Physica 5D (1982) 335-347 North-Holland ~.~ublishing Company

PROOF OF NON-INTEGRABILITY FOR THE H]~NON-HEILES HAMILTONIAN NEAR AN EXCEF_HONAL INTEGRABLE C . ~ E

Philip HOLMES Department of Theoraical and Applied Mechanics and Center for Applied Mathematics, Comell University, Ithaca, NY 14857, USA

Received 12 November 1981 Revised 5 February 1982

In this paper we describe an analytical method for establishing non-integrability in Ham~ltonian syttems which are small perturbations of integrable ones. We illustrate the method with an application to a system of H6non-Heiles type.

I. Intreduetlon

The physics !iterature is replete with examples of numerical solutions of Hamiltonian systems with two or more degrees of freedom which suggest non-integrability. The original work of H6non and Heiles [ 15] is by now well known and since then many others have studied ~imilar systems :4nc' related Poincar6 maps. Lichtenberg and Lieberman [21] give many examples and some background in- formation; also see Cbi:ikov [7]. However, no matter how persuasive the majority of numerical results might be, there are few cases in which error estimates have been performed with sufficient rigor to elevate such results to the ~tatus of proofs.

In this note we outline recent analytical results of Holmes and Marsden [17, 18, 19] which enable non-integrability to be established rigorously for a class of Hamiltonian systems close to integrable cases. Our methods are based on earlier work of Melnikov [22] and Arnold [2]. We apply our results to ;~ Hamiltonian of H~non-Heiles type near the integrable case:

H(X, y)~, ~) - .~2 .... +2 ~2 +e02 (x2 +2 y2) - .'c2y -- y'~/3, (1.1)

discussed by Aizawa and Sato [I]* and pointed out earlier by Ford [13]. We show that all neighboring systems of H6non-Heiles type are non-integrable and estimate the width of tile primary stochastic layer which is the cause of non-integrability. We note that, while (1. I) differs only in one sign from the 'standard' H6non-Heiles [15] Hamiltonian, its behavior is remarkably different. It is therefore sometimes referred to as the "anti-H6non-Heiles" system.

While numerical integration is a valuable tool for studying multi degree of freedom systems, we fee! that. its widespread use has led_ to some mieenne~rtinn~ with r~gard t n the behavior of ~_na!ytic perturbations of integrgble systems, the most serious of which is the belief that 'sufficiently smalF perturbations remain integrable. It is implicit in our results that generic, analytic perturbations are non-integrable, in the sense that ~:o matter how small the perturbation, no ar~lytic integrals are preserved other than the energy. Moreover, perhaps more significantly, our methods yield a test for non-integrability which can be applied to specific systems.

"The Aizawa.-Saito case is obtained from (I.1) under the transforn~ation y - , - y, and with e0 = I.

0167-2789/82/0000-0000/$02.75 © 1982 North-Holland

33~ It'. Holmes/Establishing non-integrability in Hamiltonian syslans

Section 2 of this paper contains a brief outline of the theory, the application is in section 3 and we add some comments and discuss extensions to the results in section 4. General information on Hamiltonian mechanics, homoclinic phenomena and chaotic solutions can be found in Arnold [3], Arnold and Avez [4]. Birkhoff [5], Lichtenberg and Lieberman [21] Moser [23], and Poincar6 [24].

Other examples in which analytic proofs of non-integrability have been carried out include the restricted three body problem (Sitnikov, Alekseev) and the case of generic perturbatio:.s of integrable systems near elliptic orbits (Zehnder [25]). Both problems are discussed by Moser [23]. ~ s o , R.C. Churchill, D.L. Rod [9. 10, I !] and their colleagues have used topological methods to establish the non-integrability of the H6non-Heiles system for "large' energies (h > ca216 in the present context). In all these, as in the present example, non-integrability is shown to occur as a result of transverse homoclinic orbits. For a quite different approach to the problem, s~.e Ziglin [30].

