

In the singular case, M/G is not even a manifold in general.Therefore, the natural setup (see Proposition 1.3.1) is to study therelative equilibria lying in M(H)/G. We will call them the set ofrelative equilibria of orbit type (H).

§ 4.3. Manifolds of Relative Equilibria 145

Theorem 4.3.1 Let (M, !, h) be a Hamiltonian dynamical systemand let G be a Lie group acting properly and in a globally Hamiltonianfashion on (M, !) with associated equivariant momentum map J :M % g". Let m " M be a non degenerate relative equilibrium of(M, !, h) with J(m) = µ and Gm = H, such that the normalizerN(H) is compact. If µ " g" is such that +# = (#" &1)(µ) = KL(m) "l" is a regular element of l" and its velocity - " Lie(NGµ(H)) can bechosen to be regular in the Lie algebra Lie(N(H)) of N(H), then therelative equilibria of orbit type (H) near m form a smooth symplecticsubmanifold of dimension dimL+rank L = dimN(H)+rank N(H)$2 dimH in M .

Proof The equivariant injection MH )%M(H) induces an equivariantdi!eomorphism between MH/L and M(H)/G, hence, looking for therelative equilibria of orbit type (H) is equivalent to searching forthe relative equilibria around m in the L–space (MH , !H , h|MH ).Since this action produces no singularities, we can apply in this setupPatrick’s result, modulo some regularity assumptions that we fix inthe following lemma.

Lemma 4.3.1 If - is a regular element in the compact Lie algebran := Lie(N(H)), then Te/(-) " l = Lie(N(H)/H), with / : N(H) %N(H)/H the canonical projection, is a regular element in l.

Proof The proof is done in two steps.Step 1 : Let k be a compact Lie algebra. By the structure theorem(see for instance [Kna96, Corollary 4.25]) we can write the Lie algebradirect sum

k = k0 ( k1 ( · · ·( kp,

where k0 = Z(k) is the center of k, and k1, . . . , kp are all the simple(non-commutative) ideals of [k, k]; thus

[k, k] = k1 ( · · ·( kp. (4.3.1)

Decompose - " k according to this direct sum decomposition as - =-0 + -1 + · · · + -p. We shall prove that

k# = k0 ( (k1)#1 ( · · ·( (kp)#p . (4.3.2)


Since k0 is the center of k, it follows that all its elements commutewith - and hence k0 ' k#. If 5 " (ki)#i , that is, 5 " ki and [5, -i] = 0,then [5, -j ] = 0 because (4.3.1) is a direct sum of Lie algebras andhence the bracket of any two elements in distinct components is zero.Thus, [5, -] = 0 and hence 5 " k# thereby showing that (ki)#i ' k#.We have hence k# / k0 ( (k1)#1 ( · · ·( (kp)#p . Conversely, let . " k#,that is, [., -] = 0, or, using the decomposition (4.3.1),

0 = [.0 + .1 + · · · + .p, -0 + -1 + · · · + -p] = [.1, -1] + . . . [.p, -p],

which implies that [.1, -1] = · · · = [.p, -p] = 0. Therefore, .i " (ki)#i ,for all i = 1, . . . , p. This shows that . = .0 + .1 + · · ·+ .p " k0( k1(· · ·( kp, whence k# ' k0( (k1)#1 ( · · ·( (kp)#p and the equality (4.3.2)is proved.

From (4.3.2) it follows that - " k is regular if and only if each-i " ki is regular in the simple Lie algebra ki. Indeed, if - is regular ink, there is a unique maximal Abelian Lie subalgebra t of k, such thatt = k# = k0((k1)#1( · · ·((kp)#p . We prove now that each -i is regularin ki, for all i = 1, . . . , p. Any Abelian Lie subalgebra of ki containing-i is clearly included in (ki)#i . We shall prove now, by contradiction,that any maximal Abelian Lie subalgebra of ki containing -i equals(ki)#i for all i = 1, . . . , p, which will prove its uniqueness. Supposethat there was some index i for which there would exist a maximalAbelian Lie subalgebra ti of ki containing -i, such that ti ! (ki)#i .If .i " (ki)#i \ ti, then .i " k# \ t because if .i was in t, then .i "t * (ki)#i ' ti, which is clearly a contradiction. The conclusion ishence that .i " k# \ t which is impossible since k# = t. This showsthat if ti is any maximal Abelian Lie subalgebra of ki containing -i,then (ki)#i = ti, for all i = 1, . . . , p. Conversely, if each -i is regular,the maximal Abelian Lie subalgebra of ki containing -i equals (ki)#i .Therefore, by (4.3.2), k# = k0 ( (k1)#1 ( · · ·( (kp)#p is an Abelian Liesubalgebra of k0 ( k1 ( · · · ( kp = k containing -. It is also maximalsince k0 is the center of k and (ki)#i is maximal in ki.Step 2 : Let us return to the notations and conventions of the lemma.The Lie algebras h, n, and l are all compact. Therefore, by the struc-ture theorem ([Kna96, Corollary 4.25]), we can write

n = n0 ( n1 ( · · ·( np and h = h0 ( h1 ( · · ·( hq

where n0 = Z(n) is the center of n, h0 = Z(h) is the center of h, and

[n, n] = n1 ( · · ·( np and [h, h] = h1 ( · · ·( hq

§ 4.3. Manifolds of Relative Equilibria 147

is the Lie algebra direct sum decomposition into all the simple idealsof the semisimple Lie algebras [n, n] and [h, h] respectively. However,h is an ideal in n, so [h, h] is an ideal in [n, n], and hence each hi fori = 1, . . . , q, is also a simple ideal in [n, n]. Thus, all the ideals hi

are among the nj , for i = 1, . . . , q. To fix notations, suppose thathi = ni for i = 1, . . . , q. Let us analyze now h0 which is an Abelianideal. Note that [h0, nj ] ' h0 * nj , since both h0 and nj are ideals inn. For j = 1, . . . , p, we conclude that [h0, nj ] = 0, because h0 * nj isan Abelian ideal of the simple ideal nj . Therefore, the elements of h0

commute with all elements of n and thus h0 ' n0. Therefore,

l = l0 ( nq+1 ( · · ·( np,

where n0/h0 = l0 = Z(l) is the center of l, is the decomposition of lgiven by the structure theorem. In particular, we have the Lie algebradirect sum

n = h( l.

We use now the first step to show that if - " n is regular, thenTe/(-) = - " l is also regular. Indeed, since Te/ : n % l is theprojection, by the first step, - is regular if and only if its o! centercomponents -j , for j = 1, . . . , p, are regular in nj . But this clearlyimplies that the components -i, for i = 1, . . . , p are regular in hi = ni,so that by the first step again, - " l is regular. #

The previous lemma guarantees that m " MH is a non degener-ate relative equilibrium of the L–space (MH , !H , h|MH ) with regularvelocity - " l and regular momentum value +# " l. The result followsfrom the application of [Pat95, Theorem 17] in this setup. !

Chapter 5

Stability of Equilibria andRelative Equilibria

Capıtulo X: Donde se cuenta la industria queSancho tuvo para encantar a la senora Dulcinea,y de otros sucesos tan ridıculos como verdaderos.Cervantes, Don Quijote de la Mancha, II

5.1 Introduction

In Section 1.2 we took the first steps in the study of the stabilityof Hamiltonian systems. In the symplectic context we saw that, forinstance, the non degeneracy of the symplectic form, allows the iden-tification the equilibria of the system with the critical points of itsHamiltonian function. Moreover, Lagrange’s Theorem provides a suf-ficient condition for the stability of the equilibrium in terms of theHessian of the Hamiltonian at the equilibrium. The generalizationof these ideas to Poisson manifolds makes possible the analysis ofequilibria that are not critical points of the Hamiltonian, due to thedegeneracy of the Poisson tensor. Since the Hessian is only definedat critical points (see Section 5.2), a Lagrange Theorem in this sce-nario is not possible if one uses only the Hamiltonian function. Nev-ertheless, when there are conserved quantities present, one can usearguments in the spirit of the Lagrange theorem (the so called “en-

150 Chapter 5. Stability of Equilibria

ergetics”) to prove stability criteria; this is the energy–Casimirmethod (see [A65, A66b, Hal85]).

The stability analysis of relative equilibria is a very well developedtopic that goes at least as far back as Poincare and was significantlygeneralized and extended in [SLM89, SLM91, Lew92, Mar92] underthe name of energy–momentum method . This method is com-pletely algorithmical in the sense that it gives a step by step proce-dure to be followed in order to analyze the Hessian of an augmentedHamiltonian function. If this Hessian is definite, Lyapunov stabilityrelative to a subgroup follows. In this chapter we will present theenergy–momentum method in the context of general Poisson systemswith symmetry. Our treatment will allow the existence of singulari-ties and will take advantage of the eventual availability of additionalnon symmetry related conserved quantities.

The key technique used in proving results of this type comes fromPatrick [Pat92] and is based on the use of certain “penalty func-tions”. It presents the advantage of being easy to use, is physicallyintuitive, and matches very well with reduction theory and the localtechniques introduced in the previous chapters, which will allow us touse it in the study of a great variety of phenomena (see Chapter 6).Patrick [Pat92] dealt only with free actions. The generalization tothe singular case has occured in several steps: Lewis in [Lew92] hastreated the singular Abelian case. The general non Abelian case hasbeen covered, under di!erent sets of hypotheses, by [MO97, LS, OR2].

5.2 Hessians and Patrick’s Lemma

A concept that will be used ubiquitously is the Hessian, whose def-inition and properties we recall in what follows. We will use thedefinition of the Hessian from di!erential topology (see [Mil69]). IfM is a smooth manifold and f " C!(M), suppose that m " M is acritical point of f , that is, df(m) = 0.

Definition 5.2.1 The Hessian of f " C!(M) at the critical pointm "M is the symmetric bilinear form d2f(m) : TmM ) TmM % R,given by:

d2f(m)(v, w) := v[W [f ]],

where v, w " TmM and W " X(M) is an arbitrary extension ofw " TmM to a vector field on M , that is, W (m) = w.

§ 5.2. Hessians and Patrick’s Lemma 151

Remark 5.2.1 The requirement that m "M is a critical point of fis crucial and guarantees the correctness of Definition 5.2.1, that is,the value of d2f(m)(v, w) does not depend on the extension W of w.In addition, m being a critical point of f , allows one to easily provethe symmetry of d2f(m). "

The proof of the following proposition follows directly from thedefinitions.

Proposition 5.2.1 Let m " M and n " N , with M and N smoothdi!erentiable manifolds. Let 4 : M % N be a smooth map such that4(m) = n and let f " C!(N) with df(n) = 0. Then d2(4"f)(m) =T "

m4(d2f(n)), that is, for any v, w " TmM :

d2(4"f)(m)(v, w) = d2f(n)(Tm4 · v, Tm4 · w).

In particular, if S is a submanifold of M , f " C!(M), and m " Sthen,

d2f(m)|TmS)TmS = d2(f |S)(m).

We quote the following classical result for future reference.

Lemma 5.2.1 (Morse) Let M be a manifold and f " C!(M) asmooth map with m " M a nondegenerate critical point, that is,df(m) = 0, and the quadratic form d2f(m) is nondegenerate. Then,there is a coordinate chart about m in which m is mapped by f to 0and the local representative of f satisfies

f(z) = f(0) +12d2f(0) · (z, z).

In particular, nondegenerate critical points of f are isolated.

Proof See [Mil69, Lemma 2.2] or [AM78, Lemma 3.2.3]. !Finally, the proofs of our main results will require the use of a

lemma due to Patrick [Pat92].

Lemma 5.2.2 (Patrick) Let A and B be bilinear forms on a finitedimensional vector space. Suppose that A is positive semidefinite andthat B is positive definite on ker A. Then there exists r > 0 such thatA + *B is positive definite for all * " (0, r).

Proof See [Pat92, Lemma 3]. !


5.3 The Energy–Momentum Method

We present in this section the energy–momentum method. Our setupwill be a Poisson system (M, {·, ·}, h) with a symmetry given bythe proper and globally Hamiltonian action of a Lie group G, withrespect to which the Hamiltonian h is invariant, and with associatedequivariant momentum map J : M % g". Suppose that m " M isa relative equilibrium of (M, {·, ·}, h), with canonical velocity + "Lie(L) = Lie(N(H)/H), where H := Gm. We will now define theconcept of orthogonal velocity of the relative equilibrium m. Ifour relative equilibrium m " M is such that J(m) = µ " g", wesaw in Theorem 4.1.1 that the canonical velocity + belongs actuallyto Lie(NGµ(H)/H) ' Lie(L). The properness of the G–action, andtherefore the compactness of H, allow us to choose an AdH–invariantinner product in nµ := Lie(NGµ(H)) and hence we have an orthogonaldirect sum decomposition

nµ = h( pµ, (5.3.1)

where pµ is the orthocomplement of h in nµ relative to the innerproduct on nµ.

If we consider the quotient Lie group NGµ(H)/H, the canonicalprojection

/ : NGµ(H)% NGµ(H)/H

is a surjective submersion and therefore kerTe/ = (Te/)%1([e]) = h,which implies that

Lie(NGµ(H)/H) 7= nµ/h 7= pµ, (5.3.2)

where the first isomorphism is that of Lie algebras and the second isonly a vector space isomorphism.

Let - " pµ ' nµ be the unique image of the canonical velocity+ " Lie(NGµ(H)/H) by the isomorphism in (5.3.2). Since / is agroup homomorphism, we can write

Ft(m) = expL t+ · m = exp t- · m.

Definition 5.3.1 The unique element - " pµ just defined is calledthe orthogonal velocity of the relative equilibrium m "M , relativeto the splitting (5.3.1).

§ 5.3. The Energy–Momentum Method 153

Remark 5.3.1 Note that the orthogonal velocity depends on thesplitting (5.3.1) and is unique only if this splitting is specified. There-fore, it is not uniquely determined by the relative equilibrium. In ap-plications, probing the stability of the system with all its possible or-thogonal velocities, that is, considering all possible splittings (5.3.1),is the way to obtain optimal stability conditions. See Section 5.5 foran illustration of this comment. "

We can now state the main result of this section.

Theorem 5.3.1 (Energy–Momentum–Method) Let(M, {·, ·}, h) be a Poisson system with a symmetry given bythe Lie group G acting properly on M in a globally Hamiltonianfashion, with associated equivariant momentum map J : M % g".Assume that the Hamiltonian h " C!(M) is G–invariant. Letm " M be a relative equilibrium such that J(m) = µ " g", Gµ iscompact, H := Gm, and - " Lie(NGµ(H)) is its orthogonal velocity,relative to a given AdH–invariant splitting. If there is a set ofGµ–invariant conserved quantities C1, . . . , Cn : M % R for which

d(h$ J# + C1 + . . . + Cn)(m) = 0,

and

d2(h$ J# + C1 + . . . + Cn)(m)|W)W

is definite for some (and hence for any) W such that

kerdC1(m) * . . . * kerdCn(m) * ker TmJ = W ( Tm(Gµ · m),

then m is a Gµ–stable relative equilibrium. If dimW = 0, then m isalways a Gµ–stable relative equilibrium. The quadratic form d2(h $J# +C1 + . . .+Cn)(m)|W)W , will be called the stability form of therelative equilibrium m.

Proof We first suppose that W += {0}. It is easy to show that theresult does not depend on the choice of m in the relative equilibriumIndeed, if d(h$ J# + C1 + . . . + Cn)(m) = 0 and Ft is the flow of theHamiltonian vector field Xh, then for any t > 0 and any v, w " TmM


we have

d(h$ J# + C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))

= F "t (d(h$ J# + C1 + . . . + Cn)(m))(v, w)

= d(F "t (h$ J# + C1 + . . . + Cn))(m)(v, w)

= d(h$ J# + C1 + . . . + Cn)(m)(v, w),

since F "t & d = d & F "

t and h, J#, C1, C2, . . . , Cn are invariant un-der Ft. If W is a complement to Tm(Gµ · m) in kerdC1(m) * . . . *kerdCn(m) * kerTmJ, then for any t > 0, TmFt(W ) is a comple-ment to Tm(Gµ · Ft(m)) in kerdC1(Ft(m)) * . . . * kerdCn(Ft(m)) *ker TFt(m)J. Moreover, d2(h$J#+C1+. . .+Cn)(m)|W)W is definite i!d2(h$J# +C1+ . . .+Cn)(Ft(m))|TmFt·W)TmFt·W is definite, since theconservation of h, J#, C1, . . . , Cn, implies that for any v, w " TmM :

d2(h$ J# + C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))

= d2(F "t h$ F "

t J# + F "t C1 + · · · + F "

t Cn)(m)(v, w)

= d2(h$ J# + C1 + . . . + Cn)(m)(v, w).

Therefore, the statement of the theorem does not depend on thechoice of the point m in the relative equilibrium.

The choice of W is also irrelevant since

d2(h$ J# + C1 + · · · + Cn)(m)(v, w) = 0,

whenever v " Tm(Gµ · m), and w " ker TmJ * kerdC1(m) * . . . *kerdCn(m). Indeed, if we take v = .M (m) with . " gµ, then

d2(h$ J# + C1+ · · · + Cn)(m)(v, w)

= w[XJ" [h$ J# + C1 + · · · + Cn]]

= w[{h, J$}$ J[#, $] + {C1, J$} + · · · + {Cn, J$}]

= w+J[#, $]

,= 0,

where we used the Gµ–invariance of h, C1, . . . , Cn, and that w "ker TmJ.

Let now T := Gµ )H W be a tube around the orbit Gµ · m,associated to the Hamiltonian action of Gµ on M . We denote by Sthe submanifold of T given by

S := H )H W.


It can be easily shown that

TmM = TmS ( Tm(Gµ · m). (5.3.3)

Let now be

Z := TmS * ker TmJ * kerdC1(m) * . . . * kerdCn(m). (5.3.4)

Since Tm(Gµ·m) ' kerTmJ*kerdC1(m)*. . .*kerdCn(m), by (5.3.3)and (5.3.4), we have that

kerTmJ * kerdC1(m) * . . . * kerdCn(m) = Z ( Tm(Gµ · m);

hence Z satisfies the requirements of W in the statement of the the-orem.

We now introduce Patrick’s velocity map. We start with thefollowing lemma.

Lemma 5.3.1 Fix a splitting (5.3.1) and let - " pµ be the corre-sponding orthogonal velocity of the relative equilibrium m "M whosesymmetry group is H := Gm. Then Adh- = - for any h " H.

Proof By definition

h · - := Adh- =d

dt

$$$$t=0

h exp t- h%1 =d

dt

$$$$t=0

exp t- h$(t)h%1 (5.3.5)

where h$(t) is some element in H such that h exp t- = exp t- h$(t) andwhose existence is guaranteed by the fact that exp t- " NGµ(H). Byconstruction, h$(t)h%1 is a curve in H through the identity and hencethere is a 0 " h such that

d

dt

$$$$t=0

h$(t)h%1 =d

dt

$$$$t=0

exp t0 = 0.

Using the Leibniz rule (Lemma 1.3.1) in (5.3.5) we get

h · - =d

dt

$$$$t=0

exp t- h$(t)h%1 = - + 0.

Since - " pµ, the AdH–invariance of the splitting (5.3.1) impliesthat h · - " pµ. The above identity and the splitting (5.3.1) forces0 " h * pµ = {0} which then implies h · - = -. #


Since Gµ)HS , Gµ·S = T is an open Gµ–invariant neighborhoodof the orbit Gµ · m and there is a Gµ–equivariant retraction

r : Gµ · S $% Gµ · m[g, w] -$% g · m,

by Theorem 3.1.1, S is a slice at m for the Gµ–action. We define

' : Gµ · m $% Gµ · -g · m -$% Adg-

with - the orthogonal velocity of the relative equilibrium. The pre-vious lemma guarantees that ' is well-defined: if g · m = g$ · m theng%1g$ " H and therefore g%1g$ · - = - and so g$ · - = g · -. We definePatrick’s velocity map as ' := '&r : g·z " Gµ ·S -% Adg- " Gµ ·-.Note that '(m) = '(m) = - and that for any g " Gµ and anyz = g$ · z$ " Gµ · S,

'(g · z) = '(gg$ · z$) = Adgg!- = Adg(Adg!-) = Adg'(g$ · z$) = Adg'(z).

Also, Im' = Gµ · - and 0µ,'(z)1 = 0µ, -1, for any z " Gµ · S.Let now f1 and f2 be the functions defined by

f1 = (h$ h(m)) + (0J, '1 $ 0µ, -1) + (C1 $ C1(m))+ . . . + (Cn $ Cn(m)),

f2 = (C1 $ C1(m))2 + . . . + (Cn $ Cn(m))2 + <J$ µ<2,

where in f2, the modulus is taken using the norm associated to someAd"

Gµ–invariant inner product in g" (always available by the com-

pactness of Gµ), that makes f2 a Gµ–invariant conserved quantity.Remark that f1 is Gµ–invariant but, in general it is not conserved.Notice also that on S, h $ J# + C1 + · · · + Cn and f1|S di!er bya constant, which implies that d(f1|S)(m) = 0 and d2(f1|S)(m) iswell–defined. Moreover,

d2(f1|S)(m)|Z)Z = d2(h$ J# + C1 + · · · + Cn)(m)|Z)Z .

Since Z satisfies the requirements of W , the hypotheses of the theo-rem guarantees that d2(f1|S)(m)|Z)Z is definite.

We now prove that Z is the kernel of d2(f2|S)(m). It is easy tosee that if v1, v2 " TmS then

d2(f2|S)(m)(v1, v2)= 2 [(dC1(m) · v2)(dC1(m) · v1) + . . .

. . . + (dCn(m) · v2)(dCn(m) · v1) +<TmJ · v1< <TmJ · v2<] .


Then, v1 " kerd2(f2|S)(m) i! for any v2 " TmS, we have that

(dC1(m) · v2)(dC1(m) · v1) + . . . + (dCn(m) · v2)(dCn(m) · v1)+ <TmJ · v1< <TmJ · v2< = 0.

In particular, for v1 = v2, this identity implies that dC1(m) · v1 =. . . = dCn(m) · v1 = <TmJ · v1< = 0 and hence v1 " kerTmJ *kerdC1(m) * . . . * kerdCn(m) * TmS = Z. Conversely, if v1 " Z =TmS * kerdC1(m)* . . .* kerdCn(m)* kerTmJ the above relation issatisfied trivially for all v1 " TmS. Therefore,

Z = kerd2(f2|S)(m).

Therefore, by Lemma 5.2.2, there exists a positive constant a > 0for which

f := af1 + f2

is such that d2(f |S)(m) is positive definite. Note that f is Gµ–invariant but, in general, it is not a constant of the motion since0J, '1 is not conserved. In fact, for any z " S

f(Ft(z))$ f(z) = 0J(Ft(z)), '(Ft(z))1 $ 0J(z), '(z)1= 0J(z), '(Ft(z))$'(z)1= 0J(z)$ µ + µ, '(Ft(z))$ -1= 0J(z)$ µ, '(Ft(z))$ -1+ 0µ, '(Ft(z))1 $ 0µ, -1= 0J(z)$ µ, '(Ft(z))$ -1 ,

where we used Noether’s Theorem, '(z) = - because z " S, and0µ, '(z)1 = 0µ, -1, for any z " Gµ(S). Hence, for any z " S

0 = f(Ft(z)) = f(z) + |0J(z)$ µ, '(Ft(z))$ -1|= f(z) + <J(z)$ µ<(<'(Ft(z))<+ <-<)= f(z) + 2<-< <J(z)$ µ<, (5.3.6)

where we used that Im' = Gµ ·-, and the Gµ–invariance of the norm< · <.

With these tools, we prove the Gµ–stability of m. Let V be aGµ–invariant open neighborhood of Gµ · m. Since f(m) = 0, by thepositive definiteness of d2(f |S)(m) and the Morse Lemma, there is an


* > 0 such that f%1[0, *) * S ' V , where f%1[0, *) is an open neigh-borhood of m in S. The continuity of f and J and Corollary 3.1.1imply the existence of an open H–invariant neighborhood S$ of m inS such that S$ ' f%1[0, *) * S, and that for any z " S$, f(z) < */2and <J(z)$µ< < */4<-<. We see that U := Gµ)H S$ is the neighbor-hood that we need to conclude stability, in other words, Ft(U) ' V .By the Gµ–invariance of V and the Gµ–equivariance of Ft, it su"cesto show that Ft(S$) ' V . Let s$ " S$ be such that Ft(s$) = l · s, withl " Gµ and s " S. Recalling (5.3.6) as well as the Gµ–invariance off :

f(s) = f(Ft(s$)) = f(s$) + 2<-< <J(s$)$ µ< < *,

since s$ " S$. Then, s " f%1[0, *) * S ' V . By the Gµ–invariance ofV , Ft(s$) = l · s " V , as required.

If W = {0}, then d2(f1|S)(m) is definite. The Theorem followstaking f1 as f in the previous proof. !

Remark 5.3.2 As could be seen in the proof, the requirement in thestatement of the theorem on the compactness of Gµ is used only inthe construction of an Ad"

Gµ–invariant inner product in g". Therefore,

the compactness condition could be relaxed to the existence of suchan inner product. "

Remark 5.3.3 The energy–momentum method, as stated in theprevious Theorem, presents the improvement, over the traditionalpresentation (see for instance [Pat92]), of taking advantage of theeventual availability of additional non symmetry related conservedquantities. This is computationally relevant given that dimW , andtherefore the size of the stability form, is generically reduced by oneeach time we find a Gµ–invariant conserved quantity. "

If in Theorem 5.3.1, we consider the case G = {e}, the relativeequilibrium becomes an equilibrium and we obtain the promised gen-eralization of the Lagrange Theorem to Poisson systems.

Corollary 5.3.1 (Energy–Casimir method) Let (M, {·, ·}, h) bea Poisson system, and m " M be an equilibrium of the Hamiltonianvector field Xh. If there is a set of conserved quantities C1, . . . , Cn "C!(M) for which

d(h + C1 + . . . + Cn)(m) = 0,


and

d2(h + C1 + . . . + Cn)(m)|W)W ,

is definite for W defined by

W = kerdC1(m) * . . . * kerdCn(m),

then, m is stable. If W = {0}, m is always stable.

Remark 5.3.4 Even in the symplectic framework, the energy–Casimir method provides a finer stability condition than the LagrangeTheorem, as can be seen in the following elementary example. Con-sider the Hamiltonian system with phase space T "R2, endowed withthe canonical symplectic structure, and Hamiltonian function givenby

h(q1, q2, p1, p2) = (q1)2 $ (q2)2 + p21 $ p2

2.

This system (two–dimensional isotropic harmonic oscillator) has anequilibrium at the point (0, 0, 0, 0). The study of the stability of thispoint by means of the Lagrange Theorem is inconclusive, however theuse of energy–Casimir method with the constants of the motion C1

and C2, defined by

C1(q1, q2, p1, p2) := (q1)2 + p21 and C2(q1, q2, p1, p2) := (q2)2 + p2

2,

guarantees the stability of the point (0, 0, 0, 0). "

Example 5.3.1 Stability of the sleeping Lagrange top as aPoisson equilibrium: The Lagrange top can be described as a Pois-son system on R3)R3 7= se(3), by taking the Poisson structure givenby the (–) Lie–Poisson bracket on se(3)". If we denote by (", #) theelements in R3)R3 this bracket, also called the heavy top bracket,has the form

{F, G}(", #) = $" · (>%F )>%G)$ # · (>%F )>&G$>%G)>&F ).

The Lagrange top Hamiltonian in these variables takes the form

h(", #) =12

0*2

1 + *22

I1+

*23

I3

1+ Mgl+3,


where M is the total mass of the body, l is the distance from thefixed point of the top to its center of mass and the inertia tensor ofthe body is I = diag[I1, I1, I3].

It can be easily verified that the quantities

C1 = "1(<#<2), C2 = "2(" · #), and C3 = "3(*3),

with "1, "2, and "3 arbitrary real smooth functions, are constants ofthe motion, that we will use in Corollary 5.3.1 to study the stabilityof the sleeping top solution, that is, the equilibrium solution given by

m , (0, 0, *3, 0, 0, 1),

with *3 arbitrary. Let f = h + C1 + C2 + C3. It is easy to see that

df(m) = (0, 0,*3

I3+ "$

2(*3) + "$3(*3), 0, 0, Mgl + 2"$

1(1) + *3"$2(*3)).

Hence, taking "1, "2, and "3 such that

"$1(1) = $1

2(Mgl + k*3), "$

2(*3) = k, and "$3(*3) = $kI3 + *3

I3,

with k " R arbitrary, we have that df(m) = 0. With the notation ofCorollary 5.3.1, it may be computed that

W = kerdh(m) * kerdC1(m) * kerdC2(m) * kerdC3(m)= span{(1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0),

(0, 0, 0, 0, 1, 0)}.

Moreover,

d2f(m)|W)W =

5

667

1/I1 0 k 00 1/I1 0 kk 0 $Mgl $ k*3 00 k 0 $Mgl $ k*3

8

99: ,

whose eigenvalues are

+± =1

2I1(1$MglI1 $ kI1*3

±;

4I1(I1k2 + Mgl + k*3) + (1$MglI1 $ kI1*3)2).


The eigenvalues ++ and +% have both the same sign provided that

*23 >

0I1k +

Mgl

k

12

.

Since we are free in the choice of k, the optimal stability condition will

occur when3I1k + Mgl

k

42has a minimum with respect to k, which

happens when k =;

Mgl/I1. Hence an upright sleeping top is stableprovided that

*23 > 4MglI1, (5.3.7)

which is the classical stability condition for a fast top. "

We know from Theorem 4.1.1 that the relative equilibria of a sys-tem coincide with the equilibria of the (eventually singular) reducedspaces associated to it. A reasonable question to ask is what is therelation between the Gµ–stability of a relative equilibrium and thestability of its corresponding reduced equilibrium. A first answer tothis question is provided by the following theorem, which will be im-proved in the next section using some group representation tools. Inwhat follows, we restrict ourselves to the symplectic case and the onlyconstants of the motion that we will use will be the components ofthe momentum map.

Theorem 5.3.2 Let (M, !, h) be a Hamiltonian dynamical systemwith a symmetry given by the Lie group G acting properly on M in aglobally Hamiltonian fashion, with associated equivariant momentummap J : M % g". Assume that the Hamiltonian h " C!(M) is G–invariant. Let m "M be a relative equilibrium such that J(m) = µ "g", Gµ is compact, H := Gm, and - " Lie(NGµ(H)) is its orthogonalvelocity relative to a given AdH–invariant splitting. If the quadraticform

d2(h$ J#)(m)|W)W


kerTmJ = W ( Tm(Gµ · m),

then m is a Gµ–stable relative equilibrium. If dimW = 0, then m isalways a Gµ–stable relative equilibrium.


Moreover, there exists a particular complement of Tm(Gµ · m) inker TmJ for which the stability form block diagonalizes into two blockswith the upper one equal to the stability form of the reduced Hamilto-nian. In particular, if m satisfies the hypothesis of the theorem andhence is Gµ–stable, the equilibrium defined by m on the reduced spaceis stable.

Proof The first part of the statement is the rewriting of Theo-rem 5.3.1 for symplectic manifolds, in the absence of other additionalconserved quantities, apart from the components of the momentummap.

For the second part, the use of the MGS normal form will allowus to reinterpret the statement in terms of reduced spaces and toprove the existence of complements to Tm(Gµ · m) in kerTmJ forwhich the Hessian of the Lyapunov function block diagonalizes. Morespecifically, we shall use the model

Y := G)H (m" ) Vm),

given by the MGS normal form in which the relative equilibriumm is represented by [e, 0, 0]. Since by Theorem 3.2.2, there is a G–equivariant symplectomorphism between a G–invariant neighborhoodof G · m in M and a G–invariant neighborhood of G · [e, 0, 0] in Y ,we can work exclusively in the space Y to prove the theorem. Recallthat J(m) = µ is a fixed element of g" in all that follows.

