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Hamiltonian systemsRecommend this on Google

Ja m es Meiss (2 007 ), Sch ola r pedia ,2 (8 ):1 9 4 3 . doi:1 0.4 2 4 9 /sch ola r pedia .1 9 4 3

r ev ision #1 2 9 9 2 5 [lin k to/cite th isa r t icle]

Prof. James Meiss, Applied Mathematics University of Colorado, Boulder, CO, USA

A dynamical system of first order, ordinary differential equations

is an degree-of-freedom (d.o.f.) Hamiltonian system (when it is nonautonomous it has d.o.f.).

Here is the ''Hamiltonian'', a smooth scalar function of the extended phase space variables and time the

matrix is the Poisson matrix and is the identity matrix. The equations naturally split into

two sets of equations for canonically conjugate variables, i.e.

Here the coordinates represent the configuration variables of the system (e.g. positions of the component

parts) and their canonically conjugate momenta represent the impetus gained by movement.

Hamiltonian systems are universally used as models for virtually all of physics.

Contents

1 Formulation

2 Examples

2.1 Springs

2.2 Pendulum

2.3 N-body problem

2.4 Electromagnetic Forces

3 Geometric Structure

3.1 Conservation of Energy

3.2 Liouville's Theorem

3.3 Poincaré's Invariant

3.4 Symplectic Maps

4 Integrable Systems

4.1 Action-Angle Variables

4.2 Liouville Integrability

5 KAM Theory

6 Hamiltonian Chaos

7 References

8 External Links

2n ,

= J∇H(z, t), J = ( ) ,z0

−I

I

0(1)

n n + 1/2H z t ,

2n × 2n J I n × n

n z = (q, p) ,

= ∂H/∂p, = −∂H/∂q .q p (2)

n q

p

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9 See Also

Formulation

In 1834 William Rowan Hamilton showed that Newton's equations for a set of particles in a

conservative force field with "potential energy" could be derived from a single function that he

called the "characteristic function",

Here is the position of the particle whose mass is and is its canonical momentum The

equations of motion are obtained by (2), which can in turn be converted to Newton's second order form by

differentiating the equation

At first it seems that Hamilton's formulation gives only a convenient restatement of Newton's system---the

convenience perhaps most evident in that the scalar function encodes all of the information of the

first order dynamical equations. However, a Hamiltonian formulation gives much more than just this

simplification. Indeed, if we allow more general functions and a more general relationship between

the canonical momenta and the velocities then virtually all of the models of classical physics have a

Hamiltonian formulation, including electromagnetic forces, which are not derivable from a (scalar) potential.

Moreover, waves in inviscid fluids such as surface water waves or magnetohydrodynamic waves also have a

Hamiltonian (PDE) formulation. Quantum mechanics is formally obtained from classical mechanics by

replacing the canonical momentum in the Hamiltonian by a differential operator.

Hamiltonian structure provides strong constraints on the flow. Most simply, when does not depend upon

time (autonomous) then its value is constant along trajectories: the energy is constant, see Energy

Conservation. Similarly if the Hamiltonian is independent of one of the configuration variables (the variable is

ignorable), then (2) implies that the corresponding canonical momentum is an invariant. This gives a simple

explanation for the relation between symmetries (for example rotational symmetry) and invariants (for

example angular momentum)---see Noether's Theorem.

One of the stronger constraints imposed by Hamiltonian structure relates to stability: it is impossible for a

trajectory to be asymptotically stable in a Hamiltonian system. Even more structure applies: for each

eigenvalue of an equilibrium there is a corresponding opposite eigenvalue For example an equilibrium of

a one degree-of-freedom system must either be a center (two imaginary eigenvalues, or a saddle (two real

eigenvalues, ) or have a double zero eigenvalue.

Another geometric implication is that knowledge of invariants is enough to fully characterize a solution of the

equations for an degree-of-freedom system, i.e., the Hamiltonian is integrable. This follows from

Liouville's Integrability Theorem. Moreover, if the orbits of such a system are bounded, then almost all of them

must lie on -dimensional tori. Kolmogorov, Arnold and Moser proved that a sufficiently smooth, nearly-

integrable Hamiltonian system still has many such invariant tori (see KAM theory). This strong structural

stability of Hamiltonian dynamics was unexpected even in the middle of the century when physicists

began the first computer simulations of dynamical systems (see Fermi Pasta Ulam problem).