2. Reduction and the Melnikov method

Recently. Holmes and Marsden [17] have extended the method of Melnikov [22] (Holmes [16], Chow. Hale and Mallet-Paret [8]. Greenspan and Holmes [ 14]) to the study of multi-degree of freedom Hamiltonian systems close to integrable systems. Here, for simplicity, we discuss only the two degrees of freedom result.

Consider a Hamiltonian of the form

H ' (q ,p , 0, 1) = F ( q , p ) + G ( l ) + e H ' ( q , p , O , l ) , (2.1)

where q is a generalized coordinate, p its conjugate momentum and I, 0 are action angle variables. When ~ = 0 the unperturbed system is completely integrable, since Hamilton's equations split into two independent sets:

,~I: 3F q :~ -:,~(q.p). /~ - (q,p) : (2.2al 3q

_ 3(; dot

o l - ~q(1), i = O, (2.2b)

the former of which is integrable by quadratures, and the latter of which has the solution, based at 1 o, C b .

,. o. 1s = (f~( l°)t +o° . I ° ) . (2.3)

Thus the phase plane of (2.2b), or some compact subset of it, is filled with a continuous family of periodic orbits parametrized by the action I °, and of frequency 1"1(1°).

We make. the following assumptions on the unperturbed system: I~ ~,~ l) = t, ~1) > u, for i > u (non-zero frequency)

H2) The (q, p) phase plane of (2.2a) contains a hyperbolic saddle point (qo, P0) with a homoclinic orbit (qlrl, P(Lt))" i.e. (qO' PO) lies in a closed leve! curve of F.

We illustrate these hypotheses in fig. 1. The case in which (2.2a) has a heteroclinic orbit connecting t~vo distinct saddle points can be tr~.ated similarly.

VCe wish to study the geometrical structure of solutions of the perturbed system (~ # O) lying near the product of the homoclinic orbit (q,/5) and one of the unperturbed circular orbits I = I °. In

.... • . . . . . . . . .... P, ~ l m ~ l ~tablisMng non-mtegrabili~ in Haml~nmn isystems 3 3 7

• F-system G. sySlem

Fill , 1. T h e u n p e r t u r b e d s y s t e m .

particular, we wish to determine whether a 's tochastic layer ' exists near this homoclinic manifold and if so, to describe its characteristics. The existence of such a layer, containing transverse homoclinic orbits, implies that the perturbed system has a Smale horseshoe (Smale [26, 27]): an invariant Cantor set containing a dense orbit, and hence that the system has no analytic second integral (Moser [23]).

2.1. The r educ t ion procedure

We note that the total energy

H'(q, p, 0, I ) = h (2.4)

is still conserved for the perturbed system, and that hypothesis H 1 implies that (2.4) may be inverted to solve for I in terms of (q, p, O; h). Subsequently the independent variable t may be replaced by 0 and the two degree of freedom system reduced, on each energy surface h for which I ~ 0, to a periodically perturbed single degree of freedom oscillator. See Birkhoff [5, ch. VI.3], Whittaker [28, ch. 12], Churchill [5] or Holmes and Marsden [17, 18] for details. One obtains the system

OL t q'--'--~L-~°(q,p;h)-,~ (q,p,O.h)+O(e2), Op "-~ ' ~L ° ~L I

p' = -~--(q, p; h )+ • -f~-(q, p, 0: h ) + 0(~2), (2.5)

where ( )' = did0( ) and the leading terms of the Hami~tonian are

L°(q, ~,; h) = G- ' (h - F(q, l,)),

L'(q, p, O" h) = _ H ' ( q , P, O, L°(q, p; h)) , a(LO(q, p, h)) • (2.6)

2.2. M e l n i k o v " s m e t h o d

Hypothesis H2 implies that, for • ~-0, the reduced L ° system has a homoclinic loop filled with an infinite family of orbits ( 4 (0 -0° ) , 1~(0-0°)). Melnikov's [22] method enables one to compute the perturbation experienced by such orbits when the periodic 'forcing' term L ! is added. Briefly, the hyperbolic saddle (qo, P0) of the unperturbed system becomes a small (O(e)) 2It-periodic orbit when e ~r0 and its stable and unstable manifolds continue to lie close to the product of the homoclinic loop (~, 1~) with the orbit I - l ° in the three-dimensional energy surface H -- h. Here I ° is uetermined by the 'excess energy'

def

G(1) = F ( ~ , ~) - h = h t _ It. (2.7)

338 p. Holmes! Estabfishing noa-integmbility in Hamiltonicm systems

['he Melnikov methed measures the separation between the two-dimensional invariant manifolds in this energy surface. By computing the distance at various phase angles, O °, one finds whether or not these manifoids intersect.