Since the canonical projection

/Y : G)m" ) Vm $% G)H (m" ) Vm)(g, ., v) -$% [g, ., v],

is a surjective submersion, it follows that kerT(e,0,0)/Y = h){0}){0},and hence

T[e,0,0]Y = T[e,0,0](G)H (m" ) Vm)) 7= g/h)m" ) Vm.

Let JY : [g, ., v] " Y -% g · (µ + . + JVm(v)) " g" be the momentummap given by Theorem 3.2.2. We now characterize the subspaceker T[e,0,0]JY . If w = d

dt

$$t=0

[exp t0, t., tv] " kerT[e,0,0]JY , then

d

dt

$$$$t=0

JY ([exp t0, t., tv]) =d

dt

$$$$t=0

exp t0 · (µ + t. + JVm(tv))

= $ad") · µ + . + T0JVm · v = 0.


Since JVm is homogeneous quadratic on Vm, it follows that T0JVm ·v = 0 and hence the previous relation becomes $ad"

) · µ + . = 0.However, note that ad"

)µ " g#µ, the annihilator of gµ in g", and . " g"µ,therefore, necessarily ad"

)µ = 0 and . = 0 since g#µ * g"µ = {0} in g".Consequently,

ker TmJ 7= ker T[e,0,0]JY7= gµ/h) Vm

7= Tm(Gµ · m)) Vm.

This expression singles out Vm as a complement to Tm(Gµ · m) inkerTmJ.

Recall that by Theorem 3.4.2, (M (H)µ , !(H)

µ ) is locally symplecto-morphic to (V H

m , !Vm |V Hm

) which is a symplectic vector subspace of(Vm, !Vm). Therefore we have the !Vm–orthogonal decomposition

Vm = V Hm ( (V H

m )%Vm , (5.3.8)

where (V Hm )%Vm := {v " Vm | !Vm(v, V H

m ) = 0}. We now define

iVm : Vm $% G)m" ) Vm

v -$% (e, 0, v)

and hVm : Vm % R by hVm = hY & /Y & iVm , that is, hVm(v) =hY ([e, 0, v]), where hY is the Hamiltonian induced by h on the G–invariant neighborhood of [e, 0, 0], by the local symplectomorphismthat maps from a G–invariant neighborhood of m. By construction,hVm is H–invariant (property inherited from the G–invariance of hY ),that is, for any l " H

hVm(l · v) = hY ([e, 0, l · v]) = hY ([l, 0, v]) = hY (l · [e, 0, v])= hY ([e, 0, v]) = hVm(v).

This implies that if FhVmt is the flow on Vm of the Hamiltonian vector

field defined by hVm , then for all l " H

FhVmt & 2l = 2l & F

hVmt ,

where 2l denotes the action of the group element l " H in the left rep-resentation of H on Vm. Therefore, if u " V H

m , the associated Hamil-tonian vector field XhVm

" X(Vm) satisfies l ·XhVm(u) = XhVm

(u) forany l " H and thus

XhVm(u) " V H

m whenever u " V Hm . (5.3.9)


Note that since i : V Hm )% Vm is a symplectic map, we have !V H

m:=

!Vm |V Hm

= i"!Vm and Tui · XhV H

m(u) = XhVm

(u), for any u " V Hm ,

where hV Hm

is the restriction of hVm to V Hm . Therefore, if u, v " V H

m ,we have

!V Hm

(XhV H

m(u), v) = (i"!Vm)(Xh

V Hm

(u), v) = !Vm(XhVm(u), v).

(5.3.10)

Note also that if u and v are arbitrary elements of Vm, then

dhY ([e, 0, v]) · d

dt

$$$$t=0

[e, 0, u + tv] =d

dt

$$$$t=0

hY ([e, 0, u + tv])

=d

dt

$$$$t=0

hVm(u + tv)

= dhVm(u) · v (5.3.11)

which is equivalent, by the definition of a Hamiltonian vector field,to

!Vm(XhVm(u), v) = ![e, 0, u](XhY ([e, 0, u]),

d

dt

$$$$t=0

[e, 0, u + tv]).

Using all these remarks we prove the following intermediate result.

Lemma 5.3.2 The map hVm : Vm % R has a critical point at 0 andtherefore d2hVm(0) is a well-defined quadratic form.

Proof We first take an arbitrary v " V Hm . Then

dhVm(0) · v = !Vm(XhVm(0), v) = !V H

m(Xh

V Hm

(0), v) = 0.

The last equality is proved in the following way. By Theorem 3.4.2,the reduced space M (H)

µ is locally symplectomorphic to V Hm around

the equilibrium [m](H)µ which corresponds to 0 in V H

m . But XhV H

m(0)

is mapped by the derivative of this local symplectomorphism toX

h(H)µ

([m](H)µ ) = 0, since [m](H)

µ is an equilibrium of Xh(H)

µbecause m

is a relative equilibrium.Second, if v " (V H

m )%Vm then

dhVm(0) · v = !Vm(XhVm(0), v) = 0


because by expression (5.3.9), XhVm(0) " V H

m . The direct sum de-composition (5.3.8) guarantees then that dhVm(0) = 0. #

We are now ready to compute d2(hY $ J#Y )([e, 0, 0])

$$$Vm

. Recallthat the definition of the Hessian involves extensions of vector fields.If v, w " Vm are arbitrary, we will denote by V and W " X(Y ),the local constant extensions of d

dt

$$t=0

[e, 0, tv] and ddt

$$t=0

[e, 0, tw] tovector fields on Y . More explicitly,

V([g, ., u]) :=d

dt

$$$$t=0

[g, ., u + tv]

W([g, ., u]) :=d

dt

$$$$t=0

[g, ., u + tw].

Therefore, we can write

d2(hY $ J#Y )([e, 0, 0])(v, w)

:= d2(hY $ J#Y )([e, 0, 0])

0d

dt

$$$$t=0

[e, 0, tv],d

dt

$$$$t=0

[e, 0, tw]1

= Vm[W[hY $ J#Y ]]([e, 0, 0]) =

d

dt

$$$$t=0

(W[hY $ J#Y ])([e, 0, tv])

=d

dt

$$$$t=0

d

ds

$$$$s=0

(hY $ J#Y )([e, 0, tv + sw])

=d

dt

$$$$t=0

d

ds

$$$$s=0

hY ([e, 0, tv + sw])

$ d

dt

$$$$t=0

d

ds

$$$$s=0

J#Y ([e, 0, tv + sw]). (5.3.12)

We compute separately the two summands of the last expression.Firstly,

d

dt

$$$$t=0

d

ds

$$$$s=0

J#Y ([e, 0, tv + sw])

=d

dt

$$$$t=0

d

ds

$$$$s=0

0µ + JVm(tv + sw), -1

=d

dt

$$$$t=0

0TtvJVm · w, -1 = 0

because TtvJVm ·w " h" and the orthogonal velocity - belongs to pµ,which is the orthocomplement to h in nµ = Lie(NGµ(H)).


It is easy to verify that ddt

$$t=0

dds

$$s=0

hY ([e, 0, tv + sw]) =d2hVm(0)(v, w) and thus we get

d2(hY $ J#Y )(m)(v, w) = d2hVm(0)(v, w),

where the right hand side is well-defined by Lemma 5.3.2.We now show that d2(hY$J#

Y )([e, 0, 0]) block diagonalizes relativeto the splitting (5.3.8). We consider two cases:

• If v, w " V Hm , then

d2hVm(0)(v, w) =d

dt

$$$$t=0

!Vm(XhVm(tv), w).

The expression (5.3.10) allows us to write this as

d

dt

$$$$t=0

!V Hm

(XhV H

m(tv), w) =

d

dt

$$$$t=0

dhV Hm

(tv) · w

= d2hV Hm

(0)(v, w).

However, d2hV Hm

(0) is conjugate by the derivative of the lo-cal symplectomorphism between M and Y to d2h(H)

µ ([m](H)µ ),

where, as usual, h(H)µ denotes the reduced Hamiltonian in M (H)

µ

(which is locally symplectomorphic to V Hm ) induced by h.

• If v " V Hm and w " (V H

m )%Vm , then

d2hVm(0)(v, w) =d

dt

$$$$t=0

!Vm(XhVm(tv), w) = 0

because by (5.3.9), XhVm(tv) " V H

m .

These two points imply that the Hessian d2(h $ J#)(m) of theLyapunov function h$J# in the statement of the theorem is conjugateby the derivative of the local symplectomorphism between M and Yto the block diagonal matrix

d2(hY $ J#Y ) ([e, 0, 0])|Vm

= d2hVm(0)

=

<d2hV H

m(0) 0

0 d2hVm(0)$$(V H

m )'Vm

=, (5.3.13)


relative to the symplectic orthogonal splitting

Vm = V Hm ( (V H

m )%Vm .

Since the upper block is conjugate to d2h(H)µ ([m](H)

µ ), the formalstability of the relative equilibrium m that is, the definiteness ofd2(hY $ J#

Y )([e, 0, 0])$$$Vm

= d2hVm(0), implies the stability of [m](H)µ

on M (H)µ , but clearly not vice versa. !

Remark 5.3.5 Notice that if the action is free (H = {e}), the ma-trix in (5.3.13) becomes, up to conjugation, d2hµ([m]µ); in otherwords, the Gµ–stability of m is implied by the formal stability of theassociated equilibrium [m]µ "Mµ.

In view of this comment, the proof of the first part of Theo-rem 5.3.2 if the action is free and G is compact is remarkably simple.If m " M is the relative equilibrium for which d2(h$ J#)(m)

$$Vm

isdefinite then, the associated equilibrium [m]µ " Mµ is stable sinced2hµ([m]µ) is definite. In terms of the MGS normal form aroundG · m, M looks locally like Y = G) g"µ ) Vm and Mµ is locally sym-plectomorphic to Vm. One may reduce the proof to the case µ = 0by using symplectic slicing techniques similar to those introduced inSection 4.2 (see [LS] for a detailed exposition of this procedure), inwhich case it is enough to show G–stability. Let U = G)Ug#)UVm bea G–invariant open neighborhood of m , (e, 0, 0), with Ug# and UVm

open neighborhoods of 0 " g" and 0 " Vm respectively. Since 0 " Vm

is stable under the dynamics induced by h0, there exists an openneighborhood OVm ' UVm of 0 " Vm such that for each w " OVm ,F 0

t (w) " UVm for any t, with F 0t the Hamiltonian flow corresponding

to the reduced Hamiltonian h0. Also, let O0 ' Ug# be an open neigh-borhood of 0 " g" such that for each . " O0, the coadjoint orbit of., O$, is such that O$ ' Ug# (the compactness of G makes possiblethis choice). We now see that V = G)O0)OVm is the open set thatwe need to prove G–stability, that is, if (g, ., v) " V is arbitrary, thenFt(g, ., v) := (g(t), .(t), v(t)) " U for all t 4 0. Indeed, the stabilityof [m]0 and the choice of OVm imply that v(t) " UVm for all t 4 0.Conservation of momentum

JY (g(t), .(t), v(t)) = JY (g, ., v)

and the explicit formula for JY yield g(t) · .(t) = g · ., which impliesthat .(t) " O$ ' O0 ' Ug# . This proves that Ft(g, ., v) " U for anyt 4 0. "


Remark 5.3.6 We will see in Section 5.4 that the block diagonal-ization in (5.3.13) is a consequence of deeper group theoretical con-siderations; however, the computation carried out in the proof ofTheorem 5.3.2 illustrates very well the importance of the role playedby the stability of the associated equilibrium in the stability of therelative equilibrium and served as an excuse to introduce some toolsthat will be used in the considerations that follow. "

Under certain additional hypothesis, like for example if the coad-joint action of Gµ on g"µ is proper, Theorem 5.3.2 can be viewed as acorollary to a theorem by Montaldi [MO97, Theorem 1.2 (i)] on whichwe now comment. Let hµ : J%1(µ)/Gµ % R be the point reducedHamiltonian introduced in Theorem 2.2.2, that is, the continuousmap defined on the Hausdor! topological space Mµ := J%1(µ)/Gµ,such that the diagram

J%1(µ) )% Mh$% R

&µ* ?hµ

J%1(µ)/Gµ

is commutative.Notice that using the coordinates provided by the MGS normal

form, Theorem 3.4.1 allows us to write, locally around [m]µ = /µ(m)

Mµ 2 J%1Vm

(0)/H.

At the same time, the G–invariance of h implies that in these coordi-nates, h can be considered as a H–invariant function of m" and Vm,that is

h , h(m", Vm).

Therefore, hµ can be considered as the H-invariant restriction of hto {0}) J%1

Vm(0):

hµ , h|{0})J"1Vm

(0). (5.3.14)

Also, with this notation

hVm , h|{0})Vm. (5.3.15)


Proposition 5.3.1 In the hypothesis of Theorem 5.3.2 the point[m]µ " Mµ is an extremal equilibrium, that is, hµ has a local ex-tremum at [m]µ in Mµ.

Proof From (5.3.13), it follows that the hypothesis of Theorem 5.3.2is equivalent to saying that d2hVm(0) is definite, that is, hVm has anstrict extremum at 0 " Vm. Looking at the characterizations (5.3.14)and (5.3.15), it is clear that hµ is the restriction of hVm to {0} )J%1

Vm(0) and therefore it also has an strict extremum at 0, which in

Mµ amounts to proving that hµ has an extremum at [m]µ. !We now quote the following:

Theorem 5.3.3 Let (M, !, h) be a Hamiltonian dynamical systemwith a symmetry given by the Lie group G acting properly on M ina globally Hamiltonian fashion, and properly on g" by the coadjointaction, with associated equivariant momentum map J : M % g".Assume that the Hamiltonian h " C!(M) is G–invariant. Let m "M be a relative equilibrium such that J(m) = µ " g", and [m]µ isextremal in Mµ. Then, the equilibrium [m] = /(m) is stable in M/G,with / : M %M/G the continuous canonical projection.

Proof If the coadjoint action of G on g" is proper, g"/G is a Haus-dor! topological space (see Proposition 1.3.1). The rest follows fromRemark 1.3 (d) and Theorem 1.2 (i) in [MO97]. !

Theorem 5.3.2 follows directly from the previous result if we as-sume, in addition, that the coadjoint action is proper: firstly, theproblem may be reduced to the case µ = 0 using symplectic slicingtechnique as in [LS]. Secondly, the previous Proposition guaranteesthat in the conditions of Theorem 5.3.2, [m]0 is an extremal equilib-rium in M0. It follows then that m is G–stable (Gµ–stable in thegeneral case) by Theorem 5.3.3.

Remark 5.3.7 The approach based on [MO97] that we just com-mented on, is stronger than Theorem 5.3.2, in the sense that it re-quires less conditions on the relative equilibrium to conclude stability.This can be seen by looking at (5.3.14) and (5.3.15); Montaldi’s resultrequires extremality on a smaller set (namely J%1

Vm(0)) than the one

for Theorem 5.3.2 (the entire Vm); however, the latter result presentsthe advantage of computability using di!erential calculus (that is notavailable on Mµ since it is not a smooth manifold in general) and,


more importantly, it has a natural generalization to periodic andrelative periodic orbits, as we will see in the next chapter. The com-putability feature is particularly important in the Lagrangian andcotangent bundle scenarios, where very sophisticated tools like thereduced energy-momentum method of [SLM89, SLM91] and theLagrangian block diagonalization method [Lew92] have been de-veloped using stability formulations parallel to Theorem 5.3.2. "

5.4 Block Diagonalization of the StabilityForm

In Theorem 5.3.2 we proved that one way to insure the stabilitymodulo Gµ of a relative equilibrium was showing the definiteness ofthe Hessian of its associated Lyapunov function in a complement toTm(Gµ ·m) in kerTmJ. This is what we called the stability form. Inapplications, the dimensionality of this form may be large therefore,it will be convenient to express it in block diagonal form, if possible,to make the definiteness analysis simpler. In the second part of The-orem 5.3.2, we took the first step in this direction. Additionally, itwas shown using an elementary block diagonalization, how the sta-bility of the associated equilibrium in the reduced space is related tothe stability modulo Gµ of the relative equilibrium. The main limita-tion of this result (expression (5.3.13)) is that it is a purely abstractstatement, in other words, using the MGS normal form we provedthat such a block diagonalization exists but since the symplectomor-phism that links the model Y with M is not explicit (it comes fromthe Constant Rank Embedding Theorem), it is not always possibleto implement it in particular cases.

In what follows, we give a method to achieve block diagonalizationin a completely explicit manner.

Our starting point will be a splitting due to Arms, Fischer, andMarsden [AFM75] (AFM splitting) generalizing the so called Mon-crief decomposition which will be recalled below. In all that fol-lows, we fix m " M , denote Gm := H, and assume that N(H) iscompact (and therefore, by Lemma 2.3.2, so is L = N(H)/H). SinceH is compact, we can always find an H–invariant metric 0·, ·1 on M .We define T "

mJ : g % TmM to be the dual map of TmJ : TmM % g"

followed by the identification of T "mM with TmM by the metric, that

§ 5.4. Block Diagonalization of the Stability Form 171

is,

0T "mJ(-), vm1 = TmJ(vm) · - for all - " g , vm " TmM. (5.4.1)

Note that this definition implies that

kerT "mJ = (rangeTmJ)#,

where

(rangeTmJ)# = {- " g | TmJ(vm) · - = 0 for all vm " TmM}

is the annihilator in g of rangeTmJ ' g". In particu-lar, dim (rangeT "

mJ) = dim (M) $ dim (kerT "mJ) = dim (M) $

dim ((rangeTmJ)#) = dim (rangeTmJ). In addition, a simple ver-ification shows that rangeT "

mJ @ (kerTmJ)(, where (ker TmJ)( isthe orthogonal complement of kerTmJ in TmM relative to the H–invariant metric 0·, ·1. These two facts immediately imply

rangeT "mJ = (kerTmJ)( (5.4.2)

and hence

TmM = kerTmJ( rangeT "mJ . (5.4.3)

We also have the linear map

$m : gµ $% TmM- -$% -M (m)

and proceeding as before, we conclude

TmM = range$m ( ker$"m (5.4.4)

where $"m : TmM % g"µ is defined by $"m(vm) · . = 0vm,$m(.)1 for all. " gµ, vm " TmM . The infinitesimal equivariance of J states thatTmJ · -M (m) = $ad"

#J(m), which implies range$m ' ker TmJ. Thus(5.4.4) implies the orthogonal direct sum decomposition

ker TmJ = range$m ( (kerTmJ * ker$"m) (5.4.5)

and hence the finer orthogonal decomposition of TmM [AFM75,Theoreme 2.3],

TmM = range$m ( (kerTmJ * ker$"m)( rangeT "mJ

= Tm(Gµ · m)( (kerTmJ * Tm(Gµ · m)()( rangeT "mJ(5.4.6)

7= Tm(Gµ · m)( kerTmJ/Tm(Gµ · m)( Tm(G · m) ,


where the last line should be understood as an external direct sum.We define

W := kerTmJ * ker$"m = kerTmJ * Tm(Gµ · m)(.

By the Reduction Lemma, we have

Tm(G · m)% = kerTmJ,

Tm(Gµ · m) = Tm(G · m) * ker TmJ, (5.4.7)

which implies that

2m((kerTmJ)() = !m(Tm(G · m)),

2m(Tm(G · m)() = !m(kerTmJ), (5.4.8)

where 2m : TmM % T "mM and !m : TmM % T "

mM are the iso-morphisms associated to the metric and the symplectic structurerespectively. Indeed, by (5.4.2), any element of 2m(ker(TmJ)()is of the form 0T "

mJ · -, ·1, for some - " g. By (5.4.1) and thedefinition of the momentum map, for any vm " TmM , we have0T "

mJ · -, vm1 = TmJ(vm) · - = !(m)(-M (m), vm), which is equiva-lent to the first identity above. The second is proved in a similarway.

Proposition 5.4.1 The subspace W = kerTmJ*ker$"m = kerTmJ*Tm(Gµ ·m)(, constructed using the H–invariant metric 0·, ·1, has thefollowing properties:

(i) (W, !(m)|W ) is a symplectic subspace of (TmM, !(m)).

(ii) W is H–invariant.

(iii) The symplectic vector subspace (WH ,!(m)|W H ) isnaturally symplectomorphic to the tangent space(T

[m](H)µ

M (H)µ , !(H)

µ ([m](H)µ )) of the reduced space (M (H)

µ , !(H)µ ).

Proof (i) Let v " W such that !(m)(v, w) = 0 for all w " W .If u = .M (m) " range$m for some . " gµ, then !(m)(v, u) = 0,since v " kerTmJ = Tm(G · m)%, by (5.4.7). This implies that v "(range$m)% *W% = (range$m (W )% = (kerTmJ)% = Tm(G · m),by (5.4.5) and (5.4.7). Since v " W ' ker TmJ, we therefore have


v " kerTmJ* Tm(G ·m) = Tm(Gµ ·m) = range$m, again by (5.4.7).But range$m *W = {0} and hence v = 0.(ii) Let v " W and h " H ' Gµ be arbitrary. By H–invariance ofthe metric 0·, ·1, for any .M (m) " range$m, that is, . " gµ, we have

0h · v, .M (m)1 = 0v, h%1 · .M (m)1 = 0v, (Adh"1.)M (m)1 = 0(5.4.9)

since v " W ' (Tm(Gµ · m))(, h · m = m, and Adh"1. " gµ. Thush · v " (range$m)(.

Analogously, if w " (kerTmJ)(, by (5.4.8), there is a + " g forwhich

0h · v, w1 = !(m)(h · v,+M (m)) = !(m)(v, (Adh"1+)M (m))= $TmJ(v) · Adh"1+ = 0, (5.4.10)

by H–invariance of !, the definition of the momentum map, andsince v " W ' ker TmJ. Thus, h · v " kerTmJ and we conclude thath · v " ker TmJ * (range$m)( = W .(iii) Since, by hypothesis, the subgroup N(H) is compact, the sin-gular symplectic point reduced space (M (H)

µ , !(H)µ ) is naturally sym-

plectomorphic to (K%1L (+#)/L($ , !($). Since the canonical projection

/($ : K%1L (+#)% K%1

L (+#)/L($ is such that kerTmKL = Tm(L($ ·m),we can identify T

[m](H)µ

M (H)µ with TmK%1

L (+#)/Tm(L($ · m). By the

definition of WH and recalling that

kerTmJ * TmMH = Tm(J%1(µ) *MH) = Tm(K%1L (+#)) = kerTmKL,

we write

WH = kerTmJ * TmMH * Tm(Gµ · m)(

= TmK%1L (+#) * Tm(Gµ · m)( ' TmK%1

L (+#).

We will show that the linear map

( : WH $% TmK%1L (+#)/Tm(L($ · m)

w -$% w + Tm(L($ · m)

is an isomorphism.


If w + Tm(L($ · m) = Tm(L($ · m), then w = +M (m) for some+ " l(0 . Since L($ = NGµ(H)/H, there is a , " Lie(NGµ(H)) ' gµ

which projects to +. Therefore, by the definition of the action of L(0 ,we have ,M (m) = +M (m) = w "W . Since W *Tm(Gµ ·m) = {0}, itfollows that ,M (m) = 0 and hence w = 0, proving that ker( = {0}and hence injectivity of (. We now show that ( is surjective. Letw + Tm(L($ · m) " TmK%1

L (+#)/Tm(L($ · m) with w " TmK%1L (+#) '

ker TmJ. By the AFM decomposition (5.4.5), w can be uniquelyexpressed as an orthogonal sum

w = .M (m) + w$ with . " gµ and w$ "W.

Since TmK%1L (+#) = kerTmJ * TmMH ' (TmMH)H the vector w is

also H–fixed, that is,

h · w = w for all h " H.

We therefore conclude that

h · .M (m) + h · w$ = .M (m) + w$ for all h " H. (5.4.11)

Because h ·.M (m) = (Adh.)M (m) " Tm(Gµ ·m) and, by part (ii), Wis H–invariant, formula (5.4.11) implies that h ·w$ = w$ for all h " H,that is, w$ " WH ' TmK%1

L (+#), and that h · .M (m) = .M (m) forall h " H. This latter equality is equivalent to Adh. $ . " h for allh " H, which by Lemma 1.3.2 implies that . lies in the Lie algebraof N(H). Therefore, .M (m) " Tm(NGµ(H) · m) = Tm(L($ · m) sothere exists some , " l($ such that .M (m) = ,M (m). Consequently,by the definition of (, we have ((w$) = w +Tm(L($ ·m). ( is hencesurjective.

Finally, we prove that ( is a symplectic linear map. Let [v]($ :=v + Tm(L($ · m) and [w]($ := w + Tm(L($ · m) be arbitrary ele-ments in TmK%1

L (+#)/Tm(L($ · m) 7= T[m]#$(K%1

L (+#)/L($). Noticethat if we use the latter isomorphism (which depends on the choiceof the representative m " [m]($) the quotient map TmK%1

L (+#) %TmK%1

L (+#)/Tm(L($ ·m) can be naturally identified with Tm/(0 . Thedefinition of ( shows that it is the restriction of this quotient mapto WH . In particular, since ( is surjective, v and w can always bechosen in WH . The definition of the reduced symplectic form !(0

implies then that

!($([m]($)(((v), ((w)) = !($(/($(m))(Tm/($(v), Tm/($(w))= !(m)(v, w),


which shows that ( is symplectic. !

Definition 5.4.1 The subspace W = kerTmJ*Tm(Gµ ·m)( is calledthe orthosymplectic subspace through m " M associated to theH–invariant metric 0·, ·1.

Theorem 5.4.1 Assume the hypotheses and notations of Theo-rem 5.3.2 and that, in addition, the group N(H) is compact. Let W bethe orthosymplectic subspace through m "M , associated to some H–invariant metric g = 0·, ·1. Then, the restriction d2(h$ J#)(m)

$$W)W

of the Hessian of the Lyapunov function h$ J# to W has the form

d2(h$ J# )(m)|W)W

=

5

6667

d2(h$ J#)(m)$$W H)W H 0

0A1 · · · 0... . . . ...0 · · · Ar

8

999:, (5.4.12)

where A1, . . . , Ar are the restrictions of d2(h$J#)(m) to the isotypiccomponents of the H–action on W1, . . . , Wr of (WH)%(m)|W . In ad-dition, d2(h$ J#)(m)

$$W H)W H = d2h(H)

µ ([m](H)µ ), if T

[m](H)µ

M (H)µ is

identified with WH via the natural symplectomorphism in Proposi-tion 5.4.1, where d2h(H)

µ ([m](H)µ ) is the stability form associated to

the equilibrium [m](H)µ "M (H)

µ .

Proof We prove first that all the elements in the statement are well–defined. By Proposition 5.4.1, the orthosymplectic space W is aHamiltonian H–space and, by Proposition 1.3.2, the set of H–fixedvectors WH of W , is a symplectic subspace of W . Thus we can writethe following symplectic direct sum

W = WH ( (WH)%(m)|W . (5.4.13)

Both summands of (5.4.13) are H–invariant and WH is the triv-ial isotypic component of W . Since (WH)%(m)|W is a H–invariantcomplementary subspace of WH , it contains the remaining isotypiccomponents of W . Therefore

W = WH (W1 ( · · ·(Wr and (WH)%(m)|W = W1 ( · · ·(Wr,(5.4.14)


are the isotypic decompositions of W and (WH)%(m)|W respectively.The block diagonal nature of the stability form is a consequence

of the following lemma.

Lemma 5.4.1 The stability bilinear form d2(h$ J#)(m)$$W)W

, con-sidered as a linear map A : W %W , is H–equivariant.

Proof Let {e1, . . . , en} be a basis of W , and {*1, . . . , *n} be thecorresponding dual basis of W ". The linear map A : W % W isdefined by the relation

0*i, A(ej)1 = d2(h$ J#)(m)(ei, ej),

where 0·, ·1, denotes the natural pairing between W " and W .Clearly, the matricial expressions for the bilinear form

d2(h$ J#)(m)$$W)W

and the linear map A in the basis {e1, . . . , en}coincide; moreover, A is H–equivariant i! d2(h$ J#)(m)

$$W)W

is H–invariant. Indeed, for any h " H and any i, j " {1, . . . , n} we havethe equivalences:

d2(h$ J#)(m)(h · ei, h · ej) = d2(h$ J#)(m)(ei, ej)

AB 0h · *i, A(h · ej)1 = 0*i, A(ej)1AB 0*i, h%1 · A(h · ej)1 = 0*i, A(ej)1AB A(h · ej) = h · A(ej).

We now prove the H–invariance of d2(h$ J#)(m)$$W)W

. First,note that the G–invariance of h trivially implies its H–invariance.Second, recall that - is an orthogonal velocity of m relative to someAdH–invariant Riemannian metric. Lemma 5.3.1 guarantees thenthat Adh- = - for all h " H. Therefore, if z " M and h " H arearbitrary, we have

J#(h · z) = 0J(h · z), -1 = 0Ad"h"1J(z), -1

= 0J(z),Adh"1-1 = 0J(z), -1 = J#(z).

Thirdly, Hessians commute with pull–backs. Thus, for v, w "W andh " H, we have

d2(h$ J#)(m)(h · v, h · w) =32"h(d2(h$ J#)(m))

4(v, w)

= d2(2"hh$ 2"hJ#)(h · m)(v, w)

= d2(h$ J#)(m)(v, w) ,


since m " MH ; 2h denotes both the H–action on M (in the secondequality) and its linearization on W (in the first equality). #

Part (iii) of Theorem 1.3.1 together with Lemma 5.4.1 and (5.4.14)imply the block diagonal form (5.4.12). Recall that the reducedspaces (M (H)

µ ,!(H)µ ) and (K%1

L (+#)/L($ , !($) are naturally symplec-tomorphic. We shall identify from now on these two symplectic man-ifolds. Thus, the proof of the Theorem is completed if we showthat d2(h$ J#)(m)

$$W H)W H = d2h(H)

µ ([m](H)µ ), via the identifica-

tion of T[m](H)

µM (H)

µ7= TmK%1

L (+0)/Tm(L(0 · m) with WH . Recall

from Proposition 5.4.1 (iii) that the quotient map TmK%1L (+0) %

TmK%1L (+0)/Tm(L(0 · m) restricted to WH is an isomorphism and

hence we have

T[m]

(H)µ

M (H)µ = {Tm/

(H)µ · v | v "WH}.

We will now prove the following technical result.

Lemma 5.4.2 Any v " WH can be expressed as v = ddt

$$t=0

F vt (m),

with F vt the Hamiltonian flow of a N(H)–invariant function g "

C!(M)N(H).

Proof Recall that, from propositions 2.3.1 and 2.3.2, and by thecompactness of N(H), the action of the compact Lie group L onMH is free and globally Hamiltonian, hence, the MGS normal formin this situation says that the symplectic manifold MH is locally L–equivariantly symplectomorphic around the relative equilibrium m "MH to YH := L) l"(0

)VL , where VL := kerTmKL/Tm(L(0 ·m) is thesymplectic normal space at m to the orbit L ·m , l(0 is the coadjointisotropy of L at +0 = KL(m) " l", and l = l(0 ( s is an orthogonaldecomposition relative to an AdN(H)–invariant inner product on l.The point m " MH corresponds under this symplectomorphism to(e, 0, 0) " YH .