F = ma

F = −∇V V

H(q, p) = + V ( , , … , ) .∑i=1

n |pi|2

2mi

q1 q2 qn (3)

qi ith ,mi pi = .pi mi q i

= / .q i pi mi

H(q, p) 2n

H(q, p, t)q

H

E = H(q, p)

λ −λ .±iω

±λ

n

2n n

n

20th

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Figure 1 : Coupled Springs

Examples

For many mechanical systems, the Hamiltonian takes the form where is the

kinetic energy, and is the potential energy of the system. Such systems are called natural Hamiltonian

systems. The simplest case is when the kinetic energy is of the form in (3) for a set of particles with kinetic

momenta and masses More generally, when the extent of the bodies is taken into account the

kinetic energy can depend upon the configuration of the system, but it is typically a quadratic function of the

momenta, so that where the mass matrix represents the shape as well

as the inertia of the system, and the vector includes both linear momenta and angular momenta.

Springs

A harmonic spring has potential energy of the form where is

the spring's force coefficient (the force per unit length of extension) or

the spring constant, and is the length of the spring relative to its

unstressed, natural length. Thus a point particle of mass connected

to a harmonic spring with natural length that is attached to a fixed

support at the origin and allowed to move in one dimension has a

Hamiltonian of the form and thus its

equations of motion are

If the spring is hanging vertically in a constant gravitational field, then the new equations are obtained by

simply adding the gravitational potential energy to

A set of point masses that are coupled by springs has potential energy given by the sum of the potential energies

of each spring in the system. For example suppose that there are two masses connected to three springs as

shown in (Figure 1). The Hamiltonian is

One advantage of the Hamiltonian formulation of mechanics is that the equations for arbitrarily complicated

arrays of springs and masses can be obtained by simply finding the expression for the total energy of the system

(However, it is often easier to do this using the Lagrangian formulation of mechanics which does not require

knowing the form of the canonical momenta in advance).

Pendulum

The ideal, planar pendulum is a particle of mass in a constant gravitational field, that is attached to a rigid,

massless rod of length as shown in (Figure 2). The canonical momentum of this system is the angular

momentum and the potential energy is the gravitational energy where is the angle

from the vertical. The Hamiltonian is

H(q, p) = T(q, p) + V (q) , T(q, p)V (q)

∈pi R3 .mi

T(q, p) = M(q p ,12 pT )−1

n × n M(q)p ∈ Rn

,k

2 x2 k

x

m

L

H(q, p) = + (q − L12m

p2 k

2 )2

= p/m , = −k(q − L) .q p

mgq H .

H(q, p) = + + + ( − + (L − .1

m1p2

11

m2p2

2k1

2q 2

1k2

2q2 q1)2 k3

2q2)2

m

L ,p = mL2 θ −mgL cos θ , θ

H(θ, p) = − mgL cos θ .p2

2mL2(4)

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Figure 2:

Planar

Pendulum

Figure 3: Phase Space of the Pendulum

This gives the equations of motion

While these equations are simple, their explicit solution requires elliptic functions. However, the trajectories of

the pendulum are easy to visualize since the energy is conserved, see (Figure 3). When

the energy is below the angle cannot exceed and the pendulum oscillates. Since

the energy is conserved, the orbit must be periodic. For energies larger than the

pendulum rotates, and the angle either monotonically grows with time (if the angular

momentum is positive) or decreases (negative ). The critical level set is the separatrix;

the two orbits on this level set asymptotically approach the equilibrium as

These are called homoclinic orbits.

N-body problem

A set of point masses interacting by Newton's gravitational force is also a Hamiltonian

system of the natural form (3) with potential energy

where is the position of the body. In

addition to the conserved energy this system

has additional conserved quantities. Since is a

function only of the difference between particle

positions, the total momentum

is conserved. Since is a function only of the distance

between the bodies, the total angular momentum is

also conserved

For the case of two bodies, the Hamiltonian has six degrees of freedom (the three components of the position

and momentum for each body), however, the conservation of total momentum means that if we choose

coordinates moving with the center of mass

= , = −mgL sin θ .θp

mL2p

mgL π

mgL ,

p

(±π, 0)t → ±∞ .