In Holmes and Marsden [17], using the results of Melnikov [22] and Greenspan and Holmes [14] and the reduction process sketched above, we prove the following (of. Arnold [2]):

Theorem L Let hypotheses H I - 2 hold and h ' = F(~], #) be the energy of the homoclinic manifold of the F system. Pick h > h ~ and let 1 ° = G-t(h - h t). Let {F, H'}(t - to) denote the Poisson bracket of F and H t evaiuated along an orbit ( ~ l ( t - to ) , / 5 ( t - to), fl(l°)t, I °) in the homoclinic manifold. Then, for

+ 0 sufficiently small, if the Melnikov function

: g -

s,fth, t,,) = f {F, H'igt - to) dt (2.8) . ~ , -

has simple zero~,i, the Hamiitonian system (2.1) has a Smale horseshoe in its dynamics on the energy surface H ' = h, and consequently possesses no analytic second integral.

In the next section we apply this result to the H6non-Itei les Hamiltonian (H6non and Heiles [ 15]), but first some comments are i,n order.

!) We note that the Melnikov function is expressed entirely in t e r n s of the original F and H ' functions, rather than the reduced L °, L' Hamiltonian. This ntakes for ease of computation. (See comment 3, below.)

2) As our application in the next section shows, the unperturbed G system need not necessarily be explicitly transformed to action angle coordinates: all one requires is that it have a continuous family e+~ " periodic orbits in some (compact) subset of its phase space.

3) The width of the stochastic layer can be estimated. The Melnikov theory of Greenspan and Holmes [14] shows that the splitting of the manifolds, and hence the width of the stochastic layer near the point ~/5(0), c~(0), 0 °, I°). is given by

~ f ~ {L°,L~}(O -O°)dO

= ! (_ .... o,o,,. +

(of. Greenspan and Holmes [14, eq. (3.11)], where the {L°,L '} bracket is integrated along the homoclinic orbit (q,/5) expressed in terms of the transformed time-like variable O. Using (2.3), and tee×pressing (:..'..9) in terms of {F, H ~}, we obtain (cf. Holmes and Marsden [ 17])

!3F 3F '

"' i " f f ' ( l °) . . . . . la(l") + 0(~2) x

, oF p(o) ) ) l + (2.10)

Thus, the maximum width of the splitting, and hence of the stochastic layer near the closed curve

339

layer

layer.

(¢(0), tY(0), flto, I °) on the energy surface H" = h is given by

d , , - ~ sup Mlh , OF (c](0),f~(0)),-~-(q(0),p(0)) +O(e2). (2.11)

We note that, to satisfy the conditions necessary for perturbation analysis, the base point (q(0), ~(0)) of the homoclinic orbit must be chosen outside a neighborhood of the saddle (qo, po) (Greenspan and Holmes [14]). Fig. 2 illustrates these ideas.

Since (gF/Op,-OF/0q)c~.pc0) is a fixed vector, it follows that the splitting of the homoclinic manifold is 0(t) , in contr~t to the more delicate c~ses of higher order resonance, when the unperturbed system possesses only invariant tori, and no homoc.linic orbits exist for • - 0 ; cf. Arnold [3, Appendix 7], and the remarks in section 4, below.

However, in these cases, as in the present case, th: integrable systems are isolated in the space of all systems, since there is no second integral for any small e# 0 for which theorem I holds. It is therefore incorrect to speak of the existence of 'isolating integrals' for sufficiently small e (or energies), and of 'critical energies' at which these integrals vanish. The stochastic layer is always present for e # 0, although it may be be.~eath the range of numerical resolution.