Let v " WH = [kerTmJ * Tm(Gµ · m)(]H = kerTmJ * TmMH *[Tm(Gµ · m)]( = kerTmKL * [Tm(Gµ · m)](. We can replace aL–invariant neighborhood of m " M with a symplectomorphic L–invariant neighborhood of (e, 0, 0) " YH . Under the derivativeof this symplectomorpism, the tangent space TmMH correspondsto l($ ) s ) l"($

) VL, the momentum map KL : MH % l" toKLYH

: (g, ., v) " YH -% g · (+0 + .) " l", and the vector v " WH to


(1, 0, 0, u) for some 1 " l($ and u " VL. The reconstruction equa-tions corresponding to this particular MGS normal form guaranteethat, taking a N(H)–invariant function gH " C!(l"($

) VL) on YH

such that Dl##$gH(0, 0) = 1 and DVLgH(0, 0) = !VL(u, ·), whose exis-

tence may readily be verified, the flow F gH

t of the Hamiltonian vectorfield on YH defined by gH , satisfies

d

dt

$$$$t=0

F gH

t (e, 0, 0) = (1, 0, 0, u) 2 v.

However, we need a flow satisfying this condition but whoseHamiltonian function is defined on the entire manifold M and isN(H)–invariant. First, we induce via the local symplectomorphismaround m between MH and YH a N(H)–invariant function on aN(H)–invariant neighborhood of m in MH and then use Proposi-tion 1.3.1 to extend this function to a N(H)–invariant function onMH . Second, using again Proposition 1.3.1, we extend this functionto a N(H)–invariant function g " C!(M)N(H) on M . Let g|MH

de-note the restriction of this function to MH . By the previous relation,the flow of the Hamiltonian vector field of g|MH

on MH satisfies

Xg|MH(m) =

d

dt

$$$$t=0

Fg|MH

t (m) =d

dt

$$$$t=0

F gH

t (e, 0, 0) = v.

On the other hand, by the N(H)–invariance of g on M , F gt satisfies

F gt & 2n = 2n & F g

t for any n " N(H),

where 2n denotes the N(H)–action on M . This guarantees thatF g

t (m) "MH . Since MH is an open subset of MH and m "MH , bycontinuity, there is a t# > 0 such that

F gt (m) "MH whenever t < t#.

This relation, the symplectic character of the natural inclusion i :MH )%M , and g|MH

= g & i, imply

Tmi · Xg|MH(m) = Xg(i(m))

and hence

Xg(m) =d

dt

$$$$t=0

F gt (m) = Tmi · Xg|MH

(m) = v


as required. #

The entries of the (1, 1)–block that we want to compute have theexpressions

d2(h$ J#)(m)(v, w), for arbitrary v, w "WH .

By Lemma 5.4.2, there is a N(H)–invariant function g on M , whoseHamiltonian flow F v

t satisfies v = ddt

$$t=0

F vt (m). We extend w to a

vector field W along F vt (m) by setting

W(F vt (m)) = TmF v

t · w.

By the definition of the Hessian we get

d2(h$J#)(m)(v, w) = v[W[h$ J#]]

=d

dt

$$$$t=0

W[h$ J#](F vt (m))

=d

dt

$$$$t=0

(dh(F vt (m)) · TmF v

t · w $ dJ#(F vt (m)) · TmF v

t · w).

We compute the two summands of this expression separately. Thesecond term equals:

d

dt

$$$$t=0

dJ#(F vt (m)) · TmF v

t · w

=d

dt

$$$$t=0

!(F vt (m))(XJ!(F

vt (m)), TmF v

t · w)

=d

dt

$$$$t=0

!(F vt (m))(-M (F v

t (m)), TmF vt · w).

(5.4.15)

Since - " Lie(NGµ(H)) and F vt is N(H)–equivariant,

-M (F vt (m)) =

d

dt

$$$$t=0

exp t- · F vt (m)

=d

dt

$$$$t=0

F vt (exp t- · m)

= TmF vt · -M (m)


hence the expression (5.4.15) equals

d

dt

$$$$t=0

!(F vt (m))(TmF v

t · -M (m), TmF vt · w)

=d

dt

$$$$t=0

((F vt )"!) (m)(-M (m), w)

=d

dt

$$$$t=0

!(m)(-M (m), w) = 0,

because F vt is a symplectomorphism. Thus, the second term vanishes.

Hence,

d2(h$ J#)(m)(v, w) =d

dt

$$$$t=0

dh(F vt (m)) · TmF v

t · w. (5.4.16)

We will use this identity to prove that

d2(h$ J#)(m)(v, w) = d2h(H)µ ([m](H)

µ )(Tm/(H)µ · v, Tm/

(H)µ · w),

for all v, w " WH . Recall that h(H)µ is defined by the relation

h(H)µ & /(H)

µ = h & i(H)µ , where, if we identify (M (H)

µ ,!(H)µ ) with

(K%1L (+#)/L($ ,!($), we can take for /(H)

µ and i(H)µ (abusing the no-

tation) the canonical projection /(H)µ : K%1

L (+#) % K%1L (+#)/L($

and immersion i(H)µ : K%1

L (+#) )% M . The vector Tm/(H)µ · w "

T[m](H)

µM (H)

µ can be extended, using F vt , to a vector field W(H)

µ along

/(H)µ (F v

t (m)) by

W(H)µ (/(H)

µ (F vt (m))) = Tm(/(H)

µ & F vt ) · w.

Then, since dh(H)µ ([m](H)

µ ) = 0, we get

d2h(H)µ ([m](H)

µ )(Tm/(H)µ · v, Tm/

(H)µ · w)

=d

dt

$$$$t=0

dh(H)µ ((/(H)

µ & F vt )(m))(Tm(/(H)

µ & F vt ) · w)

=d

dt

$$$$t=0

d(h(H)µ & /(H)

µ )(F vt (m)) · TmF v

t · w

=d

dt

$$$$t=0

d(h & i(H)µ )(F v

t (m)) · TmF vt · w.

which coincides with expression (5.4.16) proving our claim. !


SUMMARY OF THE METHOD

We have shown that taking the orthosymplectic subspace

W = kerTmJ * Tm(Gµ · m)(

constructed with a H–invariant metric, the relative equilibrium m isGµ–stable if the symmetric matrix

5

6667

d2(h$ J#)(m)$$W H)W H 0

0A1 · · · 0... . . . ...0 · · · Ar

8

999:, (5.4.17)

is definite, where A1, . . . , Ar are the restrictions of d2(h $ J#)(m)to the isotypic components W1, . . . , Wr of the H-space (WH)%(m)|W ,with - an orthogonal velocity of m with respect to certain AdH -invariant splitting of Lie(NGµ(H)). In addition, relative to the natu-ral symplectomorphism in Proposition 5.4.1 (iii), the (1, 1)–block isgiven by

d2(h$ J#)(m)$$$W H)W H

= d2h(H)µ ([m](H)

µ ), (5.4.18)

where d2h(H)µ ([m](H)

µ ) is the stability form associated to the equilib-rium [m](H)

µ . Notice that this result is an explicit implementationof the abstract result introduced in Theorem 5.3.2 that allows us todetermine the stability of a relative equilibrium in terms of the sta-bility of the associated reduced equilibrium, that is, given a relativeequilibrium m " M , theorems 5.3.2 and 5.4.1 guarantee that m isstable modulo Gµ if the following three conditions are satisfied:

(i) The bilinear form d2h(H)µ ([m](H)

µ ) is definite and, therefore, theassociated singular reduced equilibrium is stable;

(ii) The bilinear forms d2(h$ J#)(m)$$Wi)Wi

are definite, for anyi " {1, . . . , r}, with - an orthogonal velocity of m;

(iii) All the definite bilinear forms in (i) and (ii) have the same sign.

Finally, as we already remarked, the orthogonal velocity of a rel-ative equilibrium is only uniquely determined if the AdH–invariant


inner product on Lie(NGµ(H)) is specified. Since there are many suchinner products, in analyzing the Gµ–stability of the relative equilib-rium m one has to optimize over all such choices. This needs to bedone on a case by case basis. We will carry this out explicitly in theexample below.

5.5 An Example: the Stability of the SleepingLagrange Top

One of the simplest systems that exhibits singular relative equilibriais the Lagrange top in the upright position, that is, an axisymmetricrigid body with a fixed point moving steadily in a gravitational fieldin such a fashion that the axis of symmetry, the center of mass vector,and the axis of gravity are all parallel.

The stability of this relative equilibrium is a classical result inelementary mechanics. We already explored this system in Ex-ample 5.3.1 considering it as a Poisson reduced system and stud-ied its stability using the energy–Casimir method (see also [Hal85]and [Lal92] for a study based on the reduced energy–momentummethod and for its spectral stability analysis). We will show be-low that using the results presented in the previous section, the sameclassical optimal stability result can be obtained, thinking of the La-grange top as a system in the cotangent bundle of SO(3).

The notation used here is identical to the one in [Lal92]and [MR94]. In particular, the elements of TSO(3) in spatial repre-sentation (that is, right trivialization) will be expressed as (#,>!"#)where # " SO(3) and >!" is the skew–symmetric matrix associated to!" " R3 via the relation >!" x = !")x. Analogously, the elements ofT "SO(3) have the form (#, '##) with # " R3. The symbol g = ge3

denotes the gravity vector, where {e1, e2, e3} is a spatial orthonormalbasis of R3. We define the mass vector by M :=

?B 1ref (X)Xd3X,

where B is a reference configuration. If m is the total mass of thebody and l is the distance from the fixed point to the center of mass,then | M |= ml. The reference inertia tensor Iref is defined as

Iref :=&

B1ref (X)(| X |2 I3 $X CX)d3X

and the current spatial inertia tensor is given by

I# := #Iref#T .

§ 5.5. Example: the Sleeping Lagrange Top 183

If m := #M is the spatial representation of the mass vector, in thesevariables, the Hamiltonian of the heavy top is given by

h(#,#) := m · g +12# · I%1

# #

and its Lagrangian is

L(#,$) :=12$ · I#$ $ g · #M.

The symmetry properties of the Lagrange top allow us to choose,without loss of generality, Iref = diag [I1, I1, I3] for some constantsI1 and I3. The reference configuration will be chosen in such a waythat the axis of symmetry of the top will be parallel to the gravityvector, that is, M = mle3.

The symmetries of this system are given by the Hamiltonian ac-tion of the Abelian Lie group G = S1 ) S1 on the phase space of thesystem, that is, T "SO(3). Using spatial variables, the G–action onthe space of velocities has the form

G) TSO(3) $% TSO(3)((71, 72), (#, !")) -$% (exp(71>e3)# exp($72>e3), exp(71>e3)!")

and on the phase space

G) T "SO(3) $% T "SO(3)((71, 72), (#,#)) -$% (exp(71>e3)# exp($72>e3), exp($71>e3)#) .

The infinitesimal generators associated to the G–action on the con-figuration space Q = SO(3) are given by

(-,!)Q(#) =d

dt

$$$$t=0

exp(t->e3)# exp($t!>e3) = ->e3#$ !#>e3

= ->e3#$ !#>e3#T# = (->e3 $ !@#e3)#,

for some -, ! " Lie(S1) = R. We hence have that, in spatial coordi-nates

(-,!)Q(#) = (#, -e3 $ !#e3).

In the same spatial coordinates, the momentum map J : T "SO(3)%g" = R2, corresponding to the lifted action is given by

0J(#, #), (-,!)1 = 0(#,#), (#, -e3 $ !#e3)1 = -# · e3 $ !# · #e3,


hence

J(#,#) = (# · e3, $# · #e3).

We now show how any sleeping top is a relative equilibrium, inother words, every point in T "SO(3) of the form FL((-,!)Q(I)), forcertain (-,!) " g and I the identity element of SO(3), is a relativeequilibrium. The symbol FL denotes the fiber derivative of the func-tion L which is a vector bundle map TSO(3)% T "SO(3). If we takez = FL((-,!)Q(I)) = (I, (- $ !)I3e3) := (I,#e), by Theorem 4.1.1,z is a relative equilibrium i! there is an element ($1,$2) " g = R2

for which

d(h$ J(,1,,2))(z) = 0.

Since J(,1,,2)(#, #) = $1# · e3 $ $2# · #e3, we get

dJ(,1,,2)(#, #)((#, !#) = $1!# · e3 $ $2!# · #e3 $ $2# ·>!"#e3,

where (# := >!"#. Hence,

dJ(,1,,2)(z)((#, !#) = ($1 $ $2)!# · e3 $ $2(- $ !)I3e3 ·>!"e3

= ($1 $ $2)!# · e3,

where we have used the relation e3 · >!"e3 = e3 · (!" ) e3) = 0. Onthe other hand, since

h(#,#) = mgle3 · #e3 +12# · #I%1

ref#T #,

the first derivative is

dh(#,#)((#, !#) = mgle3 ·>!"#e3 + !# · #I%1ref#

T # + # ·>!"#I%1ref#

T #,

which evaluated at z gives

dh(m)((#, !#) = (- $ !)!# · e3 + (- $ !)2I3e3 ·>!"e3 = (- $ !)!# · e3.

Hence the derivative of the Lyapunov function equals to

d(h$ J(,1,,2))(z)((#, !#) = ((- $ !)$ ($1 $ $2))!# · e3.

Therefore, in order to have d(h$J(,1,,2))(z) = 0 and consequently toprove that z is a relative equilibrium we just need to take ($1 $ $2)


such that + := (- $ !) = ($1 $ $2). Moreover, according to ourgeneral definition, (-,!) is a velocity for the relative equilibrium z.This relative equilibrium will be called upright sleeping top. Wewill see that its stability depends on the parameter +. A top whose tippoints in the direction of the gravity vector is called a hanging topand the study of its stability is handled analogously to the uprightsleeping case.

It should be immediately noticed that z has non–trivial symme-try. Indeed if (71, 72) " G is such that (71, 72) · z = z, that is,(exp((71, 72)>e3), (- $ !)I3e3) = (I, (- $ !)I3e3), then 71 = 72 andthus

H = {(71, 72) " G | 71 = 72}.

It is also easy to check that

(T "SO(3))H = {(exp(4>e3), /e3) | 4 " Lie(S1) = R, / " R}.

Let µ = J(z) = ((- $ !)I3, $(- $ !)I3) be the momentum valueof z. Since G is Abelian, Gµ = G and NGµ(H) = G hence, in thisparticular case, L := N(H)/H = G/H. Now, using the notationof singular reduction, it is easy to see by dimension count that thesingular reduced space

T "SO(3)(H)µ 2 K%1

L (+#)/L($ = K%1L (+#)/(G/H)

is trivial. Therefore the (1, 1) block of the stability form vanishes.We now determine ker TzJ. Let ((#, !#) " Tz(T "SO(3)) be such

that >(# = ddt

$$t=0

#(t) and !# = ddt

$$t=0

#(t) with #(0) = I and#(0) = (- $ !)I3e3 = #e. Then

(0, 0) = TzJ((#, !#) =d

dt

$$$$t=0

(#(t) · e3, $#(t) · #(t)e3)

= (!# · e3,$!# · e3 $ #e · >(#e3),

which implies that

!# · e3 = 0 and #e · >(#e3 = 0. (5.5.1)

However #e ·>(#e3 = #e · ((#) e3) = 0, hence the second expressionin (5.5.1) does not impose any restriction on (# and consequently

ker TzJ = {((#, !#) " Tz(T "SO(3)) | !# · e3 = 0}.


One computes similarly

Tz(Gµ · z) = Tz(G · z) = span {(>e3, 0)} .

Looking at the spaces that we have already computed, it seemsthat the most convenient metric that one can take to construct theorthosymplectic space, is the one that gives us the natural Euclideaninner product in Tz(T "SO(3)) 2 R3 ) R3, for which

W = kerTzJ * Tz(G · z)(

= {(!$, !#) " Tz(T "SO(3)) | !$ · e3 = !# · e3 = 0}.

It is easy to see that W is the direct sum of two bidimensionalH-irreducible spaces. These two spaces are trivially H-isomorphic,therefore W has a trivial isotypic decomposition and, in principlethere is no blocking of the kind predicted by Theorem 5.4.1.

With all these ingredients, we compute the Hessian of the Lya-punov function at z, restricted to W . If (#1 = >!"1# and (#2 = >!"2#are elements in T#SO(3) we have

d2h(#, #) · (((#1, !#1), ((#2, !#2))

= mgle3 ·>!"1>!"2#e3 + !#1 · #I%1

ref#T !#2

+ !#2 ·>!"1#I%1ref#

T # + # ·>!"1#I%1ref#

T !#2

+ !#1 ·>!"2#I%1ref#

T # $ !#1 · #I%1ref#

T>!"2#

+ # ·>!"1>!"2#I%1

ref#T # $ # ·>!"1#I%1

ref#T>!"2#.

We evaluate this expression at z, using the parameter + = - $ !.Since !"i · e3 = !#i · e3 = 0 we get that I%1

ref!#i = 1I1

!#i andI%1ref!"i = 1

I1!"i for i " {1, 2}. Therefore,

d2h(z) · (((#1, !#1), ((#2, !#2))

= mgle3 ·>!"1>!"2e3 +

1I1

!#1 · !#2 + +!#2 ·>!"1e3

++I3

I1e3 ·>!"1!#2 + +!#1 ·>!"2e3 $

+I3

I1!#1 ·>!"2e3

+ +2I3e3 ·>!"1>!"2e3 $

+2I23

I1e3 ·>!"1

>!"2e3. (5.5.2)

In order to use Theorem 5.3.2, we need to write the secondsummand of the Lyapunov function using an orthogonal velocity,


that is, the projection of (-, !) on the orthogonal complement ofh = Lie(H) = span {(1, 1)} with respect to an AdG–invariant metricon g. Since G is Abelian, any metric is AdG–invariant, hence themost general situation consists of taking the inner product in g givenby the quadratic form

g =0

a bb c

1

subject to the condition det g = ac $ b2 > 0, which ensures thepositive definiteness of g. Using the notation introduced in the ex-pression (5.3.1), the orthogonal complement pµ of h with respect tog is

pµ = span {(1, $k)}

where k = (a + b)/(b + c). This implies that the orthogonal velocityvc of z with respect to the splitting determined by g is

vc(k) = +

01

1 + k,$k

1 + k

1.

At the same time it is easy to see that for ($1, $2) " R2 arbitrary

d2J(,1,,2)(#,#) · (((#1, !#1), ((#2, !#2))

= $$2(!#1 ·>!"2#e3 $ !#2 ·>!"1#e3 $ # ·>!"1>!"2#e3),

which evaluated at z takes the form

d2J(,1,,2)(z) · (((#1, !#1), ((#2, !#2))

= $$2(!#1 ·>!"2e3 $ !#2 ·>!"1e3 $ +I3e3 ·>!"1>!"2e3). (5.5.3)

Now set ($1,$2) = vc(k) and combine the expressions (5.5.2)and (5.5.3) to obtain the following matrix of the Hessian d2(h $Jvc(k))(z) restricted to W

5

66667

$mgl $ +2I3

31

1+k $I3I1

40 0 M

0 $mgl $ +2I3

31

1+k $I3I1

4$M 0

0 $M 1I1

0M 0 0 1

I1

8

9999:,


where

M = +

0I3 $ I1

I1+

k

1 + k

1,

whose eigenvalues are

0± = A ±;$4I1(1 + k)2B + A2,

with

A = (1 + k)2 $mglI1(1 + k)2 + I3+2(I3(1 + 2k)$ I1(1 + k))

B = +2(I3k + I3 $ I1)$mgl(1 + k)2.

It is clear that d2(h$ Jvc(k))(z) is positive definite i! B > 0, that is

+2 > mgl(1 + k)2

I3k + I3 $ I1,

hence for each k (for each orthogonal velocity) we have a lower boundfor the values of + for which the sleeping top is stable. The optimalstability condition will be achieved when

(1 + k)2

I3k + I3 $ I1

reaches a minimum. Taking the first and second derivatives of thisfunction, one checks that this happens when

k =2I1 $ I3

I3

and therefore, the optimal stability condition is

+2 >4mglI2

1

I3, (5.5.4)

which coincides with the classical one, obtained in expression (5.3.7)via the energy–Casimir method.

Note that

g =0 2I1%I3

I30

0 1

1

is an inner product whose k is 2I1%I3I3

and it is positive definite becausefor any diagonal inertia tensor Iref = diag [I1, I1, I3], the inequalities

Ij + Ik > Ii for all i, j, k " {1, 2, 3}, j+= k

always hold. This guarantees the consistency of the optimal stabilitycondition (5.5.4).

Chapter 6

Stability of Periodic andRelative Periodic Orbits

—Hermano Sancho, aventura tenemos. . .—No quiero yo decir —respondio Don Quijote—que esta sea aventura del todo, sino principiodella; que por aquı se comienzan las aventuras.Cervantes, Don Quijote de la Mancha, II. Cap. XII.

6.1 Introduction

The study of the existence and stability of periodic orbits in dynam-ical systems has a very long history that goes back to Poisson, Lya-punov, Poincare, and Birkho!. In the case of Hamiltonian dynamicalsystems, which constitute the case of most interest to us, the sta-bility problem is particularly appealing, not only for its physical (inparticular astronomical) relevance but also because, as we justifiedin Section 1.2, the traditional linear methods based on the analysisof the characteristic multipliers provide a necessary but not su"cientcondition for Birkho! or orbital stability in particular systems. Nev-ertheless, just for the sake of this necessary condition, the study ofthe linear tools (first–order stability in the language of Pars [Pars])has received a great deal of attention (see Whittaker [Whit] for ex-amples of this kind of approach). A particularly complete study of

190 Chapter 6. Periodic and Relative Periodic Orbits

linear stability for periodic orbits in Hamiltonian systems has beencarried out by Montaldi, Roberts, and Stewart [MOal], who deal withthe case in which the system is in addition endowed with a symmetry.

Non linear stability implies linear stability. Unfortunately, theconverse implication is false. The situation is analogous to that en-countered in the stability study of relative equilibria (see [Hal85] andthe counterexamples therein). Since non linear stability is most oftenof interest in specific cases, for example, in celestial mechanics, sev-eral recursive and approximative methods have been developed basedmainly on perturbation theory and on the use of series expansions(see [HAG57] and [HAG75] for an account of many of these meth-ods). The most remarkable result that uses perturbative methods isthe KAM Theorem (see [A78, A63, AKN], and references therein)which has been instrumental in the proof of the orbital stability invery relevant physical systems. For example, in the restricted threebody problem, this theorem is able to prove the stability of the Tro-jans and predicts the gaps in the distribution of asteroids betweenMars and Jupiter (see [T78] and [LLN]). Over the years the KAMtheorem has been extended to a very wide class of systems. However,the persistence of the tori that it predicts guarantees stability onlyin the case in which the dimension of the configuration space equalstwo; in higher dimensions, phenomena like the Arnold di"usionmay spoil the stability.

In this chapter, we will show a su"cient condition (we will call itthe energy–integrals method) for the orbital stability of periodicorbits in Hamiltonian systems that, in philosophy, generalizes La-grange’s Theorem (Theorem 1.2.2) to periodic orbits, in other words,our line of thought will be based on energetics, that is, the use of theconserved quantities of the system.

6.2 Orbital Stability and the Energy–Integrals Method

In this section we introduce the energy–integrals method, as well asa first example of its application.

§ 6.2. The Energy–Integrals Method 191

6.2.1 The Energy–Integrals Method

Theorem 6.2.1 (The energy-integrals method) Let ' be a pe-riodic orbit of the Poisson dynamical system (M, {·, ·}, h) through thepoint m " M . Let C1, C2, . . . , Cn " C!(M) be a set of conservedquantities (integrals of the motion) for which

d(C1 + . . . + Cn)(m) = 0.

If the quadratic form

d2(C1 + . . . + Cn)(m)|W)W

is definite for some (and hence for any) subspace W ' TmM suchthat

kerdC1(m) * . . . * kerdCn(m) = W ( span{'$(m)},

then ' is orbitally stable. If W = {0} (in particular, if dimM =2), then ' is always orbitally stable. The matrix d2(C1 + . . . +Cn)(m)|W)W will be referred to as the stability form of the periodicorbit '.

Proof We first prove the case W += {0} and we begin by showing thatthe result does not depend on the choices of m " ' and W . Indeed, ifd(C1 + . . .+Cn)(m) = 0 and Ft is the flow of the Hamiltonian vectorfield Xh, then for any t > 0 and any v, w " TmM we have

d(C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))= F "

t (d(C1 + . . . + Cn)(m))(v, w)= d(F "

t (C1 + . . . + Cn))(m)(v, w)= d(C1 + . . . + Cn)(m)(v, w),

since F "t & d = d & F "

t and C1, C2, . . . , Cn are invariant under Ft. IfW is a complement to span{'$(m)} in kerdC1(m)* . . .*kerdCn(m),then for any t > 0, TmFt(W ) is a complement to span{'$(Ft(m))}in kerdC1(Ft(m)) * . . . * kerdCn(Ft(m)). Moreover, d2(C1 + . . . +Cn)(m)|W)W is definite i! d2(C1 + . . . + Cn)(Ft(m))|TmFt·W)TmFt·Wis definite, since Proposition 5.2.1 and conservation of C1, . . . , Cn,imply for any v, w " TmM :

d2(C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))

= d2(F "t C1 + · · · + F "

t Cn)(m)(v, w)

= d2(C1 + . . . + Cn)(m)(v, w).


The statement of the theorem does therefore not depend on the choiceof the point m " '.

The choice of W is also irrelevant since d2(C1 + . . . +Cn)(m)(v, w) = 0 whenever v " span{'$(m)}. Indeed, since we canassume without loss of generality that v = Xh(m), the definition ofthe Hessian implies

d2(C1 + . . . + Cn)(m)(v, w) = w[Xh(C1 + . . . + Cn)]= w[{C1, h} + . . . + {Cn, h}] = 0.

Thus, the statement of the theorem does not depend on the choicesof m " ' and W ' TmM as long as kerdC1(m)* . . .* kerdCn(m) =W ( span{'$(m)}.

We proceed to the proof of the theorem by defining, for a fixedm " ',

f1 = C1 $ C1(m) + . . . + Cn $ Cn(m),

f2 = (C1 $ C1(m))2 + . . . + (Cn $ Cn(m))2.

The hypothesis of the theorem clearly implies that

df1(m) = df2(m) = 0.

Let S be a local transversal section to ' at m " '. Let Z be thesubspace

Z := TmS * kerdC1(m) * . . . * kerdCn(m).

By the properties of the local transversal section we have '$(m) /"TmS and hence

Z * span{'$(m)} = {0}.

Since span{'$(m)} ' kerdC1(m) * . . . * kerdCn(m) and TmM =TmS ( span{'$(m)} we get

kerdC1(m) * . . . * kerdCn(m)= TmM * kerdC1(m) * . . . * kerdCn(m)= (TmS ( span{'$(m)}) * kerdC1(m) * . . . * kerdCn(m)= (TmS * kerdC1(m) * . . . * kerdCn(m))( span{'$(m)}= Z ( span{'$(m)},


that is, Z is a complement to span{'$(m)} in kerdC1(m) * . . . *kerdCn(m). Since f1 and C1 + . . . + Cn di!er by a constant, the hy-pothesis of the theorem implies that the form d2f1(m)|Z)Z is definite.Using Proposition 5.2.1 we have

d2f1(m)|Z)Z =-d2f1(m)|TmS)TmS

2$$Z)Z

= d2(f1|S)(m)|Z)Z .

Hence, the hypothesis of the theorem is equivalent to saying thatd2(f1|S)(m)|Z)Z is definite.

We prove now that Z is the kernel of d2(f2|S)(m). Let v1, v2 "TmS, such that vi = d

dt

$$t=0

ci(t), with ci(t) " S for any t and i "{1, 2}. Let Xvi " X(S) be an extension of vi to a vector field on Swhose flow is denoted by F vi

t . Then, by definition:

d2(f2|S)(m)(v1, v2) = v1 [Xv2 [f2]]

=d

dt

$$$$t=0

d

ds

$$$$s=0

f2(F v2s (c1(t)))

=d

dt

$$$$t=0

d

ds

$$$$s=0

-C1(F v2

s (c1(t)))$ C1(m)22 + . . .

+-Cn(F v2

s (c1(t)))$ Cn(m)22

=d

ds

$$$$s=0

2-C1(F v2

s (m))$ C1(m)2dC1(F v2

s (m)) · TmF v2s (v1) + . . .

+d

ds

$$$$s=0

2-Cn(F v2

s (m))$ Cn(m)2dCn(F v2

s (m)) · TmF v2s (v1)

= 2[(dC1(m) · v2)(dC1(m) · v1) + . . . + (dCn(m) · v2)(dCn(m) · v1)].

Hence v1 " kerd2(f2|S)(m) i! for any v2 " TmS, we have

(dC1(m) · v1)(dC1(m) · v2) + . . . + (dCn(m) · v1)(dCn(m) · v2) = 0.

Taking in this relation v2 = v1, this implies that dC1(m) · v1 = . . . =dCn(m)·v1 = 0 and hence v1 " kerdC1(m)*. . .*kerdCn(m)*TmS =Z. Conversely, if v1 " Z = TmS * kerdC1(m)* . . .* kerdCn(m) theabove relation is satisfied trivially for all v1 " TmS. Therefore,

Z = kerd2(f2|S)(m).

Using all these remarks and the obvious positive semi definitenessof d2(f2|S)(m), Lemma 5.2.2 guarantees the existence of some a > 0for which the function f defined by

f := af1 + f2 (6.2.1)


is such that d2(f |S)(m) is positive definite.Let now V be an open neighborhood of '. With the notation of

Definition 1.2.1, the Morse Lemma allows us to choose S and * > 0such that f 4 0 on S and

f%1[0, *) * S @ V *W0 *W1.

Notice that since f is a conserved quantity, if z "W0*W1*f%1[0, *)then F"%/(z)(z) " W0 *W1 * f%1[0, *) (see Figure 6.2.1). Let DV =

Figure 6.2.1: Construction of A.

inf{d(x, ') | x " V \ V }, where d is the distance function on Massociated to some Riemannian metric on M (we assume that M isparacompact and hence there is always some Riemannian metric onit). The compactness of ' and the openness of V guarantee that DV

is never 0.If A = W0 *W1 * f%1 [0, *), we define the map:

D : A $% Rz -$% D(z) := max

t'[0, "%/(z)]d(Ft(z), ').


Note that D(m) = 0. By the continuity of D, we can choose * > 0(and therefore A) small enough so that D(z) < DV /2, for any z " A.Define the open neighborhood U of ' by

U := {Ft(z$) | z$ " A, t 4 0}.

We shall prove below that Ft(U) ' V for all t 4 0. In order to seethis, note that, by construction, U is invariant under the flow Ft andhence the claim is proved if we show that U ' V . Let’s supposethe contrary, namely that there is an element Ft(z$) " U, z$ " A,such that Ft(z$) /" V . Without loss of generality we can assume thatt " [0, & $ ((z$)] which then implies that d(Ft(z$), ') = D(z$) <DV /2. However, since we assume that Ft(z$) /" V , it follows thatd(Ft(z$), ') 4 DV , by the definition of DV .

In the case W = {0}, Z = kerd2(f2|S)(m) = {0} and, there-fore d2(f2|S)(m) is positive definite, hence we don’t need to applyLemma 5.2.2 and, the rest of the proof follows just by taking f2 in-stead of f . !

Remark 6.2.1 The method is called energy–integrals since onecan always use the Hamiltonian of the system, that is, its total en-ergy, as one of the conserved quantities Ci in the hypothesis of The-orem 6.2.1. "

Remark 6.2.2 (Energy–integrals versus energy–momentum)The statement of the energy–integrals method is remarkably closeto that of the energy–momentum method and so, the relationshipbetween them is a question that arises naturally. In addition, insome examples, by embedding the given Hamiltonian system in anappropriate space with a suitable group action, the periodic orbitsof the original system become relative equilibria for a new system;the extra conserved quantities needed to apply the energy–integralsmethod turn out to be the components of the momentum mapassociated to the symmetry group of this new system.