V ( , … ) = −q1 qn ∑i<j

Gmimj

|| − ||qi qj

∈qi R3 ith

H = E

H

P = ∑i=1

n

pi (5)

H

L = ×∑i=1

n

qi pi

Q =n

i i

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where is the total mass, then the Hamiltonian is independent of so that its conjugate

momentum (5) is constant. Thus the system is reduced to three degrees of freedom, depending only upon the

inter-particle vector and its conjugate momentum, where is the reduced

mass. In these coordinates the Hamiltonian becomes

The total angular momentum splits as well Since is constant, and the first

term is itself individually conserved, so is also constant, a fact that can also be seen from (6) directly.

The dynamics of three or more bodies can be extremely complex.

Electromagnetic Forces

A nonrelativistic charged particle in an electromagnetic field has the equations of motion

where is the electric field, and is the magnetic field and we use Gaussian (cgs) units. This system is

Hamiltonian, with

where the scalar and vector potentials and are defined through

The momentum occurring in (7) is not the kinetic momentum but rather a canonical momentum defined

by For systems that also have a Lagrangian formulation, the canonical momentum is defined

by

Note that the first term in the Hamiltonian (7) is simply the kinetic energy as usual, and the last term is the

electrical potential energy.

Geometric Structure

Q =1

M∑i=1

n

miqi

M = ∑ mi Q ,

q = − ,q1 q2 p = μ ,q μ = m1m2

M

H(q, Q, p, P) = + −P 2

2M

p2

2μ

Gm1m2

||q||(6)

L = Q × P + q × p . P = MP ,Q

l = q × p

m = eE(q, t) + × B(q, t)qe

cq

E B

H(q, p, t) = + eϕ,1

2m(p − A)e

c

2

(7)

ϕ A

E = ∇ϕ + , B = ∇ × A.∂A

∂t

m ,q

p = m + A .q e

c

p = .∂L(q, )q

∂q

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Much of the elegance of the Hamiltonian formulation stems from its geometric structure. Hamiltonian phase

space is an even dimensional space with a natural splitting into two sets of coordinates, the configuration

variables and the momenta For most physical systems the momenta are similar to velocities, which are

tangent vectors to trajectories, but the difference--emphasized in the electromagnetic example--is that they are

cotangent vectors, as we will explain further below. In this case the Hamiltonian phase space is the cotangent

bundle of the configuration space.

More abstractly, the phase space of a Hamiltonian system is an even dimensional manifold that is endowed

with a nondegenerate two-form, This two-form allows us to define a pairing between vectors and covectors.

Given a Hamiltonian function the Hamiltonian vector field is defined by

This is just a coordinate-free version of (1). Indeed, a famous theorem of Darboux implies that near each point

in there exists a set of canonical variables such that

where is the "wedge product". In terms of these coordinates, where is the Poisson

matrix (1), and the equations (8) become

which is a restatement of (1).

Conservation of Energy

If a Hamiltonian does not depend explicitly on time, then its value, the energy, is constant. Indeed

differentiating along a trajectory gives

by (2). Thus

While Hamiltonian systems are often referred to as conservative systems, these two types of dynamical

systems should not be confounded. In the autonomous case, a Hamiltonian system conserves energy, however,

it is easy to construct nonHamiltonian systems that also conserve an energy-like quantity. Moreover, in the

nonautonomous case, the Hamiltonian depends explicitly on time and there is no conserved energy.

Liouville's Theorem

One direct consequence of the form (2) is that the divergence of a Hamiltonian vector field is zero

since is antisymmetric and the Hessian matrix is symmetric. This immediately implies that the volume

of any bundle of trajectories is preserved. That is, suppose is a set of initial conditions with volume

q p .

M

ω .H : M → R , = X(z)z

ω ≡ ω(X, ⋅) = dH.iX (8)

M z = (q, p) ,

ω = dq ∧ dp ,

∧ ω(v, w) = Jw ,vT J

X = ∇H,J T

= + = 0,dH

dt

∂H

∂q

dq

dt

∂H

∂p

dp

dt

H(q(t), p(t)) = H(q(0), p(0)) = E .