3. The H&non--Heiles system

We shall consider the Hamiltonian of H~non-Heiles type:

H(.t, ~, x, y) = 't2 +2 ~,z+ ¢o=(x 22 + ),2)_ x2Y _ y~/3 + e(ay 2 - ,By'~), 0<~ • ~ I', a,/3 -'- ~(1), (3.1)

i.e. close to the exceptional integrable case e = 0 (cf. Aizawa and Sato [I]). Note that variation in a detunes the linear I: l resonance, while variation in/3 changes one of the cubic terms. Employing the symplectic transformation

l I q, = ~ (x + y), q2 = ~ ( x - y),

v ~ V K ,

I I (3.2)

(3.1) becomes

(3.3)

MO P. Holmes I Establishing non-integrability in Hamiltoniaa systems

D~

w2>.,~. o )

Fig. 3. The uncoupled (integrable) phase portraits.

For • = 0 it is clear that the system decouples into two single degree o f freedom oscillators each of which is integrable by quadratures. In fact a second symplectic transformation, leaving (p,, q)) invariant and changing (p:, q:) to action angle variables (L 0), can be made to put (3.3) into the form ( 2 . 1 ) :

,<: ,) • & ] 2 3 qi + G ( I ) + ~ a ( q , - Q2(I, 0 ) ) : - ( q , - Q:(I, 0)) ~ , (3.4)

where the function q2 = Q:(I, o) is defined by the change of coordinates. For ~ = 0 the two uncoupled systems have the phase portraits of fig. 3, each having a homoclinic

k, op, Fi, to a hyperbolic saddle point. Clearly the transformation (q2, P,)-'*(I, 0) only applies in the open region bounded by the homoclinic orbit F2. As we shall see below, we do not need to use this coordinate change explicitly.

As outlined above, we shall be interested in perturbing solutions lying near the product flow F~ x "r~, where V~ = l ~ x [0. 2~r] is one of the periodic orbits of system 2 within I':. The reader can easily verify that the unperturbed orbit is thus

( (o: ( 3 /wt , , 3,o ~ . , ,wt , (~t_) ) (~,./~,. I. t)) = ~7-~. I --2 s e c h : [ , 2 ) ) . 2 ~ sech'~-).~, tanh _ • ! k, f l ( I k ) t + 0 ° . (3.5)

where f l ( l ~) = 8G(Ik)/O! and we have based the orbit at

tO ko ) . ~q. p. I, O) : 2V2' O. I 0 ° 13.6)

Ralher than computing f l ( l k) explicitly, we shall merely require the expression for the periodic orbit of system 2 in terms of q2, which may be written in the form

_ , , , t (3.7)

where sn(4,, k) denotes the elliptic sine function and the (energy dependent) constants &, k, c are given

24fi k c~s ~ . . . . -~ .... i,

(o

3 cos(tb/3)- X/3 sin((b/3) k : = 3 cos(d>/3) + V3 sin((M3)'

c: = 8/(3 cos((b/3) + V3 sin((b/3)).

(3.7b)

We note that, as the excess energy/~k varies from 0 to w+/12, +13 varies from 1rl3 to 0 and k from 0 to

i ~:P~Holmesl ~ t a b l a k l n g a ~ i n t ~ r a ~ i t y i ~n HamJltontan'~ ~"~" '~ ' . . . . . . . . s y s t e m s

1. The total energy of orbits within thehomocfinic m ~ o l d is therefore

6 h * h'+l i k ffi = ~22 +/~* ¢ (to6112, to6/6).

¢~(t ) . . V ~ , ,, c o s ~ t ,

341

(3.8)

(3.9)

In both cases (3.7a) and (3.9) the orbits are based at q2 • (0, e02/2~/2), p : - 0. Hypotheses HI -2 are clearly satisfied for the unperturbed system, and it remains merely to

compute the Melnikov integral:

M(h, to) = f (F, Ht}(t - to) dt

fie

all1.._._, t/(t - t0)dt clPt ttql J

qo

2V~ - . (3.10)

from the Hamiltonian (3.4). Since

Q~(I ~, ll(l~)t ) " q~(t), (3.11)

we only need the solution (3.7) to the unperturbed G system, and need not work explicitly with the action angle variables.