An example where we will see this happen is the isotropic har-monic oscillator (see Subsection 6.3.2). For simplicity, we talk abouttwo oscillators, even though the same can be said for an arbitrarynumber. The orbits of this system in the configuration space are,generically, ellipses. Among them, the circles, are the relative equi-libria for the SO(2) symmetry of the system. Therefore, even though


all the orbits are periodic, generically, they are not relative equilib-ria and one needs the energy–integrals method to prove their orbitalstability. However, if one rewrites the system in complex variables,the bigger symmetry group SU(2) of the system becomes apparentand it turns out that the set of relative equilibria associated to thisnew symmetry covers all the possible dynamical orbits of the system:the evolution consists entirely of relative equilibria. Moreover, theenergy–momentum method applied to these periodic orbits (relativeequilibria) proves their orbital stability. This is hence a case wherethe traditional methods are enough to study the stability of the sys-tem. If the oscillator is not isotropic but its frequencies are resonant,all solutions of the system are still periodic orbits (the Lissajous fig-ures). However, the SU(2) symmetry of the system is destroyed, and,consequently, the energy–momentum method cannot be used in thestudy of the stability of these periodic orbits. As we shall see in Sub-section 6.3.2, some conserved quantities are still easily identified andthese can then be used to apply the energy–integrals method.

Another situation where this idea seems very promising is in thestability analysis of the closed Keplerian orbits. In the next sectionwe will prove that, taking as conserved quantities in Theorem 6.2.1the energy, one of the components of the angular momentum, and oneof the components of the Laplace–Runge–Lenz vector, these periodicorbits turn out to be orbitally stable. Moser [M70] has shown thateach Keplerian periodic orbit can be realized as a certain geodesic onS3 and, therefore, it arises as an SO(4) relative equilibrium. Usingthe energy–momentum method for SO(4), it should be possible toprove a stability result that, mapped back to T "R3, yields some kindof stability information of the original Keplerian orbit. We are notaware of any work that carries out this program. Besides, recall that,in general, the kind of stability predicted by the energy–momentummethod is weaker than orbital stability: a drift in the direction ofthe coadjoint isotropy orbit of the value of the momentum map onthe periodic orbit is allowed. This drift along some subgroups ofSO(4) should appear in the stability analysis of the Keplerian orbitsbecause, generically, the coadjoint isotropy subgroups of SO(4) arebidimensional. Thus, the energy–momentum method can, at best,give Lyapunov stability of the periodic orbits modulo the drift along atoral orbit, a result much weaker than the (Birkho!) orbital stabilitypredicted by the energy–integrals method.


In addition, this process that esthetically looks very appealing, isreally di"cult to implement concretely. What one has in general, areconserved quantities. Finding the symmetry for which they are thecomponents of the associated momentum map is, in general, highlynontrivial and amounts to finding a converse to Noether’s Theorem.Partial steps have been taken in this direction; see for example [FM75,CI85, CLM] and references therein. However, the results in theseworks are not strong enough to produce the necessary symmetriesthat generate the conserved quantities that appear in some classicalsystems, such as the Kowalewski top, for example (see [Ko, Boal]).Even in the Keplerian case, Moser’s results [M70] notwithstanding,it is not trivial to realize the Laplace vector as part of an SO(4)momentum map.

Hence, the energy–integrals method presents the advantage ofusing directly the conserved quantities to establish stability, withoutresorting to the symmetry that generates them. "

6.2.2 The Energy–Integrals Method and IntegrableSystems

The conserved quantities present in the statement of the energy–integrals method make integrable systems the first candidates for itsapplication. Moreover, one could think that whenever we find a sys-tem where all the integrals required to guarantee orbital stability arepresent, the system is necessarily integrable. We will see in Subsec-tion 6.3.4 that this is not the case.

We will take the traditional characterization of integrable systemsgiven by the Liouville–Arnold Theorem [A78, AM78, Fa], thatis, a Hamiltonian system, whose phase space is ) ) Tn, where theactions space ), whose elements are denoted by I = (I1, . . . , In),is an open subset of Rn and the angles space Tn = Rn/(2/Z)n hasits elements denoted by 7 = (71, . . . , 72). The symplectic form forthese systems is expressed in these coordinates as

) =n!

i=1

d7i # dIi ,

and the Hamiltonian h is a function only of the actions: h =


h(I1, . . . , In). Hamilton’s equations

7i =#h

#Ii=: !i(I1, . . . , In)

Ii = 0,

for i " {1, . . . , n}, show that I1, . . . , In are n commuting integrals ofmotion and that, once these constants have been fixed, all the pos-sible motions take place on a given torus, in a periodic or quasiperi-odic manner, depending on the rational dependence or independenceof the real n–tuple (!1(I1, . . . , In), . . . ,!n(I1, . . . , In)). Of course,we will be interested in the periodic case, that is, we will supposethat there are some values of the actions for which the associatedfrequencies (!1(I1, . . . , In), . . . ,!n(I1, . . . , In)), are rationally depen-dent and, therefore, all motions on the torus corresponding to thoseactions are periodic.

Let’s remark that, unless the dimension of the integrable systemis two, the fact that the periodic orbit lies on a torus full of other pe-riodic orbits does not imply its orbital stability: there could be valuesof the actions, arbitrarily close to the ones corresponding to the givenperiodic orbit, which produce quasiperiodic orbits and, therefore, notallowing orbital stability. Take, for instance, a four dimensional inte-grable system with Hamiltonian given by

h(I1, I2) =12-I21 + I2

2

2.

When I1 = I2 = 1, the motion takes place on a 2–torus full of periodicorbits. However, this torus is surrounded by the tori corresponding tothe actions I1, I2 having the ratio I1/I2 irrational; these surroundingtori are therefore filled with quasiperiodic motions, which makes theorbital stability of the abovementioned periodic orbits impossible. Adi!erent case is when we know in advance, as it is the case with someof the examples that we will present, that all the motions in all thetori are periodic. In this situation, the continuity of the foliation ofthe phase space by invariant tori implies the orbital stability of allthe periodic orbits of the system in a straightforward manner.

As we just saw, even in the integrable case, orbital stabilityof periodic orbits is not a trivial issue. In what follows, we willsee how the energy–integrals method gives an idea of what prop-erties an integrable system should have in order to exhibit or-bitally stable periodic orbits. In order to simplify the exposition,


we will consider an n–dimensional integrable system such that, forsome values of the actions (I0

1 , . . . , I0n), the associated frequencies

(!1(I01 , . . . , I0

n), . . . ,!n(I01 , . . . , I0

n)), are rationally dependent and allof them di!erent from zero. We shall study the orbital stability ofthe periodic point m := (I0

1 , . . . , I0n, 0, . . . , 0). Since the Hamilto-

nian depends only on the actions (which are conserved quantities),the first part of the requirements in the energy–integrals method canalways be satisfied by choosing a set of n real functions 21, . . . ,2n "C!(R) such that 2$i(I

0i ) = 0h

0Ii(I0

1 , . . . , I0n) = !i(I0

1 , . . . , I0n), for all

i " {1, . . . , n}, one has

d(h$ C1 $ . . .$ Cn)(m) = 0,

with Ci = 2i(Ii). Since all the frequencies are di!erent from zero, weclearly have

kerdh * kerdC1 * . . . * kerdCn = {0}) T0Tn.

Since the Hamiltonian and the conserved quantities in the previousexpression depend only on the actions, using the notation of Theo-rem 6.2.1, it follows that d2(h$C1 $ . . .$Cn)(m)|W)W = 0, unlesseither n = 1 or the hypotheses of the following proposition are satis-fied.

Proposition 6.2.1 Let ())Tn, !, h) be an integrable Hamiltoniansystem and m = (I0

1 , . . . , I0n, 0, . . . , 0) be a periodic point, that is, the

frequencies (!1(I01 , . . . , I0

n), . . . ,!n(I01 , . . . , I0

n)) are rationally depen-dent and all of them assumed to be nonzero. Then, if f " C!(Tn)is an integral of the motion for the integrable system, such thatdf(m) = 0 and d2f(m)|U)U is definite for some subspace U sat-isfying T0Tn = Rn = U ( span{'$(m)}, then m is an orbitally stableperiodic point.

Proof It follows by using the previous remarks and h$C1$. . .$Cn+f as the combination of conserved quantities in the energy–integralsmethod. !

Remark 6.2.3 The previous Proposition states that a necessarycondition for an integrable system to exhibit orbitally stable peri-odic orbits through the energy–integrals method, is the existence ofintegrals of motion involving the angles. Such functions are usually


called resonances and, in some instances, they force the motion totake place on subtori of Tn. These systems are often called super in-tegrable or non commutatively integrable systems (see [Fa] andreferences therein). "

6.3 Applications of the Energy–IntegralsMethod

In the following simple examples, we show how to use the energy–integrals method concretely.

6.3.1 The Stability of the Closed Rigid Body Orbits

As a first illustration of the energy–integrals method, we study thestability of the closed orbits of the free rigid body. We will considerthe three dimensional rigid body in body representation (see Chapter15 of [MR94] for a detailed exposition). In this framework, the rigidbody is a Poisson system, more specifically a Lie–Poisson system,whose phase space is the dual so(3)" 2 R3 = {" = (*1, *2, *3) |*1, *2, *3 " R} of the Lie algebra so(3) 2 R3 of the rotation groupSO(3). The Lie–Poisson bracket corresponding to this Lie algebra is

{F, G}(") = $" · (>F ("))>G(")),

with F and G arbitrary functions in C!(R3). The rigid body in theprincipal axis body frame is the Hamiltonian system defined by theenergy function:

h(") =12

0*2

1

I1+

*22

I2+

*23

I3

1,

where the constants I1, I2, I3 are the principal moments of iner-tia . We shall assume in what follows that the rigid body is general,that is, no two principal moments of inertia are equal. Concretely,we make the assumption

I3 > I2 > I1.

§ 6.3. Applications of the Energy–Integrals Method 201

Hamilton’s equations for this system are the standard Euler equa-tions:

*1 =I2 $ I3

I2I3*2*3

*2 =I3 $ I1

I3I1*1*3

*3 =I1 $ I2

I1I2*1*2

The total angular momentum in the body frame , definedby

*2 :=< " <2= *21 + *2

2 + *23

is a Casimir function, that is, it Poisson commutes with any otherfunction F " C!(R3). This implies that the symplectic leaves ofthis system are the spheres of radius *. The dynamics induced bythe Euler equations on these spheres is illustrated in Figure 6.3.1.As may be observed, the dynamical behavior on each symplectic leafconsists of six equilibria, four heteroclinic orbits, and closed orbitswhose stability we will establish using the energy–integrals method.We will denote in what follows C(") := (*2

1 + *22 + *2

3)/2.It is clear from Figure 6.3.1 that if we fix the energy value E (that

is, the value of h) and the total body angular momentum *, thereare two closed orbits corresponding to these values (we assume thatwe are avoiding the heteroclinic orbits and the equilibria). We willchoose for our stability analysis the orbit ' through the point

m :=

5

7A

I1(*2 $ 2I3E)I1 $ I3

, 0,

AI3(2I1E $*2)

I1 $ I3

8

: .

The study of the other orbit is completely analogous.We now apply the energy–integrals method. We will use as con-

served quantities the energy h, and

C" (") = $&0

12

0*2

1

I1+

*22

I2+

*23

I3

11,

C1 = $20

12(*2

1 + *22 + *2

3)1

,


Figure 6.3.1: A symplectic leaf of the rigid body.

with &, 2 : R % R arbitrary real valued functions that will be deter-mined by the requirements of the method.

Before we go any further, notice that the stability of the closedorbits considered as integral curves on the symplectic leaves is auto-matic since the dimension of these leaves is two, which falls into thelast situation considered in Theorem 6.2.1. We will study the stabil-ity of the orbits given by the Poisson system, that is, we consider theorbits as lying in R3. Since

dh(") =0*1

I1,*2

I2,*3

I3

1, dC(") = (*1, *2, *3)

it follows that

dh(m) =

<A*2 $ 2I3E

I1(I1 $ I3), 0,

A2I1E $*2

I3(I1 $ I3)

=, (6.3.1)

dC(m) =

5

7A

I1(*2 $ 2I3E)I1 $ I3

, 0,

AI3(2I1E $*2)

I1 $ I3

8

: . (6.3.2)


From (6.3.1) and (6.3.2) it follows that

d(h + C" + C1)(m) =5

71$ & $(E)$ I12$

3%2

2

4

I1

AI1(*2 $ 2EI3)

I1 $ I3, 0,

1$ & $(E)$ I32$3

%2

2

4

I3

AI3(2EI1 $*2)

I1 $ I3

8

: .

If & and 2 are such that

2$0*2

2

1= 0 and & $(E) = 1, (6.3.3)

then d(h+C" +C1)(m) = 0 and the first requirement of the energy–integrals method is satisfied. With the values in (6.3.3) it followsthat

d2(h + C" + C1)(m) =

5

7A 0 B0 0 0C 0 D

8

: , (6.3.4)

with

A =2EI3 $*2

I1(I1 $ I3)

0& $$(E) + I2

12$$0*2

2

11,

B = $& $$(E) + I1I32$$

3%2

2

4

I1I3

AI3(2EI1 $*2)

I1 $ I3

AI1(*2 $ 2EI3)

I1 $ I3,

C = $& $$(E) + I1I32$$

3%2

2

4

I1I3

AI3(2EI1 $*2)

I1 $ I3

AI1(*2 $ 2EI3)

I1 $ I3,

D =2EI1 $*2

I3(I3 $ I1)

0& $$(E) + I2

32$$0*2

2

11.

In order to establish the stability of ', it su"ces to show that (6.3.4),restricted to a complement W of span{'$(m)} in kerdh(m) *kerdC" (m) * kerdC1(m) = kerdh(m) is definite. From (6.3.1) itfollows that

kerdh(m) = span

B(0, 1, 0),

<1, 0, $

AI3(*2 $ 2I3E)I1(2I1E$*2)

=C.


Given that '$(m) " span{(0, 1, 0)}, we can take

W = span

B<1, 0, $

AI3(*2 $ 2I3E)I1(2I1E $*2)

=C.

With this choice,

d2 (h + C" + C1)(m)|W)W =(I1 $ I3)(2EI3 $*2)2$$

3%2

2

4

I1.

Hence d2 (h + C" + C1)(m)|W)W is definite as long as

I1 += I3, 2EI3 += *2 and 2$$0*2

2

1+= 0.

The first condition is satisfied by the hypothesis that the rigid bodyis general, and the second is implied by the first. Indeed, since thesecond coordinate of the point m is zero we have

2EI3 $*2 =I3 $ I1

I1*2

1;

hence if I3 += I1, then 2EI3 += *2 unless *1 = 0, which would placem at the equilibrium (0, 0, *) that was eliminated by hypothesis.

To sum up, by the energy–integrals method, the closed orbit 'through m is orbitally stable provided that I1 += I3, and that we canfind real functions & and 2 such that

2$0*2

2

1= 0, 2$$

0*2

2

1+= 0, and & $(E) = 1.

The functions:

&(x) = x, and 2(x) =0

x$ *2

2

12

satisfy these conditions.

6.3.2 Stability of the Orbits of the Bidimensional Har-monic Oscillator

In this section we apply the energy–integrals method to prove thestability of the periodic orbits produced by bidimensional resonant


harmonic oscillators. This study generalizes trivially to arbitrarydimensions. The phase space of this system is T "R2, endowed withthe canonical symplectic form

! = dq1 # dp1 + dq2 # dp2.

The dynamics of the harmonic oscillator is associated to the Hamil-tonian function

h(q1, q2, p1, p2) =!1

2(p2

1 + (q1)2) +!2

2(p2

2 + (q2)2).

If the frequencies !1 and !2 are resonant, that is, !1/!2 = j1/j2,with j1 and j2 relatively prime integers, the phase portrait of thissystem consists of an equilibrium and periodic orbits (the Lissajouscurves, see Figure 6.3.2), whose stability we will study. We willassume that !1 and !2 are both positive.

Figure 6.3.2: Some Lissajous curves created by the harmonic oscilla-tor in the configuration space, for di!erent resonance conditions.

As we mentioned in Remark 6.2.2, the case !1 = !2 can be studiedusing the energy momentum method. Indeed, if the system is writtenin complex variables, it exhibits a Hamiltonian SU(2) symmetry suchthat all the orbits of the system are relative equilibria with respect to


it. Moreover, the energy–momentum method, in this particular case,proves that this orbits are orbitally stable.

In the general anisotropic resonant case, the SU(2) symmetrydoes not exist. However, some of the conserved quantities associatedto the components of the momentum map of the isotropic case (Hopffibration) survive and can be used to prove the orbital stability ofthe Lissajous curves.

It will be computationally convenient to write the system inaction–angle coordinates (I1, I2, 71, 72) which, at the same time, willenable us to use the results in Section 6.2.2. One way to constructthese coordinates is taking

pi(Ii, 7i) =;

2Ii cos 7i, qi(Ii, 7i) =;

2Ii sin 7i .

In these coordinates, the Hamiltonian is h(I1, I2) = I1!1 + I2!2 andthe equations of the motion are

Ii = 0, 7i = !i.

We study the orbital stability of the periodic solution that goesthrough the point

m := (I1, I2, 0, 0).

Any 2/–periodic function of j172 $ j271 is an integral of the motion;in particular, we will see that the function

f(71, 72) = cos(j172 $ j271),

can be used as the resonance that, through Proposition 6.2.1, makesthe periodic point m stable. Indeed, df(m) = (0, 0) and

d2f(m) =0$j2

2 j1j2

j1j2 $j21

1.

Since '$(m) = (0, 0, !1, !2), the vector subspace span{(1, 0)} =: Ucan be taken as the complement to span{'$(m)} in T0T2 = R2. Hence

d2f(m)|U)U = $j22 += 0,

which, by Proposition 6.2.1 guarantees the orbital stability of m.


6.3.3 Stability of the Elliptical Orbits of the Three Di-mensional Kepler Problem

The Kepler problem is the study of the motion of two gravitationallyattracting massive particles. The translational invariance of the com-plete system allows us to perform a reduction to the center of mass.The problem is then transformed to the study of a particle with massm (the reduced mass), subject to a potential of the form k/r. Thus,the Kepler system may be formulated as the Hamiltonian dynamicalsystem with phase space T "R3 and Hamiltonian function given bythe Legendre transform of the Lagrangian:

L =12m3r2 + r272 + r2 sin2 7"2

4+

k

r,

where (r, 7, ") are the spherical coordinates on R3 (7 is the colatitudeand " is the azimuth).

As it is well known, conservation of angular momentum forces thesolutions of the Lagrange equations for L to lie in a plane. For thesake of simplicity and without loss of generality we will study solu-tions lying in the plane xy. Depending on the relative values of theangular momentum and the energy, these solutions may be ellipses,parabolas, or hyperbolas in configuration space. The first case, oc-curring for negative values of the energy, produces periodic orbits inphase space whose orbital stability we will study. Specifically, we willanalyze the periodic orbit ' given by

r =a(1$ e2)1 + e cos"

, 7 =/

2. (6.3.5)

As seen from Figure 6.3.3, this curve corresponds in configurationspace to an ellipse of eccentricity 0 = e < 1, which is determined bythe total energy E < 0 and z-angular momentum l of the orbit, moreprecisely,

e =

D1 +

2El2

mk2=

D1$ l2

mka, a = $ k

2E.


Figure 6.3.3: Projection on the configuration space of the periodicorbit.

The canonical momenta associated to L are

pr =#L

#r= mr (6.3.6)

p2 =#L

#7= mr27 (6.3.7)

p! =#L

#"= mr2 sin2 7", (6.3.8)

which allows us to write the Hamiltonian h associated to L:

h(r, 7, ", pr, p2, p!) =p2

r

2m+

p22

2mr2+

p2!

2mr2 sin2 7$ k

r.

Note that the z–component of the angular momentum Jz, coincideswith p! whose value along ' is constant and equal to l.

We will carry out our stability analysis at the point m0 " ' givenby

m0 :=3a(1$ e),

/

2, 0, 0, 0, l

4,

where the entries in the previous vector are in the order(r, 7, ", pr, p2, p!).


Since one of the ingredients needed in the energy–integrals methodis '$(m0), we compute it. The r-coordinate reaches the minimuma(1$ e) at m0 and 7 = //2 along '; it follows that r = 7 = 0 at m0.By equation (6.3.8)

" =p!

mr2 sin2 7,

which at m0 yields

"(m0) =l

ma2(1$ e)2.

Recall that p! = 0 since " is a cyclic coordinate of L. Since

p2 = $#h#7

=p2!

mr2

cos 7sin3 7

= mr2 sin 7 cos 7"2,

at m0, where 7 = //2, we conclude that p2(m0) = 0. Also,

pr = $#h#r

= mr72 + mr sin2 7"2 $ k

r2,

which at m0 takes the value:

pr(m0) =l2 $mka(1$ e)

ma3(1$ e)3=

ke

a2(1$ e)2.

Hence,

'$(m0) =0

0, 0,l

ma2(1$ e)2,

ke

a2(1$ e)2, 0, 0

1.

Next, we recall the expression of the conserved quantities that weneed in the application of the energy–integrals method. These willbe supplied by the energy, the third component Jz of the angularmomentum, and the first component Ax of the Laplace–Runge–Lenz vector (Laplace vector in what follows). We will computethe expression of these two quantities in spherical coordinates. Therelation between the canonical momenta in Euclidean and sphericalcoordinates is given by:

px =p2 cos 7 cos"+ prr cos" sin 7 $ p! csc 7 sin"

r,

py =p! cos" csc 7 + sin"(p2 cos 7 + rpr sin 7)

r,

pz = pr cos 7 $ p2 sin 7r

.


Therefore, in spherical coordinates, the expression of the componentsof the angular momentum is:

Jx = ypz $ zpy = $p! cos" cot 7 $ p2 sin",

Jy = zpx $ xpz = p2 cos"$ p! cot 7 sin",

Jz = xpy $ ypx = p!.

Next, we compute the expression of the Laplace vector

A = p) J$mkr

r,

in spherical coordinates:

A =

<$ 1

4r

3csc 7-(p2

2 $ kmr) cos(27 $ ")

$ 2(p22 + 2p2

! $ kmr) cos"+ p22 cos(27 + ")

$ kmr cos(27 + ") + prp2r sin(27 $ ")$ 4prp!r sin"

+ prp2r sin(27 + ")24

,

14r

3csc 7-$ prp2r cos(27 $ ")$ 4prp!r cos"+

prp2r cos(27 + ") + p22 sin(27 $ ")

$ kmr sin(27 $ ") + 2p22 sin"+ 4p2

! sin"$ 2kmr sin"

$ p22 sin(27 + ") + kmr sin(27 + ")

24,

(p22 $ kmr) cos 7 + p2

! cot 7 csc 7 + prp2r sin 7r

=.

The rotational symmetry of the problem guarantees that p! andconsequently

C1 := $2(p!),

with 2 : R % R an arbitrary function, is a conserved quantity. Thesame holds for any arbitrary function & : R % R of the first compo-


nent of the Laplace vector:

C2 := $&<$ 1

4r

3csc 7-(p2

2 $ kmr) cos(27 $ ")

$ 2(p22 + 2p2

! $ kmr) cos"+ p22 cos(27 + ")

$ kmr cos(27 + ") + prp2r sin(27 $ ")$ 4prp!r sin"

+ prp2r sin(27 + ")24=

.

It is easy to see that

d(h + C1 + C2)(m0)

=0

k

a2(1$ e)2$ l2

a3(1$ e)3m+

l2& $(ekm)a2(1$ e)2

, 0, 0, 0, 0,

l

ma2(1$ e)2$ 2l& $(ekm)

a(1$ e)$ 2$(l)

1.

Therefore, taking 2 and & satisfying

2$(l) =k

aland & $(ekm) =

e

am(1$ e2),

we obtain that

d(h + C1 + C2)(m0) = 0.

In order to prove the orbital stability of ' we just need to establishthat

d2(h + C1 + C2)(m0)|W)W

is definite, with W a complement to span{'$(m0)} in kerdh *kerdC1 * kerdC2 = kerdh * kerdC1. Since

dh(m0) =0$ ke

a2(1$ e)2, 0, 0, 0, 0,

l

ma2(1$ e)2

1,


we have,

kerdh(m0) =

span

B(0, 1, 0, 0, 0, 0) , (0, 0, 1, 0, 0, 0) , (0, 0, 0, 1, 0, 0) ,

(0, 0, 0, 0, 1, 0) ,

01, 0, 0, 0, 0,

mke

l

1C.

Also,

dC1(m0) =0

0, 0, 0, 0, 0, $ k

al

1,

hence

kerdC1(m0) =

span

B(1, 0, 0, 0, 0, 0) , (0, 1, 0, 0, 0, 0) , (0, 0, 1, 0, 0, 0) ,

(0, 0, 0, 1, 0, 0) , (0, 0, 0, 0, 1, 0)

C.

We may then take,

W = span {(0, 1, 0, 0, 0, 0) , (0, 0, 1, 0, 0, 0) , (0, 0, 0, 0, 1, 0)} .

It may be checked that

d2(h + C1 + C2)(m0)|W)W =

5

67

ka(1%e2) 0 0

0 ke2

a(1%e2) 00 0 1

a2m(1%e2)

8

9: ,

which, since e < 1, is positive definite. Hence, taking for instance

2(x) =k

alx,

&(x) =e

am(1$ e2)x,

the quadratic form d2(h+C1+C2)(m0)|W)W is well defined and posi-tive definite which, by Theorem 6.2.1, guarantees the orbital stabilityof the closed Keplerian orbit '.


6.3.4 Orbital Stability Via the Energy–IntegralsMethod in a Non Integrable System

This example proves the orbital stability of some of the periodic orbitsin a non integrable system, using the energy–integrals method.

We first take a heavy top with moments of inertia I1, I2, and I3,such that the center of mass lies on the principal axis of inertia witheigenvalue I3, and I1 += I2, I1 += 2I3, I1 += 4I3 (the top is neithera Lagrange’s top, a Kowalewski’s top, nor a Goryachev–Chaplygintop). This system is Hamiltonian with phase space the cotangentbundle T "SO(3). Parameterizing T "SO(3) by Euler angles and theirconjugate momenta (7, 2, 4, p2, p1, p3), the heavy top Hamiltonianhas the expression

h(7, 2, 4, p2, p1, p3) =(p2 cos4 + (p1 csc 7 $ p3 cot 7) sin4)2

2I1

+(p2 sin4 + (p3 cos 7 $ p1) cos4 csc 7)2

2I2+

p23

2I3+ mgl cos 7,

where m is the mass of the top, g is the gravitational acceleration,and l is the distance between the center of mass and the fixed pointof the top.

It can be easily checked that the group SO(2) acts in a Hamil-tonian fashion on T "SO(3) by rotations about the third spatial axis,that is

SO(2)) T "SO(3) $% T "SO(3)(ei,, (7, 2, 4, p2, p1, p3)) -$% (7, 2+ $, 4, p2, p1, p3) .

The associated conserved quantity is p1. A simple computation showsthat the curve

'T (t) =07#, at, 0, 0, I3a cos2 7# + I2a sin2 7#,

mglI3

a(I3 $ I2)

1,

is a relative equilibrium of our system, provided that

cos 7# =mgl

a2(I3 $ I2).

This relation giving 7# in terms of the velocity a determines an axisof rotation of the heavy top. These axes are known as axes ofStaude [Ma91]. It has been shown (see [Lal92, page 16]) that these


relative equilibria are SO(2)–stable in the sense of Definition 1.6.3,provided that I2 is the maximal axis of inertia or that I3 > I2 > I1

and 1 > cos2 7# > I23(I3%I2) . As we will see in the next chapter, that

is equivalent to saying that the reduced system with Hamiltonian

hµ(7, 4, p2, p3) =(p2 cos4 + (µ csc 7 $ p3 cot 7) sin4)2

2I1

+(p2 sin4 + (p3 cos 7 $ µ) cos4 csc 7)2

2I2+

p23

2I3+ mgl cos 7,

where

µ = aI3 cos2 7# + aI2 sin2 7# =a4I2(I2 $ I3)$m2g2l2

a3(I2 $ I3),

has a stable equilibrium at mT =37#, 0, 0, mglI3

a(I3%I2)

4, for which

d2hµ(mT ) is definite. By the works of Kozlov [Koz78] andZiglin [Zi80], and by the choice of the moments of inertia, the re-duced system just described is non integrable.

Consider now the Hamiltonian system on T "S2 ) T "R, withHamiltonian

H(7, 4, p2, p3, q, p) = hµ(7, 4, p2, p3) +12(q2 + p2).

This system is non integrable and has a periodic orbit

'(t) = (mT , cos t, $ sin t),

whose orbital stability we will prove using the energy–integralsmethod. It is clear that for any real valued map & : R % R, thefunction

C" := &

012(q2 + p2)

1

is a conserved quantity for the system with Hamiltonian H. We usethe energy–integrals method with C" at m = (mT , 1, 0). Firstly,

d(H $ C" )(m) =00, 1$ & $

012

1, 01

.

Hence, taking for example &(x) = x, we have d(H $ C" )(m) = 0.A simple computation shows that span{'$(m)} = span{(0, 0, 1)}

§ 6.4. Stability of Regular Relative Periodic Orbits 215

and that kerdH(m) * kerdC" (m) = kerdH(m) = TmT (T "S2) )span{(0, 1)}. Hence, in the notation of Theorem 6.2.1, we can takeas W the subspace

W = TmT (T "S2)) {(0, 0)}.

Now,

d2(H $ C" )(m)|W)W = d2hµ(mT ),

which is definite, since at the equilibrium mT , the second variationof the reduced Hamiltonian is known to be definite. This shows that' is orbitally stable.

6.4 Stability of Regular Relative Periodic Or-bits

In this section, we will see that, using a combination of the energy–integrals method with reduction theory, one can prove a su"cientcondition for the stability of the relative periodic orbits correspondingto the proper free action of a compact Lie group G. The idea behindthis condition, that we will call the symmetric energy–integralsmethod is that, by Theorem 1.6.2, the RPOs of a symmetric systemcoincide, in this particular case, with the periodic orbits in the associ-ated Marsden–Weinstein reduced spaces. Since for periodic orbits wehave at hand the energy–integrals method introduced in the previoussection, we will work with it in the reduced space and we will seethat the orbital stability in the reduced space implies Gµ–stability,in the sense of Definition 1.6.3, of the original RPO, where µ " g"µ isthe momentum value corresponding to the RPP in question. Remarkthat in this section there are no singularities present. The behaviorof the general singular case is more complicated, in the sense thatthe orbital stability of the singular reduced periodic orbit does notguarantee the Gµ–stability of the original RPO, therefore, it will betreated in a separate section

Theorem 6.4.1 (The symmetric energy–integrals method)Let (M, !, h : M % R) be a Hamiltonian dynamical system witha symmetry given by the Lie group G acting freely and properly onM in a globally Hamiltonian fashion with associated equivariant


momentum map J : M % g". Assume that the Hamiltonianh " C!(M) is G–invariant and that J is equivariant. Let m " Mbe a RPP such that J(m) = µ " g" is a regular value of J and Gµ iscompact. Then, if there is a set of Gµ–invariant conserved quantitiesC1, . . . , Cn : M % R for which

d(C1 + . . . + Cn)(m) = 0,

and

d2(C1 + . . . + Cn)(m)|W)W


kerdC1(m) * . . . * kerdCn(m) * TmJ%1(µ)= W ( (span{Xh(m)} + Tm(Gµ · m)), (6.4.1)

then m is a Gµ–stable RPP.If dimW = 0 (in particular, if dimMµ = 2), then m is always

a Gµ–stable RPP. The matrix d2(C1 + . . . + Cn)(m)|W)W will bereferred to as the stability form of the RPP m.

Proof We first study the case dim W > 0. Carrying out a compu-tation similar to the one in Theorem 6.2.1, it is easy to see that theresult does not depend on the choice of the point m in the RPO.Moreover, the choice of W is also irrelevant since d2(C1 + . . . +Cn)(m)(v, w) = 0, whenever v " span{Xh(m)} + Tm(Gµ · m). In-deed, if we take, without loss of generality, v = Xh(m) + -M (m) =Xh(m) + XJ!(m), with - " gµ, the definition of the Hessian impliesthat

d2(C1 + . . . + Cn)(m)(v, w)

= w[(Xh + XJ!)(C1 + . . . + Cn)]

= w[{C1, h} + . . . + {Cn, h} + {C1, J#} + . . . + {Cn, J#}] = 0,

given that {Ci, h} = 0 since Ci is a conserved quantity for i "{1, . . . , n}, and {Ci, J#}(z) = dCi(z) · -M (z) = 0, for any z " M ,since Ci is Gµ–invariant. The Gµ–invariance of the conserved quan-tities C1, . . . , Cn, when restricted to J%1(µ), implies the existence ofthe functions C1

µ, . . . , Cnµ : Mµ % R, uniquely defined by the relation

Ciµ & /µ = Ci & iµ, with iµ : J%1(µ) )% M the natural inclusion, and


i " {1, . . . , n}. Since d(C1 + . . . + Cn)(m) = 0, we necessarily havethat

0 = (i"µd(C1 + . . . + Cn))(m) = d((C1 + . . . + Cn) & iµ)(m)

= d(C1 & iµ + . . . + Cn & iµ)(m)

= d(C1µ & /µ + . . . + Cn

µ & /µ)(m)

= d(C1µ + . . . + Cn

µ )([m]µ) & Tm/µ.

Since /µ is a surjective submersion, this implies that

d(C1µ + . . . + Cn

µ )([m]µ) = 0. (6.4.2)

Recall that by Theorem 1.5.2,

Xhµ([m]µ) = Tm/µ · Xh(m).

Hence, applying Tm/µ on both sides of (6.4.1) one obtains

kerdC1µ([m]µ) * . . . * kerdCn

µ ([m]µ) =

Tm/µ(W )( span{Xhµ([m]µ)}. (6.4.3)

Note that the sum in (6.4.3) is direct because if there was a w "W such that Tm/µ(w) = Tm/µ(Xh(m)), then w $ Xh(m) would bein the kernel of Tm/µ and therefore there would exist an element- " gµ such that w $ Xh(m) = -M (m). This would imply thatw "W * (span{Xh(m)}+Tm(Gµ ·m)) which, by the definition of W ,implies that w = 0.

In the language of Theorem 6.2.1, expression (6.4.3) shows thatTm/µ(W ) is a complement to span{Xhµ([m]µ)} in kerdC1

µ([m]µ) *. . . * kerdCn

µ ([m]µ) for the periodic point [m]µ. Moreover, (6.4.2)implies that d2(C1

µ + . . . + Cnµ )([m]µ) is well–defined. Using Proposi-

tion 5.2.1 we write

d2(C1µ + . . . + Cn

µ )([m]µ)(Tm/µ · w1, Tm/µ · w2)

= d2(/"µ(C1µ + . . . + Cn

µ ))(m)(w1, w2)

= d2((C1µ + . . . + Cn

µ ) & /µ)(m)(w1, w2)

= d2((C1 + . . . + Cn) & iµ)(m)(w1, w2)

= d2(i"µ(C1 + . . . + Cn))(m)(w1, w2)

= d2(C1 + . . . + Cn)(m)(w1, w2).


Since w1, w2 " W are arbitrary, this equality and the hypothesis ofthe theorem guarantees that

d2(C1µ + . . . + Cn

µ )([m]µ)|Tm&µ·W)Tm&µ·W

is a definite quadratic form. Therefore the periodic point [m]µ sat-isfies the hypothesis of Theorem 6.2.1, which implies the orbital sta-bility of 'µ, the periodic orbit in Mµ through [m]µ.

Our next step will be to show that this orbit is also stable in thespace M/Gµ. Since Gµ is compact and acts freely on M , by Theo-rem 1.5.1, M/Gµ is a Poisson manifold and the canonical projection/ : M % M/Gµ is a Poisson map. The point [m] := /(m) is clearlyperiodic with respect to the Poisson dynamics induced by h on M/Gµ.We will denote by ' the periodic orbit associated to [m] " M/Gµ.Since J%1(µ) is a regular submanifold of M , the reduced space Mµ isa regular submanifold of M/Gµ. Of course, [m] = [m]µ and ' = 'µ,but we want to distinguish in what follows between objects in M/Gµ

and Mµ.We now construct, with the help of the MGS normal form around

m, associated to the G–action on M , a local transversal section S inM/Gµ to the closed orbit ' at [m], such that Sµ = S *Mµ is a localsubmanifold of Mµ and moreover, it is a local transversal section to'µ at [m]µ. The MGS normal form on M states that a G–invariantopen neighborhood of m " M is G–equivariantly di!eomorphic to aG–invariant neighborhood of (e, 0, 0) " Y = G ) g"µ ) Vm and thatthis di!eomorphism maps m " M to (e, 0, 0) " Y . This impliesthat in a neighborhood of [m], M/Gµ is locally di!eomorphic to anopen neighborhood of ([e], 0, 0) in G/Gµ ) g"µ ) Vm. In addition, byTheorem 3.4.2, the space Mµ can be locally identified around [m]µwith {[e]} ) {0} ) Vm 2 Vm. Let now Sµ ' Vm 2 Mµ be a localtransversal section to 'µ at [m]µ and let S be the local codimensionone submanifold in M/Gµ around [m] given by

S := G/Gµ ) g"µ ) Sµ.

By construction, S is a local transversal section to ' at [m] such that

Sµ = S *Mµ.

If we now use Sµ, together with d(C1µ + . . . + Cn

µ )([m]µ) = 0 and thedefiniteness of

d2(C1µ + . . . + Cn

µ )([m]µ)|Tm&µ(W ))Tm&µ(W )


we can repeat the first part of the proof of Theorem 6.2.1 in orderto prove the existence of a constant a > 0 (see (6.2.1)) for which themap fµ : Mµ % R given by

fµ := a(C1µ $ C1

µ([m]µ) + . . . + Cnµ $ Cn

µ ([m]µ))

+ (C1µ $ C1

µ([m]µ))2 + . . . + (Cnµ $ Cn

µ ([m]µ))2

satisfies dfµ([m]µ) = 0 and d2(fµ|Sµ)([m]µ) is positive definite.Note also that the Gµ–invariance of h and C1, . . . , Cn implies

the existence of functions [h], [C1], . . . , [Cn] " C!(M/Gµ), uniquelydetermined by the relations

[h] & / = h and [Ci] & / = Ci.

We clearly have [h]|Mµ = hµ, [Ci]|Mµ = Ciµ for i " {1, . . . n}. Let f

be the extension of fµ to M/Gµ, defined by

f = a([C1]$ [C1]([m]) + . . . + [Cn]$ [Cn]([m]))

+ ([C1]$ [C1]([m]))2 + . . . + ([Cn]$ [Cn]([m]))2.

To make this extension f of fµ to M/Gµ completely explicit, we shalluse the notation introduced in the following commutative diagram

J%1(µ)iµ$$$% M

&µ

""#""#&

MµiMµµ$$$% M/Gµ.

and observe that

f & iMµµ = fµ.

Moreover, since h and C1, . . . , Cn are conserved quantities for Xh,f is a conserved quantity for X[h], the vector field induced by [h] onM/Gµ defined via its Poisson structure. Note that, by construction,

f & / = a(C1 $ C1(m) + . . . + Cn $ Cn(m))

+ (C1 $ C1(m))2 + . . . + (Cn $ Cn(m))2.


Since d-(C1 $ C1(m))2 + . . . + (Cn $ Cn(m))2

2(m) = 0 and C1 +

. . . + Cn di!ers from C1 $C1(m) + . . . + Cn $Cn(m) by a constant,it is clear that d(f & /)(m) = 0, which implies that for any v " TmM

df([m])(Tm/ · v) = d(/"f)(m) · v = d(f & /)(m) · v = 0.

As / is a surjective submersion, this implies that df([m]) = 0 andhence d2f([m]) is well defined.

Let now Z be the vector subspace of T[m](M/Gµ) defined by

Z := T[m]S * T[m]µiMµµ (T[m]Mµ),

or, locally in terms of the MGS normal form,

Z := {0}) {0}) T[m]µSµ.

We now show that d2 (f |S) ([m])|Z)Z is positive definite. We willuse the notation introduced in the following commutative diagram:

SµiSµ$$$% Mµ

iSµµ

""#""#i

Mµµ

SiS$$$% M/Gµ.

Let v1, v2 " Z. These vectors can be written as vi = T[m]µiSµµ · wi,

with wi " T[m]µSµ, i " {1, 2}. Since

d2 (f |S) ([m])(v1, v2)

= d2 (f |S) (iSµµ ([m]µ))(T[m]µi

Sµµ · w1, T[m]µi

Sµµ · w2)

= d2((iS & iSµµ )"f)([m]µ)(w1, w2)

= d2((iMµµ & iSµ)"f)([m]µ)(w1, w2)

= d2(fµ|Sµ)([m]µ)(w1, w2),

the positive definiteness of d2(fµ|Sµ)([m]µ) ensures thatd2 (f |S) ([m])|Z)Z is positive definite.

Keeping this in mind, we now define a real–valued function onM/Gµ with the help of an inner product 0·, ·1 on g", invariant underthe coadjoint action of Gµ (recall that Gµ is compact by hypothesis).If we denote by | · |=

;0·, ·1 the associated norm, we define the map:

j : M/Gµ $% R[z] -$% (|J(z)|$ |µ|)2.


Locally, around [m] 2 ([e], 0, 0), j can be expressed in terms of theMGS normal form as j([g], ., v) = (|Ad"

g"1(µ+ .)|$ |µ|)2. This mapis well-defined because if z$ = g · z for some g " Gµ, the equivarianceof J and Ad"

Gµ–invariance of 0·, ·1, guarantee that

(|J(z$)|$ |µ|)2 = (|J(g · z)|$ |µ|)2 = (|g · J(z)|$ |µ|)2 = (|J(z)|$ |µ|)2.

It is easy to see that j has a critical point at [m], that is dj[m] = 0.Moreover, we will now show that d2(j|S)([m]) is positive semidefinitewith kernel Z. Indeed, if [vi] = Tm/ · vi " T[m]S with vi " TmM andi " {1, 2}, suppose that

d2(j|S)([m])([v1], [v2]) = 0, for all [v2] " T[m]S. (6.4.4)

The definition of j immediately implies that

d2(j|S)([m])([v1], [v2]) = 2|TmJ · v1| |TmJ · v2|.

Since equality (6.4.4) holds for all [v]2 " T[m]S, it holds in particularfor [v2] = [v1]. In that case we have |TmJ · v1| = 0 which implies thatTmJ · v1 = 0, that is, v1 " ker TmJ. Let’s write this conclusion interms of the MGS normal form. In general, there are elements 0 " g,. " g"µ, and v " Sµ such that

v1 =d

dt

$$$$t=0

(exp t0, t., tv).

Then,

0 = TmJ · v1 =d

dt

$$$$t=0

Ad"exp(%t))(µ + t.) = $ad"

)µ + ..

However, note that ad")µ " g#µ, the annihilator of gµ in g", and . " g"µ.

Therefore, ad")µ = 0 and . = 0 since g#µ * g"µ = {0} in g". Hence,

v1 =d

dt

$$$$t=0

(exp t0, 0, tv),

with 0 " gµ and v " Sµ, which implies that [v1] = ([0], 0, v) " Z, asrequired.

Putting together what we know about f and j, Lemma 5.2.2guarantees the existence of a constant b > 0 such that the map,F := bf + j is such that d2(F |S)[m] is positive definite. Since both f


and j are constants of the motion for the Poisson vector field X[h], Fis too. Using this map, it can be shown by repeating the last stagesof the proof of Theorem 6.2.1, that [m] is a stable periodic point andhence ' is orbitally stable. (Recall that the topology of M/Gµ ismetrizable, since it is induced by the smooth structure of M/Gµ as aregular quotient manifold (see [BC70, Proposition 9.4.1]), a technicaldevice used in the proof of Theorem 6.2.1).

Once the orbital stability of ' has been proven, it is straightfor-ward to show the Gµ–stability of m as a RPP. If V is a Gµ–invariantopen neighborhood of Gµ·{Ft(m)}t>0, then /(V ) is an open neighbor-hood of ' in M/Gµ (since the canonical projection / is a submersion,it is an open map [BC70, Proposition 6.1.5]). The orbital stability of' in M/Gµ implies the existence of another open neighborhood U of[m] " M/Gµ such that U @ /(V ) and F [h]

t (U) @ /(V ) for all t > 0,where F [h]

t denotes the flow of X[h], characterized by the equality:

/ & Ft = F [h]t .

We will use this identity in order to prove that /%1(U) is the openset that we are looking for in order to conclude the Gµ–stability ofm. Since / is surjective and U @ /(V ) we have that

/%1(U) @ /%1(/(V )) = V.

We now show that Ft(/%1(U)) @ V for positive time. If u " M issuch that /(u) " U , we know that

F [h]t (/(u)) " /(V ) for all t AB /(Ft(u)) " /(V ) for all t.

Hence, for any t > 0 there is a g(t) " Gµ such that Ft(u) = g(t) · v;but since V is Gµ–invariant, g(t) ·v = Ft(u) " V , as required and theproof of Gµ–stability of m as an RPP is finished.

We now consider the case W = {0}. The proof in this case isidentical if we take fµ = (C1

µ $ C1µ([m]µ))2 + . . . + (Cn

µ $ Cnµ ([m]µ))2

which, in this particular case satisfies that dfµ([m]µ) = 0 andd2(fµ|Sµ)([m]µ) is positive definite. Note that W = {0} includesthe case dim Mµ = 2 since, by the relation (6.4.3):

kerdC1µ([m]µ) * . . . * kerdCn

µ ([m]µ) = Tm/µ(W )( span{Xhµ([m]µ)}.

Thus, if dimMµ = 2, then necessarily dimTm/µ(W ) = 0 which im-plies that W ' Tm(Gµ · m), and so, by construction, W = {0}. !

§ 6.5. Symmetric Energy–Integrals Method: Examples 223

6.5 Applications of the Symmetric Energy–Integrals Method

This section presents some simple examples on the application of thesymmetric energy–integrals method.

6.5.1 The S1–Stability of the Precessing Orbits of theSpherical Pendulum

The spherical pendulum consists of a particle of mass m, movingunder the action of a constant gravitational field of acceleration g,on the surface of a sphere of radius l. If we use spherical coordinateswith origin the center of the sphere and polar axis pointing verticallydownwards, the Lagrangian of this system is

L(7, ", 7, ") =12ml2(72 + "2 sin2 7) + mgl cos 7 (6.5.1)

The solution of the Euler–Lagrange equations that followfrom (6.5.1) is a classical problem on elliptic functions whose solutionshows that, generically, the motion of the bob describes RPOs in thephase space with respect to the S1 symmetry of the problem. Wewill characterize explicitly these RPOs and show that they are stablemodulo S1.

In order to use Theorem 6.4.1, we use the Legendre transformto write the system in phase space variables (7, ", p2, p!) of T "S2,where the canonical symplectic form is ) = d7 # dp2 + d" # dp!.From the Legendre transformation

p! =#L

#"= ml2 sin2 7"

p2 =#L

#7= ml27,

it follows that the Hamiltonian of the spherical pendulum can bewritten as

h(7, ", p2, p!) =p22

2ml2+

p2!

2ml2 sin2 7$mgl cos 7.

It may be readily verified that this system is invariant under the liftedaction to T "S2 of SO(2) on S2 by "–rotations. This action has the


well–known associated equivariant momentum map

J : T "S2 $% so(2)" 7= R(7, ", p2, p!) -$% p!.

We will restrict ourselves to regular values of J, that is, we will choosecertain µ += 0 in so(2)", and we will reduce at it. Clearly,

J%1(µ) = {(7, ", p2, µ) | (7, ", p2, µ) " T "S2},

and, since SO(2) is Abelian, Gµ = SO(2) acts via

SO(2)) J%1(µ) $% J%1(µ)(ei,, (7, ", p2, µ)) -$% (7, "+ $, p2, µ).

The reduced space Mµ = J%1(µ)/Gµ can be naturally identified withT "S1

+, where S1+ is the upper semicircle, by taking as canonical pro-

jection

/µ : J%1(µ) $% Mµ

(7, ", p2, µ) -$% (7, p2).

The reduced symplectic form !µ, uniquely determined by the relationi"µ! = /"µ!µ, takes the form, !µ = d7#dp2. The Hamiltonian reducesto

hµ(7, p2) =p22

2ml2+

µ2

2ml2 sin2 7$mgl cos 7,

which implies that (Mµ, !µ, hµ) is a simple mechanical system (itsHamiltonian has the form kinetic+potential) with potential energygiven by

Vµ(7) =µ2

2ml2 sin2 7$mgl cos 7.

In the classical literature (see [Go80, Lan]), Vµ(7) is called the ef-fective potential of the reduced problem. Figure 6.5.1 exhibits itsmain features, which allow us to classify the di!erent kinds of mo-tions that the system may generate in terms of the value of its totalenergy.

Note that Vµ(7) has a single minimum 7#, between 0 and /, de-termined by the relation

mgl sin 7# $µ2

ml2cot 7# csc2 7# = 0.


Figure 6.5.1: One e!ective potential of the spherical pendulum.

If the total energy of the system equals Vµ(7#) := Ecirc, the pendulumdescribes a circular orbit of radius l sin 7#, whose stability can bestudied using the energy–momentum method. If the total energy ofthe system E is such that E > Ecirc, the motion of the pendulumis bounded in its 7 coordinate between certain limit values 7min(E)and 7max(E), uniquely determined by the relation Vµ(7min(E)) =Vµ(7max(E)) = E; moreover, the motion in the reduced space Mµ isperiodic as we will prove below.

Proposition 6.5.1 Let (M, !, h) be a two dimensional Hamiltoniansystem and let m "M be a point such that h(m) = E, with E a regu-lar value of the Hamiltonian h, such that the connected component ofh%1(E) that contains m is compact. Then m is a periodic point, thatis, there is a & > 0 for which F" (m) = m, where Ft is the Hamiltonianflow generated by h.

Proof Since E is a regular value of h, dh(z) += 0, for any z " h%1(E).By the Liouville–Arnold Theorem (see [A78, page 271]), the con-nected component of h%1(E) that contains m is di!eomorphic to


S1 and there are action–angle coordinates (", !(E)), in which theHamiltonian flow given by h, satisfies the di!erential equations

d"

dt= !(E), ! = !(E).

If m is given in these coordinates by m := ("#, !#), with !# = !(E),then Ft(m) = (!#t + "#, !#), which clearly implies the periodicity ofm. !

We now apply this result on the reduced space (Mµ, !µ, hµ). Ifwe choose [m]µ = (7, p2), such that hµ([m]µ) = E > Ecirc, thehypothesis of Proposition 6.5.1 on the regularity of E is satisfied. Wenow show that h%1

µ (E) is compact in Mµ. Clearly h%1µ (E) is closed.

So all we need to show is its boundedness. As we already know,the 7 variable is bounded between certain limit values 7min(E) and7max(E). By conservation of energy, 7 and p2 are related by

E =p22

2ml2+

µ2

2ml2 sin2 7$mgl cos 7

or, equivalently

p2 = ±

A

2ml20

E $ µ2

2ml2 sin2 7+ mgl cos 7

1. (6.5.2)

Since p2 is a continuous function of 7 defined on the compact set[7min, 7max], strictly included on [0, /], it reaches a minimum anda maximum and, therefore it is bounded in h%1(E). Since h%1(E)is closed and bounded, it is compact and, by Proposition 6.5.1, theHamiltonian flow corresponding to hµ on this range of energies, con-sists of periodic orbits (see Figure 6.5.2).

These periodic orbits lift, by Theorem 1.6.2, to RPOs in T "S2.Since dimMµ = 2, Theorem 6.4.1 guarantees that these orbits areSO(2)–stable.

In order to completely characterize these orbits as RPOs, itjust remains to compute their relative periods, & , and phase shifts,g " SO(2) such that, in the notation of Theorem 1.6.2, Ft+" (m) =g · Ft(m), for arbitrary t. In order to give explicit expressions forthese parameters, we will make use of the solution of the sphericalpendulum in terms of elliptic functions, as can be found in classi-cal references [Whit, Pars, Law]. Firstly, for dimensional reasons we


Figure 6.5.2: Phase portrait of the reduced spherical pendulum.

define the parameters:

b =µ

m, c =

2m

E, + =D

g

l.

Using the expressions for !µ and hµ, previously introduced, one con-cludes that the evolution of the 7 variable is governed by the followingHamilton equations:

7 =#hµ

#p2and p2 = $#hµ

#7,

whose solution is given by

7(t) = arccos0$28(+t + !3)$

c

6gl

1, (6.5.3)

where we took 7(0) = 7min(E), and where 8(t) is the Weierstraßelliptic function 8(t | g2, g3), with invariants g2 and g3 given by

g2 = 1 +c2

12g2l2and g3 =

h2

4gl3+

c3

216g3l3$ c

6gl.


We will denote by !1 and !3 the unique primitive half-periods asso-ciated to g2 and g3, that is, in the traditional notation

8(t, !1, !3) = 8(t | g2, g3).

The evolution of the p2 coordinate can be obtained from (6.5.3), byusing relation (6.5.2).

Using the addition properties of the Weierstraß function, it canbe easily checked (see [Law]) that the function 7(t) takes a time equalto !1/+ to go from 7min(E) to 7max(E) (and vice versa), hence

&(E) =4!1

+

is the relative period that we are seeking.

Figure 6.5.3: Projection of the motion of the spherical pendulum onthe equatorial plane of the configuration space and phase shift.

In order to compute the phase shift, we will have to determinehow the " variable behaves. This is done by solving the di!erential


equation

" =h

l2 sin2 7,

with 7 given by (6.5.3). We choose the unique constants a and b suchthat

8(a) = $12$ c

12gl, 8(b) =

12$ c

12gl, and 8$(a) = 8$(b) =

ih;4gl3

.

In terms of these constants, "(t) is given by

"(t) = arg

<2ilC

120(+t + !3 $ a)0(+t + !3 + b)

0(a)0(b)02(+t + !3)e(-(a)%-(b))(t

=,

(6.5.4)

where 0(t) := 0(t, !1, !3) and 5(t) := 5(t, !1, !3) are the Weierstraßsigma and zeta functions respectively, with primitive half–periods !1

and !3. The constant C can be determined by taking, for example,"(0) = 0. Expressions (6.5.3) and (6.5.4) allow us to plot the pro-jection of the motion of the spherical pendulum on the equatorialplane of the configuration space. Clearly, the phase shift that weare looking for coincides with (" in Figure 6.5.3, that is, it is theangle covered by the trajectory of the pendulum in one of its relativeperiods:

(" = "

04!1

+

1

= arg0

2ilC120(4!1 + !3 $ a)0(4!1 + !3 + b)

0(a)0(b)02(4!1 + !3)e(-(a)%-(b))4%1

1.

6.5.2 Stability of the Nutating Motion of a LagrangeTop.

The Lagrange top is an axisymmetric rigid body with a fixed point,moving steadily in a constant gravitational field of acceleration g. Wewill denote by (I1, I1, I3) its principal moments of inertia, by m itsmass, and by l the distance between its center of mass and the fixedpoint. The phase space for the Lagrange top as a Hamiltonian sys-tem is T "SO(3). If we use the Euler angles (7, ", 4) to parameterize


SO(3), and denote by (p2, p!, p3) the conjugate momenta, the cor-responding Hamiltonian function in this chart on T "SO(3) has theexpression

h(7, ", 4, p2, p!, p3) =p22

2I1+

(p! $ p3 cos 7)2

2I1 sin2 7+

p23

2I3+ mgl cos 7.

Denote by ! the canonical symplectic form in T "SO(3). It canbe readily verified that the Hamiltonian system (T "SO(3), !, h) isinvariant under the lifted action of the group S1 ) S1 over SO(3)given by

(S1 ) S1)) SO(3) $% SO(3)--ei11 , ei12

2, (7, ", 4)

2-$% (7, "+ 21, 4 + 22)

.

This action is Hamiltonian and has an associated momentum mapgiven by

J : T "SO(3) $% Lie(S1 ) S1) 2 R2

(7, ", 4, p2, p!, p3) -$% (p!, p3) .

Using this symmetry we will proceed in a fashion similar to the spher-ical pendulum. Firstly, we will reduce the system at regular valuesof J, that is, we will restrict ourselves to values µ = (1, ,) " R2 of Jsuch that 1 += 0 and , += 0. Clearly,

J%1(µ) = {(7, ", 4, p2, 1, ,) | (7, ", 4, p2, 1, ,) " T "SO(3)}

and, since S1)S1 is Abelian, it follows that Gµ = S1)S1. Thus, thereduced space (Mµ, !µ, hµ) can be naturally identified with T "S1

+,with S1

+ the open upper semicircle, by taking as the canonical pro-jection:

/µ : J%1(µ) $% Mµ

(7, ", 4, p2, 1, ,) -$% (7, p2).

With this identification, !µ = d7 # dp2, and

hµ(7, p2) =p22

2I1+

(1$ , cos 7)2

2I1 sin2 7+,2

2I3+ mgl cos 7.

Analogously to the spherical pendulum, this is a simple mechanicalsystem, with potential energy (e!ective potential) given by:

Vµ(7) =(1$ , cos 7)2

2I1 sin2 7+,2

2I3+ mgl cos 7.


Figure 6.5.4: E!ective potential for the Lagrange top.

As can be seen in the Figure 6.5.4, Vµ(7) has a single minimum,7#, between 0 and /. When we tune the energy of the system to thevalue Ecirc := Vµ(7#) the system falls into a relative equilibrium withrespect to the S1)S1 symmetry, whose stability can be studied usingthe energy–momentum method. If the energy is strictly higher thanEcirc, the variable 7 is bounded between certain values 7min(E) and7max(E) for which Vµ(7min(E)) = Vµ(7max(E)) = E, and the systemdescribes a RPO, as we prove by showing that the motion in Mµ isperiodic using a method identical to the one followed in the case ofthe spherical pendulum and based on Proposition 6.5.1. The onlydi!erence is that in this case, 7 and p2 are related by

p2 = ±E

2I1(E $ Vµ(7)).

Figure 6.5.5 shows these periodic orbits, that lift by Theo-rem 1.6.2, to RPOs in T "SO(3). Since dimMµ = 2, Theorem 6.4.1guarantees that these orbits are S1 ) S1–stable.

We now give explicit expressions for the relative periods and phase


Figure 6.5.5: Phase portrait of the reduced Lagrange top.

shifts of this system. Our argument will be based on the solution ofthe equations of motion of the Lagrange top using elliptic functions,as can be found for instance in [Whit, Pars, Law]. See also [Go80,Ki85, Lan] for a qualitative analysis of the motion. We will assumethat our RPP, m, is such that J(m) = µ = (1, ,), and h(m) = E >Ecirc(µ). Using the expressions previously given for !µ and hµ, theevolution of the 7 variable is given by the di!erential equations

7 =#hµ

#p2and p2 = $#hµ

#7.

In order to express the solution of these equations, we follow Law-den [Law] in defining the following parameters, for dimensional rea-sons

n =,

I3, K = 2E $ I3n

2, P =2I1

mgl, Q =

I1K + I23n2

6I1mgl.

With these parameters, the solution for 7(t), with initial condition


7(0) = 7max(E) is

7(t) = arccos(P8(t + !3) + Q), (6.5.5)

where 8(t) is the Weierstraß elliptic function 8(t | g2, g3) with invari-ants g2 and g3 given by

g2 =1

12I41

(I1K + ,2)2 +mgl

I31

(I1mgl $ 1,),

g3 =1

216I61

(I1K + ,2)3 +1

12I51

mgl(I1mgl $ 1,)(I1k + ,2)

+1

4I41

(mgl)2(12 $ I1K).

As customary, we denote by !1 and !3 the unique half–periods asso-ciated to g2 and g3. The addition properties of 8(t | g2, g3) guaranteethat 7(t) takes a time equal to !1 to go from 7max(E) to 7min(E).Hence

&(E) = 2!1

is the relative period that we are seeking.We now compute the phase shift. Since the symmetry group is two

dimensional, the phase will have two components, namely ((", (4).We compute first (". The behavior of "(t) is governed by the dif-ferential equation

" =, $ 1 cos 7I1 sin2 7

, (6.5.6)

where 7 is given by (6.5.5), and whose solution can be expressed by

"(t) = arg0

2ilC120(t + !3 $ a)0(t + !3 + b)

0(a)0(b)02(t + !3)e(-(a)%-(b))t

1. (6.5.7)

As in the case of the spherical pendulum, 0(t) := 0(t, !1, !3) and5(t) := 5(t, !1, !3) are the Weierstraß sigma and zeta functions re-spectively, with primitive half–periods !1 and !3, and the constant Ccan be determined by taking, for example, "(0) = 0. The constantsa and b are uniquely determined by the relations

8(a) = $Q + 1P

, 8$(a) =2L

Pi, 8(b) = $Q$ 1

P, 8$(b) =

2M

Pi,


with

L =1+ ,2I1

and M =1$ ,2I3

.

It is clear that (" is the angle covered by "(t) while 7(t) covers oneof its periods, that is,

(" = "(2!1)

= arg0

2ilC120(2!1 + !3 $ a)0(2!1 + !3 + b)

0(a)0(b)02(2!1 + !3)e(-(a)%-(b))2%1

1.

In order to find (4 we have to solve the di!erential equation

4 =1

I3$ " cos 7,

with " as in (6.5.6) and 7 as in (6.5.5). If we substitute (6.5.6) andwe write + = cos 7

4 =1

I3$ 1+$ ,+2

I1(1$ +2)

=1

I3$ ,

I1$ 1+$ ,

I1(1$ +2)

=1

I3$ ,

I1+

A

++ 1$ B

+$ 1, (6.5.8)

with

A =, $ 12I1

and B =, + 12I1

.

It is easy to check that, up to the summand *I3$ '

I1, expression (6.5.8)

is the di!erential equation satisfied by ", changing A and B by L andM respectively. Hence,

4(t) =12i

ln00(t + !3 $ a$)0(t + !3 + b$)0(t + !3 + a$)0(t + !3 $ b$)

1

+1i(5(a$)$ 5(b$))t +

01

I3$ ,

I1

1t + C,

where C can be determined by taking for example the initial condition4(0) = 0, and a$ and b$ are in this case constants uniquely determinedby the relations

8(a$) = $Q + 1P

, 8$(a$) =2A

Pi , 8(b$) = $Q$ 1

P, and 8$(b$) =

2B

Pi .


The phase shift that corresponds to the 4 coordinate is therefore

(4 = 4(2!1)

=12i

ln00(2!1 + !3 $ a$)0(2!1 + !3 + b$)0(2!1 + !3 + a$)0(2!1 + !3 $ b$)

1

+1i(5(a$)$ 5(b$))2!1 +

01

I3$ ,

I1

12!1 + C.