Rewriting (3.10), using the transformation t --, t + to, we have

M O , , to)- - - f +

2 V ~ ' ~ " " ~ " ' " - ~ " " ~ " " ~ " " " " ~ " " ~ (3.12)

Noting that/It is even while Pt is odd, we see that the terms l~t¢l and Pt(]~ do not contribute to the integral, and (3.12) reduces to

[ ' ] M(h, to)= a J t - ~ J2-~J~ ,

3.12 P. Holmes! Establishing non-integmbility in Hamiltoniaa systons

where

x

L = f f~(t)q~(t + to) dt,

.-iz

1 . ~(t J- = 1 0 1 ( t ) q l ( t ) q + to) dt , - x

x

J..,= f ~l(t)q~:(t +to)dt. - .'c

(3.13)

These integrals are eva lua ted in the appendix to yield

J~ = 8 c;,vg csch wv= sin 2/~to, m = O

J.. = 3~rto6B~c",.t '~(v~. - 4) c s c h ( ~ ) sin 2~.Jo, 64V '5- m =o

. , ' / ' t o S~ = ~-~--~ cava Bc~,-- A csch sin 2 ~ , , t o - Bc~ csch(wv=) s i n 4~mto ,

(3.14a)

where

"rid L~

A = cos (d , / 3 ) - 112, B = (3 cos(~b/3)- X/3 sin(~b/3))/2,

,-r c,, - kK(k-------) csch[(2m + I)rrK'(k)/2K(k)],

/ ~ = (2m + 1)rrco~' ces(~b/3) + sin(rbl3)lX/3-~K(k), (3.14b)

z,~ = 4~.~/w,

and K. K' are the comple te ell iptic integral of the first kind and its c o m p l e m e n t .

Note that . for small excess energies (h near h t), these e x p r e s s i o n s reduce to the express ions obta:ned when q~ of (3.7) is r ep laced by the pure cos ine wave of (3.9). For example , us ing (3.5),

fsw: . , / t o t ' ( tot) X/2(h-w6/12) ]' 2 ~ 2 " COS (to(t + to)) dt ~-- s e c n ' ~ , T ) t anh 2 to

x

z

_-~ 2 w - . w h - ~ i sech~T tanh ?(cos 2? cos to to- sin 2r sin toto) 2 d~-

t o

- - ~ sech 2 r lanh ? sin 2? d? • sin toto

t o

= - 3 w h - -i-~ (2w csch rr) sin toto

~ - 6 r r toVh - e ,6 /~ csch rr sin OJto, (3.

P., Rolmes , Bslctb|llhino non ,iKt~m~...hil;t~. ~in ~Hmilto~ian 343

Similarly, we obtain

J2~0 ' (3.15b)

24~- I~ ~ -- ~ ' ~ ( h . . . . . . - a~6112) csch 2¢r sin 2,or0. . ~ (3.15c)

Here Jt and J 2 a r e correct to O ( ' ~ / ~ ) a n d a is ; ¢orrect :i to O ( h - ¢a6/i2): We therefore obtain

M ( h to) = - 6 7 r a c o V h - to~l i2 c s c h ~r c o s eoto + O(h - ~06112) ( 3 1 6 )

Now from (2,11) the maximum splitting of the seperatrices near (¢](0),/~(0)) = (¢o212~/2, 0) for h near (o6112 is

dm~={ sup M(h, to)l( OF OF ) (¢(o). (o)), I +

and using

= 0, = + d ( o ) : = T ~O 4'

we have

For arbitrary h E (~0611~, co616) the exact integrals Ji of (3.14) can in principle be used to compute supt0~aM(h, to), but we have not attempted this formidable calculation. Rather, we shall be content with aoting that, except for possibly a set of parameter values a, /3, e, h of measure zero, the expressions of (3.12)--(3.14) and the simplified expression (3.16) clearly have simple zeros at to = 0 and thus that the conditions of theorem 1 are met, and we have

Theorem 2. For an open dense set of parameter ':alues a, /3;* 0 and • > 0 sufficiently small, the H6non-Heiles Hamiltonian system (3.1) contains Smale horseshoes in its dynamics on every energy level h E (~:6/12, c06/6), and hence possesses no analytic second integral. Moreover, for h near co6112 the width of the main stochastic region is given by (3.17).