6.5.3 Stability of the Bounded Manev Orbits. The Pre-cessing Orbit of Mercury.

It is well–known that due to general relativistic corrections, even inthe two body approximation, the planets don’t follow Kepler’s FirstLaw, that is, their orbits do not describe ellipses but precessing el-lipses. In a first approximation, this correction has the form B/r2,for some constant B, that is, truncating negligible terms, the gravi-tational potential takes the form

V (r) =k

r+

B

r2. (6.5.9)

The introduction of potentials of this form to describe the gravita-tional motion goes back to Newton and Clairaut (see [Dia] for ex-cellent historical remarks). However, it was Manev [Ma24, Ma25,Ma30, Ma30a] who, using physical principles, more specifically, ageneralized action–reaction principle, was the first to propose a po-tential like (6.5.9) as a correction to the classical Newtonian poten-tial useful in celestial mechanics. The Hamiltonian flow inducedby (6.5.9) has been extensively studied, and completely classifiedin [Dal96, Dia, LLN], moreover, Diacu, Mioc, and Stoica [Dia] haveshown that in a certain approximative regime (what they call thesolar–system approximation), Manev’s model is the natural classicalanalog of the Schwarzschild problem. In fact, using the values for kand B given by Manev, one obtains an accurate description of theapsidal motion of the Moon and the perihelion advance of Mercury.These values are:

k = GM, B =GM'

2, ' =

3GM

c2,


where G is the constant of gravitation, c is the speed of light, M isthe mass of the particle at the origin, and the mass of the rotatingparticle is taken to be one.

One of the conclusions of [Dal96] is that the bounded motions ofthis problem, that is, the solutions with negative energy, are gener-ically precessing ellipses. We will concentrate on this case, and wewill show that this part of the flow consists generically of Gµ–stableRPOs.

The phase space for this problem, as a Hamiltonian system, isT "R3. If we parameterize R3 using spherical coordinates (r, 7, ") (7denotes the colatitude and " the azimuth), the corresponding Hamil-tonian function of the system can be written as

h(r, 7, ", pr, p2, p!) =p2

r

2m+

p22

2mr2+

p2!

2mr2 sin2 7$ k

r$ B

r2,

where m denotes the reduced mass of the two bodies:

m =M

M + 1.

This system is invariant under the lifted action of SO(3) to T "R3.Moreover, this action is Hamiltonian with equivariant momentummap given by the angular momentum of the system, that is,

J(r, p) = r ) p,

whose expression in spherical coordinates is

J(r, 7, ", pr, p2, p!)= ($p! cos" cot 7 $ p2 sin", p2 cos"$ p! cot 7 sin", p!).

Given that Theorem 6.4.1 is valid only for regular values of themomentum map, we will restrict ourselves to values µ += 0 of J.More specifically, we will choose our coordinate system in such afashion that, without loss of generality, µ has the form µ = (0, 0, l)with l += 0. We compute J%1(µ). If (r, 7, ", pr, p2, p!) " J%1(µ),then necessarily p! = l, p2 tan" = $l cot 7, and l cot 7 tan" = p2.Eliminating p2 this implies that cot 7(1+tan2 ") = 0, hence cot 7 = 0,and therefore 7 = //2 and p2 = 0. Hence, J%1(µ) equals the setF

(r,/

2, ", pr, 0, l) | (r,

/

2, ", pr, 0, l) " T "R3 and µ = (0, 0, l)

G.


The coadjoint isotropy subgroup Gµ of µ is isomorphic to SO(2)and acts on J%1(µ) as

Gµ ) J%1(µ) $% J%1(µ)-ei,,-r, &

2 , ", pr, 0, l22

-$%-r, &

2 , "+ $, pr, 0, l2.

Hence, Mµ = J%1(µ)/Gµ, can be naturally identified with T "R+, bytaking as the canonical projection

/µ : J%1(µ) $% Mµ-r, &

2 , ", pr, 0, l2-$% (r, pr).

Moreover, with this identification, the reduced Hamiltonian is

hµ(r, pr) =p2

r

2m+

l2

2mr2$ k

r$ B

r2

=p2

r

2m$ k

r+

l2 $ 2mB

2mr2. (6.5.10)

As we know, the reduced symplectic form !µ is uniquely determinedby the relation

i"µ! = /"µ!µ.

Since in spherical coordinates ! is given by ! = dr#dpr +d7#dp2 +d" # dp! it follows that

!µ = dr # dpr.

This implies that (Mµ, !µ, hµ) is a simple mechanical system withpotential energy (e!ective potential):

Vµ(r) = $k

r+

l2 $ 2mB

2mr2,

that is, the Hamiltonian flow in Mµ induced by hµ is given by theequations

r =#hµ

#prpr = $#hµ

#r.

Note that the Manev reduced potential Vµ is identical to theone corresponding to the Kepler problem with momentum equal toD

l2 $ 2mB. In other words, the reduced Manev system with mo-mentum µ = (0, 0, l), is identical to the reduced Kepler system with


momentum µ$ = (0, 0,D

l2 $ 2mB). Hence, up to this momentumshift, the reduced dynamics of both systems are identical. It is thegeometrical phase that lifts the dynamics in Mµ to M that di!er-entiates between the Kepler and the Manev systems, as we will seebelow.

We will focus on the bounded motions of the reduced system.These motions occur provided that the total momentum of the systeml, satisfies

l >D

2mB.

In such a case, the e!ective potential Vµ(r), looks like the one inFigure 6.5.6.

Figure 6.5.6: One e!ective potential in the Manev problem.

The particular form of Vµ, for negative values E of the energy,forces the r variable to be bounded between the values rmin(E) andrmax(E) that are given as the solutions of the quadratic equation

E = $k

r+

l2 $ 2mB

2mr2,


that is,

rmax(E) = $ 12E

<k +

Dmk2 + 2El2 $ 4BEm

m

=(6.5.11)

rmin(E) = $ 12E

<k $D

mk2 + 2El2 $ 4BEm

m

=. (6.5.12)

Also, Vµ(r) admits a unique minimum at the value r# = l2%2mBmk , for

which

Vµ(r#) = $ mk2

2(l2 $ 2mB):= Ecirc.

If the energy E of the system is such that E = Ecirc, there is a cir-cular orbit, which is a relative equilibrium with respect to the SO(3)symmetry. Its stability can be studied using the energy–momentummethod. If the energy is such that

0 > E > $ mk2

2(l2 $ 2mB),

the system describes a RPO, as we prove by showing that the motionin Mµ is periodic using a method identical to the one followed in thecase of the spherical pendulum (based on Proposition 6.5.1). Theonly di!erence is that in this case, r and pr are related by

pr = ±

A

2m

0E +

k

r$ l2 $ 2mB

2mr2

1.

We could have proven that the reduced orbits were closed by remem-bering that they coincide, as we said, with the reduced Keplerianellipses, with shifted angular momentum.

Figure 6.5.7 shows these periodic orbits that lift, by Theo-rem 1.6.2, to RPOs in T "R3. Since dimMµ = 2, Theorem 6.4.1guarantees that these orbits are SO(2)–stable.

These RPOs have been studied by Delgado et al [Dal96], who havegiven explicit expressions for them. In order to completely character-ize these orbits as RPOs, we will compute their phase shifts and rel-ative periods. The phase shift in this case coincides with the angularshift (" that the particle experiences after reaching two consecutive,say rmax(E), radial positions (see Figure 6.5.8).


Figure 6.5.7: Phase portrait of the reduced Manev problem (E < 0).

Since p! = l = mr2" is a constant, we conclude that

d" =l

mr2dt. (6.5.13)

At the same time, expression (6.5.10) can be written in terms of r as,

E =mr2

2$ k

r+

l2 $ 2mB

2mr2,

which implies that

dr

dt=

A2m

0E +

k

r$ l2 $ 2mB

2mr2

1,

and hence,

dt =drD

2m

3E + k

r $l2%2mB

2mr2

4 , (6.5.14)


Figure 6.5.8: Projection over the configuration space of a boundedManev orbit.

which substituted in (6.5.13) gives

d" =l drD

2m3Er4 + kr3 $ (l2%2mB)r2

2m

4 .

Hence

(" = 2& rmin(E)

rmax(E)

l drD2m3Er4 + kr3 $ (l2%2mB)r2

2m

4 .

Since the square root in the denominator of the integrand vanishesat rmin(E) and rmax(E), the integral is (convergent) improper. Atedious but straightforward integration yields,

(" =2/lD

l2 $ 2mB.


Notice that, as could be expected, in the Keplerian case (B = 0),(" = 2/. Also, the orbit is closed, whenever there are integers nand m such that

(" =2/lD

l2 $ 2mB=

n

m.

The relative period & may be computed with the help of expres-sion (6.5.14), which, integrated between rmin(E) and rmax(E) yields

& = 2& rmin(E)

rmax(E)

mr drD2mr2E + 2mrk $ l2 + 2mB

=kD

m/D2 | E |3/2

.

Notice that this expression coincides with Kepler’s Third Law andthat B is absent. This is another consequence of our remark on therelation between the Manev and Kepler reduced spaces; since theyare identical and & is the period of the reduced periodic orbits, it isthe same in both cases.

6.6 The Symmetric Energy–IntegralsMethod. The General Case

In this section, we generalize the symmetric energy–integrals method,that is, Theorem 6.4.1, to Poisson systems and to the case in whichthe RPPs in question have non–trivial symmetry. The proof in thissituation will not be anymore a corollary of the energy–integralsmethod given that in this case, the correspondence between the Gµ–stability of the RPO and the orbital stability of the singular reducedperiodic orbit don’t imply each other. A block diagonalization of thestability form presented in the following section will give us accurateinformation on this point.

Theorem 6.6.1 (The symmetric energy–integrals method)Let (M, {·, ·}, G, J : M % g", h : M % R) be a Hamiltoniansystem with a symmetry given by the Lie group G acting properly onM . Assume that the Hamiltonian h " C!(M) is G–invariant andthat J is equivariant. Let m " M be a RPP that is not a relativeequilibrium, such that J(m) = µ " g" and the coadjoint isotropysubgroup Gµ is compact. Then, if there is a set of Gµ–invariantconserved quantities C1, . . . , Cn " C!(M), for which

d(C1 + . . . + Cn)(m) = 0,

§ 6.6. Singular Symmetric Energy–Integrals 243

and

d2(C1 + . . . + Cn)(m)|W)W


kerdC1(m) * . . . * kerdCn(m) * ker TmJ= W ( (span{Xh(m)}( Tm(Gµ · m)), (6.6.1)

then m is a Gµ–stable RPP. If dimW = 0, then m is always a Gµ–stable RPP. The matrix d2(C1 + . . . + Cn)(m)|W)W will be referredto as the stability form of the RPP m.

Proof We first prove the case W += {0} and we begin by showingthat the result does not depend on the choices of m in the RPO andW . Indeed, if d(C1 + . . . + Cn)(m) = 0 and Ft is the flow of theHamiltonian vector field Xh, then for any t > 0 and any v, w " TmMwe have

d(C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))= F "

t (d(C1 + . . . + Cn)(m))(v, w)= d(F "

t (C1 + . . . + Cn))(m)(v, w)= d(C1 + . . . + Cn)(m)(v, w),

since F "t &d = d&F "

t and C1, C2, . . . , Cn are invariant under Ft. If Wis a complement to (span{Xh(m)}(Tm(Gµ ·m)) in kerdC1(m)* . . .*kerdCn(m)* ker TmJ, then for any t > 0, TmFt(W ) is a complementto (span{Xh(Ft(m))} ( Tm(Gµ · Ft(m))) in kerdC1(Ft(m)) * . . . *kerdCn(Ft(m))* kerTFt(m)J. Moreover, d2(C1 + . . . + Cn)(m)|W)W

is definite i! d2(C1+. . .+Cn)(Ft(m))|TmFt·W)TmFt·W is definite, sincethe conservation of C1, . . . , Cn, implies that for any v, w " TmM :

d2(C1 + . . . + Cn)(Ft(m))(TmFt(v), TmFt(w))

= d2(F "t C1 + · · · + F "

t Cn)(m)(v, w)

= d2(C1 + . . . + Cn)(m)(v, w).

The statement of the theorem does therefore not depend on the choiceof the point m in the RPO.

The choice of W is also irrelevant since d2(C1 + . . . +Cn)(m)(v, w) = 0, whenever v " span{Xh(m)} ( Tm(Gµ · m). In-deed if we take, without loss of generality v = Xh(m) + -M (m) =


Xh(m) + XJ!(m), with - " gµ, then

d2(C1 + . . . + Cn)(m)(v, w) = w[(Xh + XJ!)[C1 + · · · + Cn]]= w[{C1, h} + · · · + {Cn, h}+ {C1, J#} + · · · + {Cn, J#}] = 0,

since the functions Ci, for i " {1, . . . , n}, are Gµ–invariant conservedquantities for the evolution induced by h and therefore {Ci, h} = 0and, for any z " M , {Ci, J#}(z) = dCi(z) · -M (z) = 0, by Gµ–invariance.

We now construct a Gµ–invariant local transversal section for Xh

at m with the help of the Slice Theorem. Since Gµ is closed in G andG acts properly on M , so does Gµ. Therefore, there is a Gµ–invariantneighborhood of Gµ ·m that can be represented as a Gµ–space by (seeTheorem 3.1.2)

Yµ = Gµ )H B,

with B = Tm/Tm(Gµ · m), and where the point m is represented by[e, 0]. For times t small enough, the flow Ft of Xh is representedin these coordinates by Ft[e, 0] = [g(t), b(t)], where g(t) " Gµ, andb(t) " BH . Indeed, by the G–equivariance of Ft, H = G[e, 0] =G[g(t), b(t)], hence [g(t), b(t)] " (Yµ)H = NGµ(H))HBH , which impliesthat g$(0) " Lie(NGµ(H)) and b := b$(0) " BH . Notice that, sincem is not a relative equilibrium, b += 0 necessarily. The subspacespan{b} of BH ' B is H–invariant in B. The compactness of H andProposition 1.3.4, guarantee the existence of a H–invariant subspaceBI ' B such that

B = span{b}(BI .

The set Gµ)H BI is a submanifold of Yµ and, by construction, thereis a Gµ–invariant neighborhood T of m , [e, 0] in Gµ )H BI , suchthat for any z " T , Xh(z) /" TzT , that is, T is a Gµ–invariant localtransversal section to Xh at m. We now define S as the submanifoldof T given by

S := (H )H BI) * T,

and prove the following


Lemma 6.6.1 With the notation previously introduced, the subman-ifold S satisfies:

TmM = TmS ( Tm(Gµ · m)( span{Xh(m)}.

Proof Note first that the sum Tm(Gµ · m) ( span{Xh(m) is indeeddirect since there is no - " gµ for which Xh(m) = -M (m), for thisequality is equivalent to m being a relative equilibrium (see Theo-rem 4.1.1) which we assume is not the case; m is a genuine RPP.

Second, we show that the sum TmS+(Tm(Gµ ·m)(span{Xh(m)})is also direct by showing that (Tm(Gµ ·m)( span{Xh(m)})*TmS ={0}. Indeed, since m is identified with [e, 0], we conclude that the h–action on gµ)B (which is the linearization of the H–action at (e, 0))is just the translation by h ) {0} in gµ ) B. Thus, we can identifyTmM 7= gµ/h)B, TmS 7= {0})BI , and Tm(Gµ ·m) 7= (gµ/h)) {0}.Finally, Xh(m) is represented by a pair of the form (- + ., b), forsome . " h, - " Lie(NGµ(H)) ' gµ, and b /" BI . Thus, an arbitraryvector w " Tm(Gµ · m) ( span{Xh(m)} is represented by (1 + ., b),for . " h, 1 " gµ, and b " B. However, if this vector is also in TmS,then this representative must lie in {0} ) BI , that is, 1 = $. " hand b " BI * span{b} = {0}. Therefore, this representative must bean element of h ) {0} which in the quotient is the zero vector; weshowed that w = 0.

Third, dim(TmS) = dimB $ 1 = dimM $ dimGµ + dimH $ 1,dim(Tm(Gµ · m)) = dimGµ $ dimH, and dim(span{Xh(m)}) = 1,hence

dim(TmS ( Tm(Gµ · m)( span{Xh(m)}) = dimM = dim(TmM),

which concludes the proof. #We now define

Z := TmS * kerTmJ * kerdC1(m) * . . . * kerdCn(m).

The Reduction Lemma (Tm(Gµ ·m) = kerTmJ*Tm(G ·m), Noether’sTheorem, and the choice of the functions C1, . . . , Cn imply that

Tm(Gµ · m)( span{Xh(m)} ' kerdC1(m) * . . . * kerdCn(m) * ker TmJ.


This inclusion and Lemma 6.6.1 allow us to write

kerdC1(m) * . . . * kerdCn(m) * ker TmJ= TmM * kerdC1(m) * . . . * kerdCn(m) * ker TmJ= (TmS * kerTmJ * kerdC1(m) * . . . * kerdCn(m))( (Tm(Gµ · m) + span{Xh(m)})

= Z ( (Tm(Gµ · m) + span{Xh(m)}),

hence Z is one of the spaces that satisfy the defining conditions of Win the statement of the theorem.

Let now f1 and f2 be the functions defined by

f1 = (C1 $ C1(m)) + . . . + (Cn $ Cn(m))

f2 = (C1 $ C1(m))2 + . . . + (Cn $ Cn(m))2 + <J$ µ<2,

where in f2, the norm is associated to some Ad"Gµ

–invariant innerproduct in g" (always available by the compactness of Gµ); this makesf2 Gµ–invariant. Since f1 and C1 + · · ·+Cn, di!er by a constant, thehypothesis of the theorem implies that df1(m) = df2(m) = 0, andthat the form d2f1|Z)Z is definite. Taking into account that

d2f1(m)|Z)Z =-d2f1(m)|TmS)TmS

2$$Z)Z

= d2(f1|S)(m) |Z)Z ,

the hypothesis of the theorem implies that d2(f1|S)(m)|Z)Z is defi-nite.

We now prove that Z is the kernel of d2(f2|S)(m). It is easy tosee that if v1, v2 " TmS then

d2(f2|S)(m)(v1, v2)= 2 [(dC1(m) · v2)(dC1(m) · v1) + . . .

. . . + (dCn(m) · v2)(dCn(m) · v1) +<TmJ · v1< <TmJ · v2<] .

Then, v1 " kerd2(f2|S)(m) i! for any v2 " TmS, we have that

(dC1(m) · v2)(dC1(m) · v1) + . . . + (dCn(m) · v2)(dCn(m) · v1)+ <TmJ · v1< <TmJ · v2< = 0.

In particular, for v1 = v2, this identity implies that dC1(m) · v1 =. . . = dCn(m) · v1 = <TmJ · v1< = 0 and hence v1 " kerTmJ *


kerdC1(m) * . . . * kerdCn(m) * TmS = Z. Conversely, if v1 " Z =TmS * kerdC1(m)* . . .* kerdCn(m)* kerTmJ the above relation issatisfied trivially for all v1 " TmS. Therefore,

Z = kerd2(f2|S)(m).

Using these remarks, Lemma 5.2.2 guarantees the existence ofsome a > 0 for which the function f defined by

f := af1 + f2 (6.6.2)

is such that d2(f |S)(m) is positive definite. Note that f is a Gµ–invariant integral of the motion such that f(m) = 0. Shrinking T ifnecessary, the Morse lemma allows us to choose S such that f 4 0on S. We now prove the following

Lemma 6.6.2 The submanifold S is a slice at m " T ' M for theGµ–action on T .

Proof By Theorem 3.1.1, it is enough to prove that Gµ · S is anopen neighborhood of Gµ ·m in T and that there is a Gµ–equivariantretraction r : Gµ · S % Gµ · m such that r%1(m) = S. Withoutloss of generality we may take S = H )H BI , and T = Gµ )H BI .Clearly Gµ · S = T , which is trivially open in T , and the equivariantretraction that we need is

r : Gµ · S = T $% Gµ · m[g, b] -$% [g, 0] , g · m.

The map r is clearly well–defined, it is Gµ–equivariant, and r%1(m) ={[h, b] " T | h " H, b " BI} = S. #

Notice that Theorem 3.1.1 guarantees that the Gµ–invariant localtransversal section T is locally di!eomorphic to Gµ )H S , Gµ · S.

We now recall that, by Theorem 4.1.3, the RPP m has a phaseshift in Gµ. The dynamic orbit through m can therefore be consid-ered as a RPO associated to the Gµ symmetry of the system, andtaking T as a Gµ–invariant local transversal section, Theorem 1.6.3guarantees the existence of a Gµ–equivariant Poincare section $, withGµ–invariant open sets W0, W1 ' T , and Gµ–invariant period func-tion & : W0 % R.

With all these tools we will prove the Gµ–stability of m. Let V bean arbitrary Gµ–invariant open neighborhood of Gµ·{Ft(m)}t>0. The


positive definiteness of d2(f |S)(m) and the Morse lemma guaranteethe existence of certain * > 0 such that

f%1[0, *) * S ' V *W0 *W1, (6.6.3)

with f%1[0, *)*S an open subset of S. Notice that the Gµ–invarianceof f implies that f%1[0, *) is Gµ–invariant, in particular H–invariant.This allows us to define the open submanifold A of T as

A := Gµ )H (f%1[0, *) * S) , Gµ · (f%1[0, *) * S).

The Gµ–invariance of V, W0 and W1 and (6.6.3) guarantee that

A = Gµ(f%1[0, *) * S) ' Gµ · (V *W0 *W1)= V *W0 *W1. (6.6.4)

We now show that if / : M % M/Gµ is the continuous canonicalprojection of M onto the Hausdor! (see Proposition 1.3.1) quotienttopological space, then the closed orbit ' in M/Gµ corresponding tothe RPP m, that is ' = /({Ft(m)}t+0), is orbitally stable.

The Gµ–invariance of V, W0, W1, T , and A, allows us to define'V = V/Gµ = /(V ) and, analogously, >W0, >W1, 'T , and 'A. Also, let'$, '( and Fµ

t , be the continuous maps uniquely determined by theequalities: '$ & / = / &$, '( & / = (, and Fµ

t & / = / & Ft, with Ft theG–equivariant Hamiltonian flow of Xh. Note that by construction'A, >W0, and >W1 are included in 'T and that, for any [z] = /(z) " >W0,'$([z]) = Fµ([z], & $ '(([z])) " >W1.

We now see how the Gµ–invariance of f guarantees that if [z] " 'A,then '$([z]) " 'A. Indeed, if z = l · a with l " Gµ and a " f%1[0, *) *S 'W0*W1*S, then $(z) = F"%/(z)(z) = l·F"%/(l·a)(a). By the Gµ–invariance of (, F"%/(l·a)(a) = F"%/(a)(a), and as a "W0, F"%/(a)(a) "W1 ' T , necessarily. Since by Lemma 6.6.2, T , Gµ )H S, there areelements n " Gµ, and b " S such that F"%/(a)(a) = n · b. Now, sincef is a Gµ–invariant conserved quantity

f($(z)) = f(F"%/(z)(z)) = f(z) = f(a);

but, at the same time

f($(z)) = f(F"%/(z)(z)) = f(ln · b) = f(b) = f(a),


which guarantees that b " f%1[0, *) * S ' W0 *W1 * S and hence'$([z]) = [b] " 'A.

Note that expression (6.6.4) implies that

'A ' >W0 * >W1 * 'V ,

since

'A = /(A) ' /(W0 *W1 * V ) ' /(W0) * /(W1) * /(V ) = >W0 * >W1 * 'V .

Also, the Gµ–invariance of V (resp. A), guarantees that 'V (resp. 'A)is open in the quotient topology of M/Gµ: 'V is open i! /%1('V ) isopen in M . We show that /%1('V ) = V . Indeed, it is always true thatV ' /%1(/(V )) = /%1('V ). Conversely, if z " /%1('V ), /(z) = /(v)for some v " V , hence z = l · v for some l " Gµ. The Gµ–invarianceof V guarantees that z " V , and therefore V = /%1('V ).

Notice that since ' = Fµ([0, & ], [m]) and Fµ is continuous, ' isa compact subset of M/Gµ. This fact, together with the openness of'V , implies that the number D'V defined by

D'V = inf{d([x], ') | [x] " Cl'V \ 'V },

is never zero, where d is the distance function associated to a metricon M/Gµ that induces its quotient topology and Cl'V is the point settopological closure of the set 'V . This metric always exists and canbe constructed as follows: take a Gµ–invariant Riemannian metric onM (always available by the compactness of Gµ). This makes M intoa metric space whose metric topology coincides with the topology ofM (see [BC70, Proposition 10.6.2]). With this metric, Gµ acts byisometries, which implies that the distance function drops to M/Gµ,endowing it with a metric structure.

We define the map:

D : 'A $% R[z] -$% D([z]) := max

t'[0, "%'/([z])]d(Fµ

t ([z]), ').

Note that D([m]) = 0. By the continuity of D, we can choose * > 0(and therefore 'A) small enough so that D([z]) < D'V /2, for any [z] "'A. Define the open neighborhood 'U of ' by

'U := {Fµt ([z$]) | [z$] " 'A, t 4 0}.


We shall prove below that Fµt ('U) ' 'V for all t 4 0. In order to

see this, note that, by construction, 'U is invariant under the flow Fµt

and hence the claim is proved if we show that 'U ' 'V . Let’s supposethe contrary, namely that there is an element Fµ

t ([z$]) " 'U, [z$] " 'A,such that Fµ

t ([z$]) /" 'V . Without loss of generality we can assume thatt " [0, & $ '(([z$])] which then implies that d(Fµ

t ([z$]), ') = D([z$]) <D'V /2. However, since we assume that Fµ

t ([z$]) /" 'V , it follows thatd(Fµ

t ([z$]), ') 4 D'V , by the definition of D'V . This contradictionguarantees that ' is orbitally stable. We now see how the orbitalstability of ' implies the Gµ–stability of m, taking as the open setthat we need U = /%1('U) that is, Ft(z) " V , for any positive time tand any z " U : we know that Fµ

t ([z]) " 'V = /(V ), for any positivet. Since Fµ

t is defined by the relation

/ & Ft = Fµt & /,

it follows that / & Ft(z) = /(v), for some v " V . Hence there existssome h " Gµ such that Ft(z) = h · v but, since V is Gµ–invariant,h · v " V and, therefore Ft(U) ' V , as required. This proves the casedimW += 0.

If W = {0}, the proof is completely analogous, but in this caseone takes f2 as f , given that {0} = Z = kerd2(f2 |S)(m), and henced2(f2 |S)(m) is positive definite. !

Example 6.6.1 The previous theorem applies to the collision solu-tions of the examples presented in Section 6.5 and that could not betreated there, given their singular nature. For instance,

(i) The spherical pendulum: If the angular momentum of thependulum is equal to zero, the spherical pendulum becomes aplanar pendulum. However, zero is not a regular value of themomentum map associated to the SO(2) symmetry of the sys-tem, hence we need Theorem 6.6.1 to deal with the stabilityof these solutions that consist of an equilibrium, when the en-ergy of the system equals $mgl, and a set of planar periodicorbits when the energy is bigger. These periodic orbits are notorbitally stable, since a small perturbation on p! makes thesystem precess. However, as trivial RPOs (RPOs whose phaseshift is the identity), they are S1–stable, which follows fromTheorem 6.6.1 by showing that in this case W = {0}.


(ii) The Manev problem: In this case the situation is analogous.When the angular momentum equals zero, we have a collisionproblem, that is, apart from the equilibrium solution, the par-ticle describes an oscillatory motion on a straight line that goesthrough the origin. If we consider these motions as periodicorbits, they are not orbitally stable. However, as trivial RPOs,Theorem 6.6.1 shows that they are SO(3)–stable (W = {0})."

6.7 Block Diagonalization and Reduced Peri-odic Orbits

We know from Theorem 4.1.3 that the projection of a RPO onto thereduced space produces a periodic orbit. In Section 6.4 we saw how,in the regular case, the Gµ–stability of the RPO is equivalent to theorbital stability of the corresponding periodic orbit in the associatedMarsden–Weinstein reduced space. In this section we will constructa block diagonalization of the stability form based on the isotypic de-composition of a linear representation of a compact Lie group, similarin philosophy to the one carried out in Chapter 5 for relative equi-libria. By virtue of the blocking obtained, it will be easy to see thegeneralization to the singular case of the link between the stability ofthe RPO and its associated (singular) reduced periodic orbit.

In all that follows, we use the notation introduced in the statementof Theorem 6.6.1. In order to use the results on singular reduction inits simplest form, we will assume that N(H), the normalizer of theisotropy of the RPO is a compact subgroup of G.

Let g be a H–invariant metric on M , always available by thecompactness of H, and define

A := Tm(Gµ · m)( span{Xh(m)}.

Obviously,

TmM = (Tm(Gµ · m)( span{Xh(m)})(A(,

where A( is the orthogonal complement to A relative to the in-ner product induced by g on TmM . Since A ' kerdC1(m) * . . . *


kerdCn(m) * kerTmJ, we have that

kerdC1(m) * . . . * kerdCn(m) * ker TmJ= (Tm(Gµ · m)( span{Xh(m)})( (kerdC1(m) * . . . * kerdCn(m) * ker TmJ *A()

:= A(W, (6.7.1)

where we use W for kerdC1(m) * . . . * kerdCn(m) * ker TmJ * A(,since it is obviously one of the spaces mentioned in the hypothesis ofTheorem 6.6.1 needed to construct the stability form.

Proposition 6.7.1 The subspace W = kerdC1(m) * . . . *kerdCn(m)*ker TmJ*A(, constructed above using the H–invariantmetric g, has the following properties:

(i) W is H–invariant as a subspace of TmM , where H acts on Wvia the natural lifted action.

(ii) The vector subspace WH of H–fixed vectors is naturally isomor-phic to W (H)

µ , where W (H)µ is such that

kerd(C1)(H)µ ([m](H)

µ ) * . . . * kerd(Cn)(H)µ ([m](H)

µ )

= W (H)µ ( span{X

h(H)µ

([m](H)µ )}.

The functions (Ci)(H)µ , with i " {1, . . . , n} are uniquely defined

by the identity

(Ci)(H)µ & /(H)

µ = Ci & ı(H)µ .

Proof (i) The space kerdC1(m) * . . . * kerdCn(m) * ker TmJ isclearly H–invariant by the Gµ–invariance of the functions Ci and theG–equivariance of J. We therefore just need to show that A( is H–invariant. Let v " A(; by definition, for any - " gµ, g(m)(v, -M (m)+Xh(m)) = 0. The vector k · v for k " H behaves similarly because,by the H–invariance of g

g(m)(k · v, -M (m) + Xh(m)) = g(m)(v, k%1 · -M (m) + k%1 · Xh(m))= g(m)(v, (Adk"1-)M (m) + Xh(m))= 0,

§ 6.7. Block Diagonalization for RPOs 253

since Adk"1- " gµ.(ii) Recall that, since N(H) is compact, (M (H)

µ , !(H)µ ) is natu-

rally symplectomorphic to the Marsden–Weinstein reduced space(K%1

L (+#)/L($ , !($). Abusing the notation, we will denote by /(H)µ

the surjective submersion

/(H)µ : K%1

L (+#) $% K%1L (+#)/L($ ,

and by i(H)µ the injection

i(H)µ : K%1

L (+#) )%M,

given that

K%1L (+#) = J%1(µ) *MH ,

and

K%1L (+#)/L($ = (J%1(µ) *MH)/(NGµ(H)/H).

Since T[m]

(H)µ/(H)

µ is surjective with kernel Tm(L($ · m), we can iden-

tify (non–canonically because of the choice of m) T[m](H)

µM (H)

µ with

TmK%1L (+#)/Tm(L($ · m), and, since

WH = W * TmMH

= kerdC1(m) * . . . * kerdCn(m) * ker TmJ *A( * TmMH

= kerdC1(m) * . . . * kerdCn(m) *A( * TmK%1L (+#)

' TmK%1L (+#),

we can define the linear map

( : WH $% T[m](H)

µM (H)

µ7= TmK%1

L (+#)/Tm(L($ · m)

w -$% w + Tm(L($ · m) = Tm/(H)µ · w.

We first show that ( is injective: if w "WH is such that w+Tm(L($ ·m) = Tm(L($ ·m) then w " Tm(L($ ·m) necessarily and hence thereis an element - " Lie(NGµ(H)) ' gµ such that w = -M (m). SinceW * (Tm(Gµ · m) ( span{Xh(m)}) = {0}, it follows that w = 0 andtherefore ( is injective and an isomorphism onto its image, which we


will prove is a vector space W (H)µ := ((WH) ' T

[m](H)µ

M (H)µ , such

that

kerd(C1)(H)µ ([m](H)

µ ) * . . . * kerd(Cn)(H)µ ([m](H)

µ )

= W (H)µ ( span{X

h(H)µ

([m](H)µ )}. (6.7.2)

We show first that the sum W (H)µ + span{X

h(H)µ

([m](H)µ )} is direct,

that is,

W (H)µ * span{X

h(H)µ

([m](H)µ )} = {0}.

Suppose, without loss of generality, that there is a v " WH suchthat ((v) = v + Tm(L($ · m) = X

h(H)µ

([m](H)µ ). By Theorem 2.4.1,

Xh(H)µ

([m](H)µ ) = Tm/

(H)µ · Xh(m). If we write ((v) as Tm/

(H)µ · v,

then Tm/(H)µ (Xh(m)$ v) = 0 and, therefore there exists an element

- " Lie(NGµ(H)) such that Xh(m)$v = -M (m), hence v = Xh(m)+-M (m) and therefore v " A *W = {0}.

We now prove the equality in (6.7.2). Since every ((v) " W (H)µ

is such that v " kerdC1(m)* . . .*kerdCn(m)*kerTmJ, this impliesthat for any i " {1, . . . , n},

d(Ci)(H)µ ([m](H)

µ ) ·((v) = Tm((Ci)(H)µ & /(H)

µ )(m) · v= d(Ci & i(H)

µ )(m) · v = 0,

and, therefore

W (H)µ ( span{X

h(H)µ

([m](H)µ )}

' kerd(C1)(H)µ ([m](H)

µ ) * . . . * kerd(Cn)(H)µ ([m](H)

µ ).

Conversely, let [v](H)µ = Tm/

(H)µ · v " kerd(C1)

(H)µ ([m](H)

µ ) * . . . *kerd(Cn)(H)

µ ([m](H)µ ) with v " TmK%1

L (+#) = kerTmJ * TmMH .Clearly, this implies that v " ker TmJ * TmMH * kerdC1(m) * . . . *kerdCn(m), and hence it can be uniquely decomposed as v = v1 +v2,with v1 " A and v2 "W . Without loss of generality we assume thatv1 = -M (m) + Xh(m), for some - " gµ. Since v " TmMH , k · v = v,for any k " H, and hence

k · -M (m) + k · Xh(m) + k · v2 = v1 + v2,


or, equivalently

(Adk-)M (m) + Xh(m) + k · v2 = -M (m) + Xh(m) + v2.

Since Adk- " gµ, the directness of the splitting (6.7.1) implies thatk · -M (m) = -M (m) and that k · v2 = v2 for all k " H and,therefore [v](H)

µ = Xh(H)

µ([m](H)

µ ) + Tm/(H)µ · v2, with v2 " TmMH *

W = WH which guarantees that [v](H)µ = X

h(H)µ

([m](H)µ ) + ((v2) "

span{Xh(H)

µ([m](H)

µ )}(W (H)µ . !

Definition 6.7.1 We call the subspace W = kerdC1(m) * . . . *kerdCn(m)*kerTmJ*A(, the stability subspace through the RPPm " M , associated to the constants of the motion C1, . . . , Cn, andthe H–invariant metric g on M .

We now state the main result of this section.

Theorem 6.7.1 Assume the hypotheses and notations of Theo-rem 6.6.1 and Proposition 6.7.1, and that, in addition the groupN(H) is compact. Let W be the stability subspace through m " M ,associated to a H–invariant metric on M . Then, the stability formof the RPP m, can be written as

d2(C1 + . . . + Cn)(m)$$W)W

=

5

6667

A 0

0A1 · · · 0... . . . ...0 · · · Ar

8

999:, (6.7.3)

where A = d2((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )$$$W (H)

µ )W (H)µ

is the or-

bital stability form associated to the periodic point [m](H)µ " M (H)

µ ,and A1, . . . , Ar are the restrictions of d2(C1 + . . . + Cn)(m) to thenon–trivial isotypic components W1, . . . , Wr of the stability subspaceW .

Proof We see first how the block structure in (6.7.3) is determinedby the isotypic decomposition of W . The subspace WH is the trivial


isotypic component of W . Therefore, by Theorem 1.3.1, there existsubspaces W1, . . . , Wr such that

W = WH (W1 ( . . .(Wr, (6.7.4)

is the isotypic decomposition of W . The H–invariance of C1+. . .+Cn

implies that d2(C1 + . . . + Cn)(m)|W)W , considered as an automor-phism of W , is H–equivariant. Theorem 1.3.1 and (6.7.4) imply theblock diagonal form of (6.7.3).

It remains to be shown that the (1, 1)–block of (6.7.3) equalsd2((C1)

(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )$$$W

(H)µ )W

(H)µ

. The entries of the

(1, 1)–block that we want to compute have the expressions

d2(C1 + . . . + Cn)(m)(v, w), for arbitrary v, w "WH .

By Lemma 5.4.2, there is a N(H)–invariant function g on M , whoseHamiltonian flow F v

t satisfies v = ddt

$$t=0

F vt (m). We extend w to a

vector field W along F vt (m) by setting

W(F vt (m)) = TmF v

t · w.

By the definition of the Hessian we get

d2(C1 + . . . + Cn)(m)(v, w) (6.7.5)= v[W[C1 + . . . + Cn]]

=d

dt

$$$$t=0

W[C1 + . . . + Cn](F vt (m))

=d

dt

$$$$t=0

(dC1(F vt (m)) · TmF v

t · w + . . . + dCn(F vt (m)) · TmF v

t · w).

(6.7.6)

Recall that the (Ci)(H)µ are defined by the relations (Ci)

(H)µ & /(H)

µ =Ci & i(H)

µ , with i " {1, . . . , n}. We identify again (M (H)µ ,!(H)

µ ) with(K%1

L (+#)/L($ ,!($), and we take for /(H)µ and i(H)

µ (abusing the no-tation) the canonical projection /(H)

µ : K%1L (+#) % K%1

L (+#)/L($

and immersion i(H)µ : K%1

L (+#) )% M . If v, w " WH , the vectorTm/

(H)µ ·w "W (H)

µ can be extended, using F vt , to a vector field W(H)

µ

along /(H)µ (F v

t (m)) by

W(H)µ (/(H)

µ (F vt (m))) = Tm(/(H)

µ & F vt ) · w.


Then, since d((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ ) = 0, we get

d2((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )(Tm/

(H)µ · v, Tm/

(H)µ · w)

=d

dt

$$$$t=0

(d(C1)(H)µ ((/(H)

µ & F vt )(m))(Tm(/(H)

µ & F vt ) · w) + . . .

+ d(Cn)(H)µ ((/(H)

µ & F vt )(m))(Tm(/(H)

µ & F vt ) · w))

=d

dt

$$$$t=0

(d((C1)(H)µ & /(H)


t · w + . . .

+ d((Cn)(H)µ & /(H)


t · w)

=d

dt

$$$$t=0

(d(C1 & i(H)µ )(F v

t (m)) · TmF vt · w + . . .

+ d(Cn & i(H)µ )(F v

t (m)) · TmF vt · w)

=d

dt

$$$$t=0

(dC1(F vt (m)) · TmF v

t · w + . . . + dCn(F vt (m)) · TmF v

t · w).

which coincides with expression (6.7.6). SInce, by definition, W (H)µ =

Tm/(H)µ · WH , the fact that

d2(C1 + . . . + Cn)(m)(v, w)

= d2((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )(Tm/

(H)µ · v, Tm/

(H)µ · w),

for arbitrary v, w "WH , proves the equality in (6.7.3). !

SUMMARY OF THE METHOD

We have shown that taking the stability subspace W =kerdC1(m) * . . . * kerdCn(m) * ker TmJ * A( through the RPPm " M , associated to the constants of the motion C1, . . . , Cn, andthe H–invariant metric g, the relative periodic point m is Gµ–stableif the symmetric matrix

5

6667

A 0

0A1 · · · 0... . . . ...0 · · · Ar

8

999:, (6.7.7)


is definite, where A = d2((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )$$$W (H)

µ )W (H)µ

and A1, . . . , Ar are the restrictions of d2(C1 + . . . + Cn)(m) to thenon–trivial isotypic components W1, . . . , Wr of the H–space W .

Summarizing, given a RPP m " M , theorems 6.6.1 and 6.7.1guarantee that m is stable modulo Gµ if the following three conditionsare satisfied:

(i) The bilinear form d2((C1)(H)µ + . . . + (Cn)(H)

µ )([m](H)µ )$$$W (H)

µ )W (H)µ

is definite and, therefore, the associated singular reduced peri-odic orbit is orbitally stable;

(ii) The bilinear forms d2(C1 + . . . + Cn)(m)$$Wi)Wi

are definite, forany i " {1, . . . , r};

(iii) All the definite bilinear forms in (i) and (ii) have the same sign.

Chapter 7

Bifurcation ofHamiltonian RelativeEquilibria

Capıtulo XXVI: Donde se prosigue la graciosaaventura del titerero, con otras cosas en verdadharto buenas.Cervantes, Don Quijote de la Mancha, II

7.1 Introduction

In this chapter we will describe a method and some of its applica-tions, to deal with bifurcations of relative equilibria in Hamiltoniansystems with symmetry. In the framework of the general theory ofsymmetric dynamical systems, bifurcation is a very well studied topic(see [GS84b] and references therein) and, even though Hamiltoniansystems can be considered as a (non generic) particular case of thistheory, the existence of additional structures suggests a particular-ized study taking into account the specific dynamical features thatarise from them. Some progress has been made in this direction (seefor instance [RdSD, MR]).

The strategy that we will follow is in certain sense similar to thatof Krupa [Kr90] for general dynamical systems, by means of which

260 Chapter 7. Bifurcation in Hamiltonian Systems

he is able to locally decompose equivariant vector fields as the sumof one component tangent to the group orbit and another one normalto it. This constitutes the so called Krupa decomposition. Thisdecomposition simplifies considerably the study of the dynamics andallows a very clear description of several phenomena. The first di"-culty when trying to adapt this ideas to the Hamiltonian case is thatthis approach is based on the splitting of the vector field, structurethat in the Hamiltonian setup is rarely used. We will solve this prob-lem with the introduction of the Witt–Artin decomposition overwhich we will construct a step by step bifurcation method, in whicheach step will be related to one of the factors of the decomposition.

This chapter describes a joint work with P. Chossat and D.Lewis [CLOR].

7.2 The Witt–Artin Decomposition

In this section we will work in a symplectic manifold (M, !) endowedwith a globally Hamiltonian symmetry, given by the proper action ofa Lie group G. As customary, we will denote by J : M % g" theequivariant momentum map associated to this action. Given a pointm " M in the manifold, the Witt–Artin decomposition provides afour way decomposition of the vector space TmM , that we describein what follows. This splitting was first proved by E. Witt [Wi37]for symmetric bilinear forms, which was followed by a remark of E.Artin [Ar63], noting that the proof also worked for symplectic forms.See [CB97] for this and other very interesting historical remarks, aswell as [BL97], for an application of this splitting in the constructionof the MGS normal form.

Let H := Gm be the isotropy subgroup of m "M . If J(m) = µ "g", the equivariance of the momentum map J implies that H ' Gµ

and therefore, the associated Lie algebras h and gµ satisfy that h ' gµ.Let q and m vector subspaces of g such that

g = gµ ( q, and gµ = h(m. (7.2.1)

In this case, no invariance properties are required on (7.2.1).

Lemma 7.2.1 The vector subspace q · m ' TmM , defined by

q · m = {-M (m) | - " q},

is a symplectic vector subspace of TmM .

§ 7.2. The Witt–Artin Decomposition 261

Proof Let - " q be such that for any . " q

!(m)(-M (m), .M (m)) = 0. (7.2.2)

By the definition of the momentum map, if 5 " gµ

!(m)(-M (m), 5M (m)) = !(m)(XJ!(m), 5(m))

= dJ#(m) · 5M (m)= 0TmJ · 5M (m), -1 = 0. (7.2.3)

So, adding (7.2.2) and (7.2.3) we get

!(m)(-M (m), .M (m) + 5M (m)) = 0,

for arbitrary . " q and 5 " gµ. Given that by construction g = gµ(q,this implies that

-M (m) " (g · m)%(m) * g · m = ker TmJ * g · m = gµ · m,

hence, - " gµ * q = {0} and consequently -M (m) = 0, as required.!

The equivariance of the momentum map implies also that gµ ·m 'kerTmJ. Let V ' kerTmJ be a complement to gµ · m in kerTmJ,that is,

ker TmJ = gµ · m( V.

Lemma 7.2.2 The vector subspace V ' TmM just introduced is asymplectic vector subspace of TmM such that V * q · m = {0}.

Proof Let v " V be such that

!(m)(v, v$) = 0,

for any v$ " V . Since V ' kerTmJ = (g · m)%(m) ' (gµ · m)%(m),

!(m)(v, -M (m)) = 0 for any - " gµ,

so,

v " V %(m) * (gµ · m)%(m) = (V ( gµ · m)%(m) = (kerTmJ)%(m) = g · m.


Hence, by the Reduction Lemma, v " V * g · m ' kerTmJ * g · m =gµ · m and, consequently,

v " gµ · m * V = {0}.

We now show that V * q · m = {0}. First, notice that

gµ · m * q · m = {0}. (7.2.4)

Indeed, let - " gµ and . " q, such that -M (m) = .M (m). Then,necessarily - $ . " h ' gµ and therefore, . " gµ * q = {0}. As aconsequence of (7.2.4) we have that

g · m = gµ · m( q · m, (7.2.5)

(g · m)%(m) = (gµ · m)%(m) * (q · m)%(m). (7.2.6)

Now,

V * q · m = V * q · m( (gµ · m * q · m) (by (7.2.4))' (V ( gµ · m) * q · m= (g · m)%(m) * q · m= (gµ · m)%(m) * (q · m)%(m) * (q · m) (by (7.2.6))

= (gµ · m)%(m) * {0} = {0}, (by Lemma 7.2.1)

as required. !

Remark 7.2.1 It can be very easily verified that the symplectic vec-tor space V is symplectomorphic to the symplectic normal space at m,(Vm, !Vm), introduced in Definition 3.2.1. Moreover, if V is chosen asthe orthogonal complement to gµ ·m in kerTmJ with respect to someH–invariant inner product in TmM , the vector space V becomes aH–representation space, and the symplectomorphism between V andVm is H–equivariant. This will be relevant in future constructionsgiven that, since the Constant Rank Embedding Theorem requiresjust equality of symplectic normal bundles modulo isomorphisms,one can construct another MGS normal form for the G–action onM around the orbit G · m, in which Vm is replaced by V (see Re-mark 3.2.2). "

The previous lemma guarantees that the sum V + q ·m is direct.Moreover, V ( q · m is a symplectic subspace of TmM , since V and


q ·m are independently symplectic subspaces of TmM . This providesthe first step in the Witt–Artin decomposition of TmM :

TmM = V ( q · m( (V ( q · m)%(m).

We can go further in this splitting by using the following lemma.

Lemma 7.2.3 The subspace gµ ·m is a Lagrangian subspace of (V (q · m)%(m).

Proof We first show that gµ ·m ' (V (q ·m)%(m), which is equivalentV ' (gµ ·m)%(m) and q ·m ' (gµ ·m)%(m). Indeed, the first inclusionholds because V ' ker TmJ = (g · m)%(m) ' (gµ · m)%(m). As to thesecond one, note that q · m ' g · m = (kerTmJ)%(m) ' (gµ · m)%(m).

We now show that gµ · m is a Lagrangian subspace of (V ( q ·m)%(m), that is,

(gµ · m)%(m)|(V %q·m)'(m) = gµ · m,

or, equivalently,

(gµ · m)%(m) * (V ( q · m)%(m) = gµ · m. (7.2.7)

In order to prove this equality we will use that the definition of thesubspace V , implies that

g · m = (gµ · m)%(m) * V %(m). (7.2.8)

Additionally, recall that if A, B, and C are vector subspaces of avector space E, such that A ' B, and A * C = {0}, then

(B * C)(A = B * (C (A). (7.2.9)

We now proof equality (7.2.7). Indeed,

(gµ · m)%(m)*(V ( q · m)%(m)

= (gµ · m + (V ( q · m))%(m)

= (g · m + V )%(m) (by (7.2.5))

= (g · m)%(m) * V %(m)

= ((gµ · m)%(m) * V %(m))%(m) * V %(m) (by (7.2.8))

= (((gµ · m)%(m) * V %(m))( V )%(m)

= ((gµ · m)%(m) * (V %(m) ( V ))%(m) (by (7.2.9))

= ((gµ · m)%(m) * TmM)%(m) (by Lemma 7.2.2)= gµ · m,


as required. !According to the previous lemma, we can choose a complement

W to gµ · m in (V ( q · m)%(m) such that

(V ( q · m)%(m) = gµ · m(W.

The choice of a subspace W completes the construction of the Witt-Artin decomposition of TmM which reads:

TmM = V ( q · m( gµ · m(W. (7.2.10)

This decomposition of the tangent space of M at m will allow usto carry out what we promised in Remark 3.2.3: we will construct alinear map with the same properties as the linearization at m of theequivariant symplectomorphism 2 given by the MGS normal form.Recall that, as we mentioned there, the technical convenience of thenormal form is obscured by the fact that the equivariant symplec-tomorphism 2 that links the manifold M with the model Y is notexplicit and, in general, cannot be implemented in a systematic wayin particular examples, however, the Witt–Artin decomposition al-lows us to construct an isomorphism

f : TmM = V ( q · m( gµ · m(W $% T[e, 0, 0]Y 7= g/h)m" ) V,

that shares all the technically convenient properties of Tm2. Notethat when we write T[e, 0, 0]Y 7= g/h)m" ) V we assume that, usingthe ideas introduced in Remark 7.2.1, the symplectic tube

Y = G)H (m" ) V ),

has been constructed taking V instead of Vm. This requires H–invariance in the construction of the splitting

kerTmJ = V ( gµ · m,

which will be assumed in all that follows.

Before we get into the description of the isomorphism f , recallthat the action of Gµ ' G on M is proper and globally Hamiltonian,by inheritance from the G–action, and with equivariant momentummap Jµ : M % g"µ, given by

0Jµ(z), -1 = 0J(z), -1,


for any z " M and - " gµ. In the remainder of this section thesplittings in (7.2.1) are assumed to be AdH–invariant, that is, theycoincide with those utilized in the construction of the MGS normalform. Moreover, if we denote by

g" = g"µ ( q", and g"µ = h" (m" (7.2.11)

the dual splittings of (7.2.1), the momentum map Jµ can be writtenas Jµ = pµ &J, where pµ is the natural projection g" = g"µ( q" % g"µ.

Lemma 7.2.4 In the conditions just presented, the momentum mapJµ : M % gµ satisfies that

TmJµ(TmM) = m".

Proof The Bifurcation Lemma in these hypotheses implies that

TmJµ(TmM) = (h)ann(g#µ),

where (h)ann(g#µ) denotes the annihilator of h in g"µ, which, accordingto (7.2.11) coincides with m". !

We are now in position to describe the isomorphism f .

Proposition 7.2.1 Suppose that the splittings (7.2.1) and (7.2.11)are H–invariant, that the vector space V is constructed as a H–invariant orthogonal complement to gµ · m in kerTmJ, and that Wis a Lagrangian complement to gµ · m in (V ( q · m)%(m). Then, thelinear mapping

f : V ( q · m( gµ · m(W $% g/h)m" ) Vv + -M (m) + .M (m) + w -$% ([- + .], TmJµ(w), v),

with - " q, . " gµ, v " V , and w " W is a H–equivariant symplec-tomorphism between (TmM, !(m)) and (T[e, 0, 0]Y, !Y ([e, 0, 0]), suchthat:

(i) f(-M (m)) = -Y ([e, 0, 0]), for any - " g.

(ii) TmJ = T[e, 0, 0]JY & f.

Proof Recall that it is always possible to choose W to be Lagrangianin (V ( q · m)%(m) since, by Lemma 7.2.3, gµ · m is Lagrangian in(V ( q · m)%(m). The existence of a Lagrangian complement W is a


straightforward consequence of the definition of Lagrangian subspace(see [AM78, Definitions 5.3.1, Proposition 5.3.3]).

The mapping f is clearly linear. In order to show that it isinjective let - " q, . " gµ, v " V , and w " W be such thatf(v + -M (m) + .M (m) + w) = ([0], 0, 0). We see immediately thatv = 0. Also, since [- + .] = [0], we have that - + . " h necessarily,and therefore, -M (m)+.M (m) = 0. Regarding w "W , we have thatTmJµ(w) = 0, hence:

w " ker TmJµ *W = (gµ · m)%(m) *W (by the Reduction Lemma)

' (gµ · m)%(m) * (V ( q · m)%(m)

= gµ · m. (by (7.2.7))

Consequently, w " gµ · m * W = {0}, and f is injective. SincedimTmM = dimT[e, 0, 0]Y , the monomorphism f is a linear isomor-phism. The ideas that we just elaborated have the following corollary.

Corollary 7.2.1 The linear map TmJµ|W : W % m" is an isomor-phism.

Proof We just showed that

ker TmJµ|W = kerTmJµ *W = {0},

and hence TmJµ|W is injective. A straightforward dimension countyields that dimW = dimm". Now, by the dimensions formula:

rankTmJµ|W = dimW + dim(kerTmJµ|W ) = dimW = dimm",

hence, the map TmJµ|W is also surjective. #

We now show that f is symplectic, that is,

f"!Y ([e, 0, 0]) = !(m).

We will follow the notation introduced in Diagram 3.2.3, taking intoaccount the change from (Vm, !Vm) to (V, !V ), where !V is definedby

!V = !(m)|V .


Let -1, -2 " q, .1, .2 " gµ, v1, v2 " V , and w1, w2 " W arbitrary.Now,

f"!Y ([e, 0, 0])(v1 + (-1)M (m) + (.1)M (m) + w1,

v2 + (-2)M (m) + (.2)M (m) + w2)= !Y ([e, 0, 0])(([-1 + .1], TmJµ(w1), v1),

([-2 + .2], TmJµ(w2), v2))= !Y (/(e, 0, 0))(T(e, 0, 0)/(-1 + .1, TmJµ(w1), v1),

T(e, 0, 0)/(-2 + .2, TmJµ(w2), v2))

= (/"!Y )(e, 0, 0)((-1 + .1, TmJµ(w1), v1),(-2 + .2, TmJµ(w2), v2))

= (/"(L")0))(e, 0, 0)((-1 + .1, TmJµ(w1), v1),(-2 + .2, TmJµ(w2), v2))

= (l"(/"0)0))(e, 0, 0)((-1 + .1, TmJµ(w1), v1),(-2 + .2, TmJµ(w2), v2))

= (l"(i"0()( !V ))(e, 0, 0)((-1 + .1, TmJµ(w1), v1),(-2 + .2, TmJµ(w2), v2))

= ()( !V )(e, 0, 0, 0)((-1 + .1, TmJµ(w1), 0, v1),(-2 + .2, TmJµ(w2), 0, v2))

= !(m)(v1, v2) + 0(TmJµ · w2, 0), (-1 + .1)1$ 0(TmJµ · w1, 0), (-2 + .2)1+ 0µ, [-1 + .1, -2 + .2]1. (7.2.12)

The equivariance of the momentum map J implies that the mappingdefined by

J : g $% C!(M)- -$% J#

is a Lie algebra homomorphism (see expression (1.4.2)). In particular,

0µ, [-1 + .1, -2 + .2]1 = 0J(m), [-1 + .1, -2 + .2]1= J[#1+$1, #2+$2](m)

= {J#1+$1 , J#2+$2}(m),


which substituted in (7.2.12) gives us

!(m)(v1, v2) + dJ$1µ (m) · w2 $ dJ$2

µ (m) · w1

+ !(m)((-1)M (m) + (.1)M (m), (-2)M (m) + (.2)M (m))= !(m)(v1, v2) + !(m)((.1)M (m), w2)$ !(m)((.2)M (m), w1)

+ !(m)((-1)M (m), (-2)M (m))

+ !(m)((.1)M (m), (.2)M (m)). (since gµ · m ' (q · m)%(m))(7.2.13)

At the same time,

!(m)(v1 + (-1)M (m) + (.1)M (m) + w1,

v2 + (-2)M (m) + (.2)M (m) + w2)= !(m)(v1 + (-1)M (m), v2 + (-2)M (m))

+ !(m)((.1)M (m) + w1, (.2)M (m) + w2),(7.2.14)

where we used that V ( q · m = (gµ · m ( W )%(m). Recall that,by construction, V ' ker TmJ = (g · m)%(m). This implies that!(m)((-1)M (m), v2) = !(m)((-2)M (m), v1) = 0, which in (7.2.14)yields

!(m)(v1, v2) + !(m)((-1)M (m), (-2)M (m))+ !(m)((.1)M (m), (.2)M (m)) + !(m)((.1)M (m), w2)

+ !(m)(w1, (.2)M (m)) + !(m)(w1, w2). (7.2.15)

Since we chose W to be symplectic, the last summand in the pre-vious expression vanishes, that is !(m)(w1, w2) = 0, and expres-sions (7.2.15) and (7.2.13) coincide, establishing the symplecticity off .

We now proof the claims (i) and (ii). The first one follows triv-ially from the definition of the isomorphism f . As to point (ii), let- " q, . " gµ, v " V , and w "W . Then,

T[e, 0, 0]JY & f(v+-M (m) + .M (m) + w)

= T[e, 0, 0]JY (([- + .], TmJµ(w), v))

=d

dt

$$$$t=0

exp t(- + .) · (µ + TmJµ(tw) + JV (tv))

= $ad"(#+$)µ + TmJµ(w). (7.2.16)

§ 7.3. The Bifurcation Method 269

Let 1 " gµ and & " q arbitrary but fixed. Now, using (7.2.16),

0T[e, 0, 0]JY & f(v + -M (m) + .M (m) + w), 1+ &1= 0$ad"

(#+$)µ + TmJµ(w), 1+ &1= 0$ad"

(#+$)µ, 1+ &1+ 0TmJµ(w), 11= $0µ, [- + ., 1+ & ]1+ 0TmJµ(w), 11= 0µ, [&, -]1+ !(m)(1M (m), w)

= {J" , J#}(m) + !(m)(1M (m), w)= !(m)(&M (m), -M (m)) + !(m)(1M (m), w)

(7.2.17)

where we used the equivariance of the momentum map J, and thatsince ., 1 " gµ, then 0µ, [1, -]1 = 0µ, [1, .]1 = 0µ, [&, .]1 = 0.

At the same time,

0TmJ(v + -M (m) + .M (m) + w), 1+ &1= !(m)(1M (m) + &M (m), v + -M (m) + .M (m) + w)= !(m)(1M (m) + &M (m), -M (m) + .M (m) + w),

since V ' (g · m)%(m),

= !(m)(1M (m), w) + !(m)(1M (m), .M (m))+ !(m)(&M (m), -M (m)),

since gµ · m(W = (V ( q · m)%(m),

= !(m)(&M (m), -M (m)) + !(m)(1M (m), w). (7.2.18)

since !(m)(1M (m), .M (m)) = dJ*(m)·.M (m) = 0TmJ·.M (m), 11 =0, given that .M (m) " gµ · m ' kerTmJ. Since expressions (7.2.17)and (7.2.18) coincide, claim (ii) follows. !

7.3 The Bifurcation Method

We now present, using the splitting introduced in the previous sec-tion, a step by step algorithm, we call it the bifurcation method,that allows us to study the relative equilibria that bifurcate from agiven one, in a Hamiltonian system with symmetry. The kinematical


setup in which we will work will be identical to the one in the pre-ceding section, and the dynamics will be induced by a G–invariantHamiltonian h " C!(M), whose associated Hamiltonian vector fieldhas a relative equilibrium at m "M . If J(m) = µ " g" and H := Gm,we will denote by - " Lie(NGµ(H)) a velocity of the relative equilib-rium m. Let

TmM = V ( q · m( gµ · m(W (7.3.1)

be the Witt–Artin decomposition of TmM , constructed using the hy-potheses of Proposition 7.2.1.

Definition 7.3.1 In the setup just described, a smooth mapping ' :m" ) V % M , is said to be a slice mapping at the point m " M ,when it is a di!eomorphism onto its image in an open neighborhoodof the point (0, 0) " m")V , and the following conditions are satisfied

(SM1) '(0, 0) = m,

(SM2) For any (., v) " m" ) V in the neighborhood of (0, 0) inwhich ' is a di!eomorphism onto its image, we have that

T'($, v)M = T($, v)'(m" ) V ) + g ·'(., v),

(SM3) TmJ & T(0, 0)' = T[e, 0, 0]JY |m#)V ,

(SM4) T(0, 0)'(m" ) V ) * g · m = {0}.

We will call the '–slice of the G–action at m, to the image under 'of the neighborhood of (0, 0) " m")V in which ' is a di!eomorphismonto its image.

The following examples show that slice mappings can be con-structed.

Example 7.3.1 Let " be the inverse of the MGS G–equivariant sym-plectomorphism 2, introduced in Theorem 3.2.2. The mapping

' : m" ) V $% M(., v) -$% "([e, ., v]),

is a slice mapping at the point m. Indeed,

'(0, 0) = "([e, 0, 0]) = 2%1([e, 0, 0]) = m,


by the construction of the MGS normal form. Property (SM3) isa consequence of the fact that J & " = JY . As to property (SM2),since " is a local di!eomorphism, for (., v) " m" ) V small enough,every vector u " T'($, v)M can be written as

u =d

dt

$$$$t=0

"([exp t1, . + t.$, v + tv$]),

with 1 " g, .$ " m", and v$ " V . Since the mapping " is G–equivariant

u =d

dt

$$$$t=0

"([exp t1, .+, v]) +d

dt

$$$$t=0

"([e, . + t.$, v + tv$])

=d

dt

$$$$t=0

exp t1 · "([e, ., v]) +d

dt

$$$$t=0

"([e, . + t.$, v + tv$])

= 1M ('(., v)) + T($, v)'(.$, v$)

" g ·'(., v) + T($, v)'(m" ) V ),

which shows that (SM2) holds. Property (SM4) is easily verified."Remark 7.3.1 The previous example justifies the term slice map-ping since, as can be easily checked, locally around (0, 0) " m" ) V ,the submanifold '(m" ) V ) 'M is a slice at m for the G–action onM . Note that since there is no systematic way to construct the map-ping ", the slice mapping just introduced cannot be implemented inparticular cases. The following example provides a way to do so, us-ing the Witt–Artin decomposition and the isomorphism f associatedto it that we introduced in Proposition 7.2.1. "Example 7.3.2 Let f : TmM % T[e, 0, 0]Y be the isomorphism intro-duced in Proposition 7.2.1, associated to the Witt–Artin decomposi-tion (7.3.1), and g : TmM %M be any local di!eomorphism around0 " TmM satisfying the following three properties

(i) g(0) = m,

(ii) T0g = idTmM ,

(iii) for any 1 " g, v " V , and w " W such that the vector v + w "TmM lies in the neighborhood where g is a di!eomorphism ontoits image, we have that

T(v+w)g(1M (m)) = 1M (g(v + w)).