Remark. Even if the zeros of M(h, to) are not simple, we still have M ~ 0 and since M takes both signs, the stable and unstable manifolds must intersect (non transversally) and one still obtains a stochastic layer. However, the usual hyperbolicity arguments necessary to obtain 'the horseshoe with its dense orbit do not go through in the absence of transversal intersections (cf. Moser [23], also see Cushman [12] and Churchill and Rod [10] for more on non-transversal intersectioas).

4. Extensions and conclusions

One can also use the Melnikov method to study the creation of su, bharmonic motions from the resonant tori of the unperturbed problem. Suppose that, for • -O, we select.a pair of periodic orbits (q~', p~') and (qk, ph) lying in the interiors of rt and I'2 respectively (cf. fig. 3). Let the period of these

344 P. Holmes l Establishing non-integrability in lqamiltonian sy: :'ms

orbits be T, and T: and choose

IT, = roT:: i, m E Z, (4.1)

we then say that the torus filled with the two parameter family of orbits including q~, and q~2 is resonant of order lira. Application of reduction and a Melnikov analysis anologous to that of section 2 (cf. Greenspan and Holmes [14], §3.2) shows that, if the subharmonic Me|nikov funct ion

M ' ; ' ( h . to) = f {F. H t } ( t - to)dt o

(4 .2 )

has 2hi ~imple zeros in to E [0, IT,l, then the torus breaks into precisely 2n periodic orbits of period IT, = roT:. n being elliptic and n hyperbolic. This is, of course, precisely what one expects from the general theory. In (3.2) the integral is computed along the unperturbed orbit (q~,, p~,, q~2, p~2).

An an'~lysis such as that sketched above enables one to establish the existence of a countable set of hyperbolic and elliptic orbits bifurcating from a dense set of tori as • increases from zero. Thus, while the Kolmogorov-Arnold-Moser theorem guarantees that a measurable set of smooth ~sutficiently ir~'ational) tori are preserved within the interior of Ft x F2 (and sufficiently far from F, × F:). the stable and unstable manifolds of the hyperbolic orbits are somehow packed between the members of this set. However, attempts to prove by Melnikov's method that these manifolds intersect transversally, and hence to demonstrate non-integrability directly, fail due to the fact that M(h, 10) is exponentially small in e(.~(e-m)) and the O(e2) remainder in ( I . i l ) becomes important. See Sanders [25] and of. Holmes [16]. Thus we cannot directly extend our results to the H6non-Heiles system to energies below h = o,6/12. See Holmes and Marsden [20] for more information on exponentially small Melnikov functions and averaging.

In the generic case, however, it is known that the countable set of (homoclinic) stochastic layers 'which bifurcate from the rational tori for e > 0 grow in size with • and thus successively 'consume' the irrational tori. Ultimately no isolating tori remain and solutions can wander stochastically (in this example) over the whole interior of F, × F2. Thus our remarks on the primary stochastic layer in section 2 should be understood to hold only for sufficiently small e; for cases in which no appreciable merging of layers has occurred and in which an iso!ating torus exists inside Ft x F= and within O(e) of it.

The theory is easily extended to the case in which the unperturbed system, while integrable, does not decouple into two independent systems. In this case the Hamiltonian takes the form

H = F ( q . p . l ) + H ( q . p , O . l )

and ~he Poisson bracket in the Melnikov integral must be replaced by (q, p) Poisson bracket

r H,~ , , , , - . H !,

(4.3)

(4.4)

where I~ = OF/OI, F and I t ' are evaluated along the unperturbed orbit (4, P, 0" = J'(~ [1((], p, I °) dr, I°). ,~ in the theory for decomposable systems witi~ a product structure sketched in section 2, generalizatior to n degrees of freedom are available. For details and an application to rigid body dynamics, s¢ Holmes and Marsden [19].

* . . . . v . . . . s ~ . I t .~ . s , ,u , ,~ r , ,n / ; .nun-s , t lq~ruu lUly I l l Jrl(Imll[OfllaN s y s g e m s 345

Acknowledgements

The author would like to thank the referees of this paper, and especially Joe Ford, for several constructive comments. This work was partially supported by NSF grant MEA 80-17570.

Appendix A

Evaluation of the Melnikov integrals (3.13)

From (3.7) we have

q~(t ) ffi 7~/,-.~ ( A -- B sn2(s, k)), (A.I)

where

A = cos 4,13 - 112, B = (3 cos(cM3)-. ~/3 sin(4q3)12,

(.,) = o.(~/3)+. v 3 3 ' ~o~(,/3)+~.(,/3)/, jy (A.2)

and

241] ~ COS 0 = - ' - ' r - - I.

gt

The elliptic sine sn(s, k) may be expanded in a Fourier series (Byrd and Friedman [197)])

] 2~ ,~ sin (2 r e+ l ) ~rs sn(s, k) ffi kK(h) m-01 - q 2K(k) ' (A.3)

where K(k) is the complete elliptic integral of the first kind, K'(k) is the complementary complete elliptic integral of the first kind, and q = exp(-~rK'(k)lK(k)) is the elliptic home.

Writing (A.3) as

sn(s, k) = ~ c, sin(Ass) " ~ c,, sin(/~t), m-.0 m-.0

(A.4)

where

¢rto /~,, = (2m + I )VC0s(~I3) + sin(4,13)/V'~J 4K(k ) ' (A.5)

(A. I ) becomes

"'[ :1 q~(t) = ~-~ A - B C,. sin(~mt)) . (A.6)

Now consider the integral o0

J~= f p~(t)q~(t + to)dr. ~ 0 0

(A.7)

Mb P. NofmesI Estabhhing non-integrability in Hamiltonian systCemS

Since p ,( t t = (3&t/j) sech’(ot/2) tanh(wC/2) is odd. the constant part @‘NV3 of 4: drops out in integration. Moreover. expanding the Foutier series, we find terms of the types

C,C, CoS p8t CQS pjt sin pito sin @jfO*

the first and fourth of which drop out since they are even in t, while terms of the second and third types survive only if i = j. due to orthogonality of the Fourier components. Thus the integral simplifies to

2~: sin p,,,t cos ,u,t dt sin &a cos p,,,lo

= - y & cl 1 sech(f) tanh($ sin 2p,,,f dt q sin 2~,,,to. *

Letting t = otl2. as in (3.15a), this becomes

sech’ r tanh T sin V,T dT l sin 2pmto. (A. 10)

Thct integrals may be computed by the method of re.cidues to yield

sin 2fl,ta, = - - sin 2*,to, (A. 11)

where

i’m = 4pJbJ = ( Zm + U~~cos(4/3) - sin(4/3)AAjiK(k).

SinGPar calculations for .I2 and .I1 give

sin 2p,,,fO - Bc~ csch(ntp,,,) sin 4p,,,lo 1

, (A.133

(A. 12)

cm 2wm+“? =---_ /&(A)( 1 __ $“‘+‘) = &j csch

nK'(k) + ‘)2K(k) I ’

WI

WI r141

WI

1161

1171

WI

[l91

ml WI

WI [231 [241 WI WI

r27) (281

1291 (301

Y. Aizawa and N. Sait6, Gn the stability’of 4salating iritegrals I, J, Phys; Sot. of Japan 32 (1972) 16361640. V.I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5 (1964) 581-585. V.I. ~~d,.~~ematicaI..Metho<fS. of,$Wqal Mechan##5pringer,~ New Y orkJ: ,,$V7): _ Ix, v,1[, ~j~.~~~~~ ,,~~~;~~~~~~~~~~~~!~~~i~~,~~~~~‘~~~~~~~~~~i~~~~~~~1~~~, . r ‘1 ^ - ’ G,D* B~~~~g~~*~n, ,~~~~~~~~~~~~~~~~~~~~~~~ :$.g@‘; :fi& ~9~~*ti&&““;$** ieiatied l*; I&),