Then, the map ', defined by

' : m" ) V $% M(., v) -$% (g & f%1)([0], ., v),

is a slice mapping at m. Indeed, property (SM1) is a consequenceof condition (i). For (SM3) note that

TmJ & T(0, 0)' = TmJ & T0g & T([0], 0, 0)f%1|m#)V

= TmJ & f%1|m#)V (by (ii) and the linearity of f)= T[e, 0, 0]JY |m#)V (by (ii) in Proposition 7.2.1).

Regarding (SM2), since the mapping g is a local di!eomorphismaround 0 " TmM and f is an isomorphism, for (., v) " m")V smallenough, any vector u " T'($, v)M can be written as

u =d

dt

$$$$t=0

g & f%1(t[1], . + t.$, v + tv$),

with 1 " g, .$ " m" and v$ " V . Using the definition of f in u weobtain

u =d

dt

$$$$t=0

g(t1M (m) + w + tw$ + v + tv$),

where w, w$ "W are the elements provided by Corollary 7.2.1, suchthat TmJµ(w) = . and TmJµ(w$) = .$. Now, using the Leibniz ruleand condition (iii), we write

u =d

dt

$$$$t=0

g(t1M (m) + w + v) +d

dt

$$$$t=0

g(w + tw$ + v + tv$)

= T(v+w)g(1M (m)) + T(v+w)g(v$ + w$)

= 1M (g(v + w)) + T(v+w)g(f%1([0], .$, v$))

= 1M ('(., v)) + T($, v)'(.$, v$)

" g ·'(., v) + T($, v)'(m" ) V ),

as required. The condition (SM4) can be verified in a straightfor-ward manner. "

Trying to follow a strategy similar to the approach taken in the bi-furcation theory of general dynamical systems with symmetry [Kr90],


we will look for relative equilibria bifurcated from m in the '–slice.More specifically, in the setup just described, the point m = '(0, 0)is a relative equilibrium with velocity - " Lie(NGµ(H)), that is

d(h$ J#)('(0, 0)) = 0.

We will look for relative equilibria around m, that is, values (., v) "m") V close to (0, 0), such that, correcting the velocity - to -+$+0 + & " g, with $, 0, and & , elements close to zero in h, m and q,respectively, we obtain a new relative equilibrium, that is

d3h$ J#+,+)+"

4('(., v)) = 0. (7.3.2)

We start by introducing an elementary lemma.

Lemma 7.3.1 Let (M, !, h) be a Hamiltonian dynamical system,and G be a Lie group acting properly on M in a globally Hamiltonianfashion, with equivariant momentum map associated J : M % g".Suppose that h is G–invariant and that its G–equivariant Hamiltonianvector field associated has a relative equilibrium at the point m "M ,with velocity - " g, then

d(h$ J#)(m)|g·m = 0 i! ad"#J(m) = 0.

Proof Since h is G–invariant, we have that, for any . " g,

d(h$ J#)(m) · .M (m) = $dJ#(m) · .M (m).

Using the equivariance of the momentum map J, we can write

$dJ#(m) · .M (m) = 0ad"$J(m), -1 = $0ad"

#J(m), .1.

Since . " g is arbitrary, the result follows. !Note that by property (SM2) in the construction of the '–slice,

the equation (7.3.2), which determines the bifurcated relative equi-libria, is implied by the two relationsHI

J

d-h$ J#+,+)+"

2('(., v))|T(", v)'(m#)V ) = 0

d-h$ J#+,+)+"

2('(., v))|g·'($, v) = 0,


which by Lemma 7.3.1 and the chain rule are equivalent toHI

J

d--

h$ J#+,+)+"2&'2(., v) = 0

ad"#+,+)+"J('(., v)) = 0.

(7.3.3)

In the context of the energy–momentum and the reduced energy–momentum methods [SLM91, Lew92], the second equality in (7.3.3)is usually refered to as the rigid condition. Let i : gµ )% g andj : q )% g be the canonical inclusions, and i" : g" % g"µ and j" :g" % q" be the respective dual maps. Using the mappings i" and j",equation (7.3.3) can be decomposed intoHKKKKKKKKI

KKKKKKKKJ

(B1) d--

h$ J#+,+)+"2&'2(., v)|{0})V = 0

(B2) d--

h$ J#+,+)+"2&'2(., v)|m#){0} = 0

(B3) j"ad"#+,+)+"J('(., v)) = 0

(B4) i"ad"#+,+)+"J('(., v)) = 0.

(7.3.4)

With this setup, we start the description of the bifurcation methodwhich, provided that certain conditions are met, will give us a familyof new relative equilibria parameterized by h and m". The methodproceeds in four steps.

STEP 1 Using the Implicit Function Theorem, we will eliminate theparameter & " q from the equations (7.3.4), by writing it in terms ofthe variables in h, m", V , and m. Indeed, consider the map

A : h)m" ) V )m) q $% q" = g#µ7= q

($, ., v, 0, &) -$% j"ad"#+,+)+"J('(., v)),

where we used the Ad"H–invariant splittings (7.2.11) in order to iden-

tify q" with g#µ. Since the point m is a relative equilibrium withvelocity -, we have that A(0, 0, 0, 0, 0) = 0. At the same time, thepartial derivative DqA(0, 0, 0, 0, 0) : q % q" = g#µ

7= q of A withrespect to the q factor, evaluated at (0, 0, 0, 0, 0) equals

DqA(0, 0, 0, 0, 0) · 5 = j"ad"-µ = ad"

-µ,

for any 5 " q. The last equality is a consequence of the fact thatad"

-µ " g#µ. The map DqA(0, 0, 0, 0, 0) : q % q" is injective. In-deed, if ad"

-µ = 0 for some 5 " q, then 5 " gµ * q = {0}. Since


dim q = dim q", the map DqA(0, 0, 0, 0, 0) is an isomorphism. TheImplicit Function Theorem (see for instance [AMR, Theorem 2.5.7])guarantees the existence of smooth map & : h ) m" ) V ) m % qdefined in an open neighborhood of (0, 0, 0, 0) " h ) m" ) V ) msuch that, for any element ($, ., v, 0) " h ) m" ) V ) m in thatneighborhood, we have that

j"ad"#+,+)+"(,, $, v,))J('(., v)) = 0. (7.3.5)

STEP 2 We now eliminate the parameter 0 " m from the equa-tions (7.3.4), by writing it in terms of the variables in h, m", and V .Indeed, consider the map

B : h)m" ) V )m $% (m")" 7= m($, ., v, 0) -$% d

--h$ J#+,+)+"

2&'2(., v)|m#){0},

where & denotes the function & , &($, ., v, 0) obtained in step 1.Note that since m is a relative equilibrium with velocity -, we havethat B(0, 0, 0, 0) = 0. As we plan to apply the Implicit FunctionTheorem, we compute the partial derivative of B with respect to them variables. Indeed, let 9 " m arbitrary. Then,

DmB(0, 0, 0, 0) · 9 = $ d

dt

$$$$t=0

d3J#+,+t4+"(0, 0, 0, t4) &'

4(0, 0)|m#){0}

= 0T(0, 0)(J &')(·), 9 + Dm&(0, 0, 0, 0) · 91|m#){0}.

(7.3.6)

The previous expression can be simplified using the following techni-cal lemma.

Lemma 7.3.2 With the notation introduced in the previous compu-tation, we have that

Dm&(0, 0, 0, 0) · 9 = 0,

for any 9 " m.

Proof If we di!erentiate expression (7.3.5) with respect to the mvariables in the direction of 9 " m, and we evaluate at (0, 0, 0, 0) weobtain

j"ad"4+Dm"(0, 0, 0, 0)·4µ = 0.


Since 9 " m ' gµ, the previous expression reduces to

j"ad"Dm"(0, 0, 0, 0)·4µ = 0.

Given that

ad"Dm"(0, 0, 0, 0)·4µ|gµ = 0

we conclude that

ad"Dm"(0, 0, 0, 0)·4µ = 0,

and, consequently, Dm&(0, 0, 0, 0) · 9 " gµ * q = {0}. #Using the previous lemma in expression (7.3.6) we obtain that

DmB(0, 0, 0, 0) · 9 = 0T(0, 0)(J &')(·), 91|m#){0}.

We now verify that DmB(0, 0, 0, 0) : m % m is an isomorphism.Since it su"ces to show that it is injective, let 9 " m such thatDmB(0, 0, 0, 0) · 9 = 0. This implies that for any : " m",

0T(0, 0)(J &')(:, 0), 91 = 0.

By property (SM3) of the slice mapping ', this amounts to

0T[e, 0, 0]JY ([0], :, 0), 91 = 0:, 91 = 0.

Since : " m" is arbitrary, 9 = 0 necessarily, as required.The Implicit Function Theorem guarantees the existence of a

smooth map 0 : h)m")V % m, defined in an open neighborhood of(0, 0, 0) " h)m")V such that, for any element ($, ., v) " h)m")Vin that neighborhood, we have that

d33

h$ J#+,+)(,, $, v)+"(,, $, v,)(,, $, v))4&'4

(., v)|m#){0} = 0.

(7.3.7)

Obviously, by construction, we also have that

j"ad"#+,+)(,, $, v)+"(,, $, v,)(,, $, v))J('(., v)) = 0.

STEP 3 The previous two steps allowed us to eliminate the m andq variables using the Implicit Function Theorem without any partic-ular requirement on the relative equilibrium m. We now eliminate


in a similar fashion the V variables, however, as we will see, thiswill be feasible only when the relative equilibrium fulfills some nondegeneracy requirements. Consider the map

C : h)m" ) V $% V " 7= V

($, ., v) -$% d33

h$ J#+,+)+˜"4&'4

(., v)|{0})V ,

where 0 is the function 0 , 0($, ., v) obtained in step 2, and ˜& de-notes the composition of & with 0, that is, ˜& , &($, ., v, 0($, ., v)).Clearly, C(0, 0, 0) = 0. Since we intend to use the Implicit FunctionTheorem in a fashion similar to steps 1 and 2, we see under whatconditions we can do so by computing the partial derivative evalu-ated at (0, 0, 0) " h ) m" ) V , of C with respect to the V variablesin the direction u " V :

DV C(0, 0, 0) · u

=d

dt

$$$$t=0

d33

h$ J#+)(0, 0, tu)+"(0, 0, tu,)(0, 0, tu))4&'4

(0, tu)|{0})V

= $d3JDV )(0, 0, 0)·u+DV

˜"(0, 0, 0)·u &'4

(0, 0)|{0})V

+ d2(h$ J#)(m)(T(0, 0)'(0, u), ·)|T(0, 0)'({0})V ). (7.3.8)

The first summand in (7.3.8) is zero. Indeed, if we define + :=DV 0(0, 0, 0) · u + DV

˜&(0, 0, 0) · u, we have that for any u$ " V

d-JDV )(0, 0, 0)·u+DV

˜"(0, 0, 0)·u &'2(0, 0) · (0, u$)

= dJ((m) · (T(0, 0)'(0, u$))

= 0(TmJ & T(0, 0)')(0, u$), +1= 0T[e, 0, 0]JY ([0], 0, u$), +1 (by property (SM3))

= 0T0JV · u$, +1 = 0. (since JV is quadratic)

Hence, we have that, for any u " V

DV C(0, 0, 0) · u = d2(h$ J#)(m)(T(0, 0)'(0, u), ·)|T(0, 0)'({0})V ).

(7.3.9)

Therefore, the invertibility of the linear map DV C(0, 0, 0) : V % V ",and hence the possibility of applying the Implicit Function Theo-rem to solve the V variables in terms of the h and m" variables, isequivalent to the non degeneracy of the quadratic form

d2(h$ J#)(m)|T(0, 0)'({0})V ))T(0, 0)'({0})V ),


which is actually the stability form of the relative equilibrium m.Indeed, property (SM3) of the slice mapping, and the quadraticnature of the momentum map JV guarantee that T(0, 0)'({0})V ) 'ker TmJ. Property (SM4) guarantees that T(0, 0)'({0})V )*gµ ·m ={0}. Finally, since T(0, 0)' is an isomorphism, a trivial dimensioncount guarantees that

ker TmJ = T(0, 0)'({0}) V )( gµ · m,

that is, T(0, 0)'({0} ) V ) is one of the spaces that can be used toconstruct the stability form. As we showed in the proof of Theo-rem 5.3.1, the choice of complement to gµ · m in kerTmJ does nota!ect the rank of the stability form hence, we conclude that:

the rank of the linear map DV C(0, 0, 0) : V % V " is equalto the rank of the stability form of the relative equilibriumm.

Since steps one and two were automatic, it is going to be in this step,that is, in the study of the stability form, where the possible bifur-cation phenomena are going to manifest themselves. In the simplestcase, namely, when the stability form is non degenerate, the ImplicitFunction Theorem guarantees the existence of a smooth mappingv : h)m" % V , defined in an open neighborhood of (0, 0) " h)m"

such that, for any element ($, .) " h)m" in that neighborhood, wehave that

d33

h$ J#+,+)+˜"4&'4

(., v($, .))|{0})V = 0, (7.3.10)

where now, 0 , 0($, ., v($, .)) and ˜& ,&($, ., v($, .), 0($, ., v($, .))).

STEP 4 If the stability form of the relative equilibrium m is nondegenerate, steps one through three produce a h)m" parameterizedfamily of points '(., v($, .)) " M , with ($, .) " h ) m", suchthat, taking as velocity - + $+ 0+ ˜& " g in (7.3.4), conditions (B1)through (B3) are satisfied. Since the four conditions in (7.3.4) aresu"cient to have a relative equilibrium, all those points in the h)m"–family satisfying

i"ad"#+,+)+˜"

J('(., v($, .))) = 0, (7.3.11)

are new relative equilibria of the system bifurcated from m. Noticethat in the Abelian case this condition is trivially satisfied.

§ 7.4. Bifurcation with Abelian Symmetry 279

7.4 Bifurcation in Hamiltonian Systems withAbelian Symmetries

We continue in the setup of the previous section, that is, the pointm " M is a relative equilibrium of the Hamiltonian system withsymmetry (M, !, h, G, J : M % g"), however, in this case, the Liegroup G will be assumed to be Abelian. Since in this case the adjointand coadjoint actions are trivial, if the relative equilibrium is suchthat J(m) = µ, then Gµ = G, and its velocity - " Lie(N(H)), whereH := Gm.

Note that in this situation the bifurcation method simplifies con-siderably since steps one and four vanish, consequently, if the relativeequilibrium m "M has a non degenerate stability form, the remain-ing steps produce a (h ) m")–family of relative equilibria around it,that we will study in the following theorem. Since we are provinga general fact about Hamiltonian systems with symmetry, we willmake use of the MGS normal form, more specifically, in the bifurca-tion method we will utilize the slice mapping constructed in Exam-ple 7.3.1 via the MGS normal form. Remark also that in the Abeliancase the splittings (7.2.1) and their duals reduce to

g = h(m, and g" = h" (m".

Theorem 7.4.1 Let (M, !, h) be a Hamiltonian dynamical system,and G be an Abelian Lie group acting properly on M in a globallyHamiltonian fashion, with equivariant momentum map associated J :M % g". Suppose that h is G–invariant and that its G–equivariantHamiltonian vector field associated has a relative equilibrium at thepoint m " M , with non degenerate stability form, such that, H :=Gm, J(m) = µ " g", and the element - " Lie(N(H)) is a velocity ofm. Then, there is a surface S of relative equilibria through m that,using the MGS normal form Y constructed around the orbit G · mcan be locally expressed as

S = {[g, ., v($)] " Y | g " G, . " m", $ " h},

where v : h % V is a smooth function such that v(0) = 0 andrankT,v = dimH $ dimHv(,). The rank, rankS[g, $, v(,)], of thesurface S at the relative equilibrium [g, ., v($)] equals

rankS[g, $, v(,)] = 2(dimG$ dimH) + (dimH $ dimHv(,)). (7.4.1)


Proof The existence of the surface S is a straightforward consequenceof the bifurcation method, using the slice mapping constructed in Ex-ample 7.3.1 via the MGS normal form around the orbit G·m. Indeed,the non degeneracy of the stability form of m guarantee through thesteps three and four of the bifurcation method, the existence of twosmooth functions 0 : h)m")V % m and v : h)m" % V , defined inan open neighborhood of (0, 0, 0) " h)m")V and of (0, 0) " h)m"

respectively, such that, any point of the form [e, ., v($, .)], with($, .) " h)m" close enough to (0, 0), is a relative equilibrium of thesystem (M, !, h), with velocity - + $+ 0($, ., v($, .)) " g, that is

d(h$ J#+,+)(,, $, v(,, $)))([e, ., v($, .)]) = 0.

Since the Lie group G is Abelian and the Hamiltonian flow Ft asso-ciated to h is G–equivariant, if the point [e, ., v($, .)] is a relativeequilibrium with velocity - + $ + 0($, ., v($, .)) " g then, for anyg " G, the point [g, ., v($, .)] is also a relative equilibrium with thesame velocity. Indeed, if

Ft([e, ., v($, .)]) = exp t(- + $+ 0($, ., v($, .))) · [e, ., v($, .)],

then

Ft([g, ., v($, .)]) = Ft(g · [e, ., v($, .)]) = g · Ft([e, ., v($, .)])= (g exp t(- + $+ 0($, ., v($, .)))) · [e, ., v($, .)]= (exp t(- + $+ 0($, ., v($, .)))g) · [e, ., v($, .)]= (exp t(- + $+ 0($, ., v($, .)))) · [g, ., v($, .)],

which, by Theorem 4.1.1, guarantees that the point [g, ., v($, .)] isa relative equilibrium of (M, !, h). If we now show that the functionv does not actually depend on the m" factor, we will have justified theexpression of the surface S of relative equilibria through m , [e, 0, 0]in the statement of the theorem. This will be done by proving thatthe partial derivative Dm#v($, .) of the function v : h ) m" % V ,with respect to the m" variables, is zero in an open neighborhood of(0, 0) " h ) m". We compute Dm#v($, .) by implicit di!erentiationin the relation (7.3.10), that defines the function v : h ) m" % V .Indeed, let u " V and .$ " m" arbitrary. We di!erentiate the pairing


of relation (7.3.10) with u " V , in the direction .$ " m", that is,

d

ds

$$$$s=0

d33

h$ J#+,+)(,, $+s$!, v(,, $+s$!))4&'4

(. + s.$, v($, . + s.$)) · (0, u) = 0.

This expression can also be written as

d

dt

$$$$t=0

d

ds

$$$$s=0

d33

h$ J#+,+)(,, $+s$!, v(,, $+s$!))4&'4

(. + s.$, v($, . + s.$) + tu) = 0.

By computing the derivatives, we obtain

$ d3J( &'

4(., v($, .)) · (0, u)

+ d233

h$ J#+,+)(,, $, v(,, $))4&'4

(., v($, .))

((.$, Dm#v($, .) · .$), (0, u)) = 0,

where + = Dm#0($, ., v($, .)) ·.$+DV 0($, ., v($, .)) ·(Dm#v($, .) ·.$) " m. The first summand in this expression is identically zero since

$d3J( &'

4(., v($, .))·(0, u)

=d

dt

$$$$t=0

0µ + . + JV (v($, .) + tu), +1

= 0Tv(,, $)JV · u, +1 = 0,

since + " m, and Tv(,, $)JV · u " h" = (m)#. Therefore, we have thatfor any u " V and any .$ " m

d233

h$ J#+,+)(,, $, v(,, $))4&'4

(., v($, .))

((.$, Dm#v($, .) · .$), (0, u)) = 0. (7.4.2)

By hypothesis, the quadratic form

d233

h$ J#4&'4

(0, 0)|({0})V ))({0})V ) (7.4.3)

is non degenerate, therefore, since non degeneracy is an open condi-tion, for any ($, .) " h)m" close enough to (0, 0), the same can besaid about

d233

h$ J#+,+)(,, $, v(,, $))4&'4

(., v($, .))|({0})V ))({0})V ).


Taking this into account, along with the fact that u " V and .$ "m" in (7.4.2) are arbitrary, we have that Dm#v($, .) : m" % V isidentically zero for any ($, .) " h ) m" close enough to (0, 0). Thisimplies that in a neighborhood of (0, 0) " h ) m", the map v doesnot depend on the m" variables, that is, v : h% V , as required.

In order to compute the rank of v : h% V at $ " h, we will calcu-late Dhv($) = T,v by implicit di!erentiation in the pairing of (7.3.10)with u " V arbitrary, in the direction $$ " h, and taking into accountthat the function v depends only on the h variables, that is,

d

ds

$$$$s=0

d33

h$ J#+,+s,!+)(,+s,!, $, v(,+s,!))4&'4

(., v($+ s$$)) · (0, u) = 0.

This expression can also be written as

d

dt

$$$$t=0

d

ds

$$$$s=0

d33

h$ J#+,+s,!+)(,+s,!, $, v(,+s,!))4&'4

(., v($+ s$$) + tu) = 0.

By computing the derivatives, we obtain

$ d3J,!+( &'

4(., v($)) · (0, u)

+ d233

h$ J#+,+)(,, $, v(,))4&'4

(., v($))

((0, Dhv($) · $$), (0, u)) = 0,

where + = Dh0($, ., v($))·$$+DV 0($, ., v($, .))·(Dhv($)·$$) " m.The first summand in this expression equals

$d3J,!+( &'

4(., v($))·(0, u)

=d

dt

$$$$t=0

0µ + . + JV (v($) + tu), $$ + +1

= 0Tv(,)JV · u, $$1,

since + " m, and Tv(,)JV · u " h" = (m)#. Therefore, we have thatfor any u " V and any $$ " h

d233

h$ J#+,+)(,, $, v(,))4&'4

(., v($))

((0, Dhv($) · $$), (0, u))= 0Tv(,)JV · u, $$1. (7.4.4)


By hypothesis, the quadratic form( 7.4.3) is non degenerate so, again,for any ($, .) " h)m" close enough to (0, 0), the same can be saidabout

d233

h$ J#+,+)(,, $, v(,))4&'4

(., v($))|({0})V ))({0})V ).

This implies that, since the elements $$ " h and u " V in (7.4.4) arearbitrary, the rank of Dhv($) equals the rank of Tv(,)JV , which, bythe Bifurcation Lemma equals

rankTv(,)JV = dim(hv(,))ann(h#) = dimH $ dimHv(,),

as required.The rank, rankS[g, $, v(,)], of the surface S at a given relative

equilibrium [g, ., v($)], is a straightforward consequence of the for-mula for the rank of Dhv($). Indeed, by definition, the rank of Sat [g, ., v($)] is the rank of the map that defines the surface S interms of the parameters G) h)m". Note that S is the image of thefunction

S : G) h)m" $% G)m" ) V $% G)H (m" ) V )(g, $, .) -$% (g, ., v($)) -$% [g, ., v($)],

whose rank at [g, ., v($)] is

rankT(g,,, $)S = rankS[g, $, v(,)]

= dimG + dimm" + rankT,v $ dimH

= 2(dimG$ dimH) + dimH $ dimHv(,),

as required. !

As a corollary to the previous theorem, we can formulate a gen-eralization of a result due to E. Lerman and S. Singer [LS], originallystated for toral actions, to proper actions of Abelian Lie groups.

Corollary 7.4.1 In the conditions of Theorem 7.4.1, there exists asymplectic manifold % of relative equilibria of h passing through m ofdimension

dim% = 2(dimG$ dimH).


Proof The manifold % can be obtained from the surface S, takingthe subsurface given by setting the parameter $ " h equal to zero, inother words

% = {[g, ., 0] " Y | g " G, . " m"}. (7.4.5)

The subsurface % is actually a smooth manifold since by (7.4.1), ithas constant rank equal to 2(dimG$ dimH), that is, the map

T : G)m" $% G)m" ) V $% G)H (m" ) V )(g, .) -$% (g, ., 0) -$% [g, ., 0],

whose image is %, is a local subimmersion around (e, 0) " G)m" withrank equal to 2(dimG$dimH), which implies (see for instance [AMR,Proposition 3.5.16 (i)]) that the surface % is locally a manifoldthrough the relative equilibrium m, of dimension 2(dimG$ dimH).

The symplectic nature of % can be easily seen in a way similarto the construction of the symplectic form of the MGS normal form(see Proposition 3.2.1), setting in that construction the symplecticnormal space equal to zero. We outline the main steps to be followed.First, by its own defining expression (7.4.5), the manifold % can belocally identified with the space G)H m", which will be shown to besymplectic. Note that, in the Abelian case, (we use the notation ofProposition 3.2.1) Y1 = G)g" 2 T "G, which is naturally symplectic.Consider now the left action R of H on Y1 given by

Rh(g, ,) = (gh%1, ,),

and the orthogonal decomposition g = h ( m. Using the symplecticform for Y1 given in (3.2.2), it is straightforward to verify that thisaction is globally Hamiltonian with equivariant momentum map JR :Y1 % h", given by

JR((g, (., 1))) = $., for any (., 1) " h" (m" = g".

Moreover, this action is free and proper and 0 " h" is clearly a reg-ular value of JR. Therefore J%1

R (0)/H is a well–defined Marsden–Weinstein reduced symplectic space which can be identified with% 2 G )H m" by means of the quotient di!eomorphism, inducedby the H–equivariant di!eomorphism l:

l : G)m" $% J%1R (0) ' G)m" ) h"

(g, 1) -$% (g, 1, 0).


!

. . . y Dios me entiende, y no digo mas.Cervantes, Don Quijote de la Mancha, II. Cap. I

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Index

'–slice, 268

actionadjoint, 11a"ne, 24canonical, 19coadjoint, 11free, 11globally Hamiltonian, 21Hamiltonian, 22of a Lie group, 10proper, 13transitive, 11variables, 197

adjoint action, 11a"ne action, 24AFM splitting, 170algebraic invariants, 66angle variables, 197angular momentum, 20Arnold di!usion, 190axes of Staude, 213

bifurcation, 125method, 267

Bifurcation Lemma, 48

canonicalaction, 19coordinates, 2distribution, 51map, 3velocity, 127

Casimir function, 3characteristic multipliers, 8clean value, 27coadjoint action, 11coboundary, 23cocycle, 23

identity, 23non equivariance, 24two–, 35

cohomology, 24conjugation, 11conserved quantity, 3Constant Rank Embedding

Theorem, 102critical

element, 4relative, 41–46

point, 4, 150current spatial inertia tensor,

182

Darboux’ Theorem, 2distribution, 49

canonical, 51integrable, 49involutive, 49Poisson, 51

e!ective potential, 224energetics, 150energy–Casimir, 150energy–Casimir method, 158

Index 301

energy–integrals method, 190,191

energy–momentum method,150, 153

equilibriumpoint, 4relative, 41, 126

equivariant mapping, 12Euclidean group, 14Euler equations, 201extension

local, 51property, 51

first–order stability, 189fixed points manifold, 15free group action, 11Frobenius Theorem, 49

Generalized Frobenius Theo-rem, 49

globally Hamiltonianaction, 21

Hamilton’s equations, 2Hamiltonian

action, 22flow, 56function, 1reduced, 26, 27, 34, 56, 58,

61, 65, 79, 85system, 1vector field, 1

hanging top, 184heavy top bracket, 159Hessian, 150Hilbert map, 66Hopf fibration, 206

infinitesimal generator, 11inner automorphism, 10

integrable distribution, 49integral of the motion, 3invariant

function, 12vector, 16

invariantsalgebraic, 66theory, 66

involutive distribution, 49irreducible representation, 18isotropy subgroup, 11isotypic

component, 19decomposition, 19

KAM Theorem, 8, 190KKS symplectic form, 36Kostant–Kirillov–Souriau

symplectic form, 36krupa decomposition, 258Kuranishi map, 122

Lagrange Theorem, 8Lagrangian block diagonaliza-

tion method, 169Laplace vector, 209Laplace–Runge–Lenz vector,

209Legendre transform, 40Lie group, 10

action, 10linear momentum, 20Liouville–Arnold Theorem,

197Lissajous curves, 205local

extension, 51local transversal section, 4

invariant, 44Lyapunov

302 Index

function, 42stability, 7

Manev problem, 234Marle-Guillemin-Sternberg

normal form, 95–104Marsden–Weinstein reduced

space, 26, 31MGS normal form, 95–104moment of inertia, 200momentum

angular, 20linear, 20mapping, 19–20

Moncrief Decomposition, 170Morse Lemma, 151

neutral direction, 45Noether’s Theorem, 21non commutatively integrable

system, 200non degenerate

relative equilibrium, 132RPO, 143RPP, 143

non equivariance cocycle, 24normal form

Marle-Guillemin-Sternberg, 95–104

MGS, 95–104

orbitof a group action, 11periodic, 4reduction, 33–34, 64–65,

84–86type, 15

orthogonal velocity, 152orthosymplectic subspace, 174

Patrick’s Lemma, 151

Patrick’s velocity map, 155,156

penalty function, 150period, 4

function, 5, 45relative, 43, 131

periodicorbit, 4

relative, 41, 126point, 4

relative, 41, 126persistence, 125phase shift, 43, 131Poincare map, 4

equivariant, 44point reduction, 26–27, 61–62,

79–80Poisson

algebra, 2bracket, 2distribution, 51dynamical system, 3manifold, 2map, 3reducible, 52reduction, 25–26, 58–59tensor, 3

presymplectic manifold, 1principal moment of inertia,

200proper action, 13

reconstruction equations, 108–109

reducedHamiltonian, 26, 27, 34,

79, 85space

Marsden–Weinstein, 26

Index 303

reduced energy-momentummethod, 169

reduced Hamiltonian, 56, 58reduction, 25

diagram, 39, 40, 87Lemma, 27of a smooth map, 55orbit, 33–34, 64–65, 84–86point, 26–27, 61–62, 79–80Poisson, 25–26, 58–59singular, 47symplectic, 25symplectic orbit, 36–39symplectic point, 31–33Universal, 61

reference inertia tensor, 182relative

critical element, 41–46equilibria

of orbit type, 144equilibrium, 41, 126

canonical velocity, 127non degenerate, 132velocity, 42, 127

period, 43, 131periodic orbit, 41, 126periodic point, 41, 126

representation, 10irreducible, 18

resonance, 200rigid condition, 272RPO, 41, 126

non degenerate, 143RPP, 41, 126

non degenerate, 143

shifting trick, 135singlet, 16singular reduction, 47sleeping top, 184

slice, 15, 90mapping, 268

Slice Theorem, 91solar–system approximation,

234stability, 6–10

asymptotic, 8first–order, 189form, 153, 191, 215, 242Lyapunov, 7modulo a subgroup, 46of equilibria, 7of periodic orbits, 8orbital, 8relative to a subgroup, 46subgroup, 11subspace, 253

stabilizer, 11stratified space, 47stratified subset, 49stratum, 49subvariety, 50super integrable system, 200symmetric energy–integrals

method, 214, 215Symplectic

Stratification Theorem, 3symplectic

KKS form, 36leaf, 3manifold, 1normal bundle, 102normal space, 95orbit reduction, 36–39point reduction, 31–33reduction, 25slicing technique, 132

Symplectic Eigenvalue Theo-rem, 9

304 Index

transitive group action, 11transversal section, 4

invariant, 44tube, 90twist

action, 90product, 90

two–cocycle, 35

Universal Reduction, 61upright sleeping top, 184

variety, 50velocity

canonical, 127of a relative equilibrium,

42, 127orthogonal, 152

Whitney functions, 50Witt–Artin decomposition,

258–262

SYMMETRY, REDUCTION, AND STABILITY IN HAMILTONIAN …Foreword This thesis presents some...

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