P.F, Byrd & && Fti~~,~i~~~~.,~~~~~~~~~~~~~~.f~~ &&&&‘~&j&&&& (s&&q, New York, 1971) 2nd ed_ B.V. Chirikovr, A ‘u~~e~~‘~~~~~~~~~‘~~~~~~~‘iti~~~s~~nal&kiktit& $&I@, P&i& :Re$&& 52 ff979), 263-379. S. N. Chow J. H$e and& F&tllebPtuet~~An ehniple .of,biircatigr to_ homoclinic,iorbits~ J. I#&. I&s. 37 (1980) 351-373. RX. Churchill, On provi@ the-noj$ntejtrabilit), of_ a Hainiltonian syslriem, .prepiint (Hunter Coliege, New York, 1980). R.C. Churchill and D.L: R&, Pathology in dynamical systems, JH. An&t~c~Hiu$lt~ni~ns, J. Difl’t. Eqns 37 (1980) 23.38. RX. ChurcMli, ‘U3. PoceW&$‘D;L. Rod; A survey of theqH&ton--H@ts Hamiitonian ~5th application to related examples, in Como Cotie~dce Pt’&dit&$, on Sftihastic Behavior inCla@&xuand, Quatitum Hamilt+an Systems, G. Casati and J. Ford, ei3s. (Springer Lecpre Notes in Physics, New Ye&, 1979). 1 R. Cushman, Examples of non-integrable analytic Hamiltonian vector fields with no small divisors, Trans. Amer. Math. Sot. 238 (1978) 45-55. J. Ford, private communication (1981). B.D. Greenspan and P.J, Holmes, Homoclinic orbits, subharmonics and global bifur&ions in forced oscillations; to appear in Nonlinear Dynamics and Turbulence, 0. Bartnblatt, 0. Iooss and D.D. Joseph, ec’s. (Pitman, New York, 1982). M. H&non and C. H&s, ‘I’he applicability of the third integral of motion: some numerical experiments, As&on. J. 69 (1964) 7344. P*J. Holmes, Averaging and chaotic motions in forced oscillations: S.I.A.M. J. Appl. Math. 38, 65-80, Errata and Addenda. S.I.A.M. J. Appl, Math. 40 (1980) 167.168. P.J. Holmes and J.E. Marsden, Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom, Comm. Math. Phys. 85 (1982a) 523-544, P,J. H&nes and J.E, Marsden, Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems, J. Math. Phys, 23 (1982b) 669-675. P.J. Holmes and J.E. Marsden, Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups, Indiana U. Math. J. (in press) (1982c). P.J. Holmes and J.E. Marsden [W2d] (in preparation) Averaging and Melnikov’s methcd in Hamiltonisn systems. A.J. Lichtenberg and M.A. Leiberman [1982] ‘Regular and Stochastic Motion’ (to appear (Lecture notes, Dep: qf Electrical Engineering, University oi’ Califorrua, Berkeley)). V.K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Sot. 12 (1063) l-57. J. Moser, Stable and Random Motions in Dynamical Systems (Princeton University Press, Princeton, 1973). H. Poincar& Les Methodes Nouvelles de la MCcanique Cdleste, Vol.9 I, II, III (Gauthier-Villars, Paris, !Fj2, 189% 1899). J.A. Sanders, Melnikov’s method and averaging, submitted for publication (1981). S. Smale, !DiReomorphisms with many periodic points; in Differential and Combinatorial Topology, S.S. Cairns, ed. (Princeton Univ. Press, Princeton, 1963) pp. 63-80. S. Smale, Differentiable Dynamical Systems, Bull. A.M.S. 73 (1967) 747-817. E.T. W&taker (l959), A treatise on the analytical dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge, 1959) 4th cd. E. Zehneder, Homoclinic points near clliptic fixed points, Comm. Pure. Appl. Math. 26 (l973), ~-182. S.L. Ziglin, Branching of solutions and nonexistence of integrals in Hamiltonian systems, Dolk. Akad. Nauk. SSR 257 (1981) 2629.