H(q, p, t)

∇ ⋅ X = ∇ ⋅ J∇H = = 0.∑i,j

Jij

∂H

∂ ∂zi zj

J HD2

A

V (A) = dz.∫

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If evolves to the flow of the vector field, then the new volume

is the same as the original volume

This is known as Liouville's theorem. It is valid for any divergence free vector field,

Note that Hamiltonian flow is volume preserving even when it is nonautonomous.

Poincaré's Invariant

In addition to preserving volume, Hamiltonian systems also preserve a loop action, or Poincaré invariant.

Given any loop in the extended phase space let

Then under a Hamiltonian flow the loop action is preserved

Even more generally, suppose is the two dimensional tube obtained from the flow of

and is any loop on that is homotopic to Then

This fact is used, for example, in the construction of a Poincaré section for Hamiltonian systems.

Symplectic Maps

A map is symplectic if it preserves the symplectic form Geometrically, we say that

which becomes in components

where is the Jacobian matrix

The preservation of the loop action (9) implies that the time- map of any Hamiltonian flow is symplectic. This

follows from Stokes's theorem and the fact that for a loop at a fixed value of time, the loop action reduces to

Note that this holds even if the Hamiltonian depends explicitly on time

V (A) = dz.∫A

z (z) ,φt

dz = δ( (z) − y)dy,∫(A)φt

∫A

φt

V (A) .

∇ ⋅ X = 0 .

L (q, p, t) ,

A(L) = pdq − H(q, p, t)dt.∮L

(9)

A( (L)) = A(L).φt

T L :T = { (L) : t ∈ R}φt L′ T L . A( ) = A(L) .L′

f : M → M ω . ω = ω ,f ∗

D JDf = Jf T (10)

Df 2n × 2n

Df(q, p) = .

⎛⎝⎜⎜⎜⎜

∂fq

∂q

∂fp

∂q

∂fq

∂p

∂fp

∂p

⎞⎠⎟⎟⎟⎟

T

pdq .∮L H(q, p, t) .

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Another way in which symplectic maps arise is for Poincaré sections of autonomous Hamiltonian flows on an

energy surface. For example if the surface is selected, then the

resulting return map to is symplectic with the form This is especially useful for the

visualization of the motion of a two-degree-of-freedom system, since the resulting map is two-dimensional.

The set of linear mappings that obey (10) is called the symplectic group; it is a Lie group. Any quadratic

Hamiltonian

where is a (constant) symmetric matrix, has a linear flow that is generated by the exponential

Each of the matrices in the curve is symplectic. Indeed, the collection forms the Lie

Algebra of the symplectic group.

Integrable Systems

A dynamical system is integrable when it can be solved in some way. One (rather restrictive) way in which this

can happen is if the flow of the vector field can be constructed analytically. However, since this can almost

never be done (in terms of elementary functions), this is not an especially useful class of systems.

However, there is a class of Hamiltonian systems, action-angle systems, whose solutions can be obtained

analytically, and there is a well-accepted definition of integrability for Hamiltonian dynamics due to Liouville in

which each integrable Hamiltonian is (locally) equivalent to these action-angle systems.

Action-Angle Variables

A Hamiltonian system is written in action-angle form if there is a set of canonical variables where

and and such that depends only upon the actions

In this case the equations of motion (1) become simple indeed:

These equations can be easily solved, giving

Thus the angles move along the invariant torus with a fixed frequency vector

For example, the simple harmonic oscillator Hamiltonian

Q = {(q, p) : = 0, > 0, H(q, p) = E}qn q n

Q ω = d ∧ d .|Q ∑n−1i=1 qi pi

H(z) = Kz ,12

zT

K Φ(t) = .etJK

Φ(t) {JK : = K}K T

(θ, I) θ ∈ Tn

I ∈ Rn H

H(I) .

= ∇H(I) = Ω(I) , = 0θ I (11)

(θ(t), I(t)) = ( + Ω( )t, )θo Io Io

I = Io Ω .

H(q, p) = ( + )12

p2 q 2

(q, p) = ( sin θ, cos θ) .√ √

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can be written in action angle form by setting The new variables are

canonical since (i.e., the transformation is canonical). In the new coordinates the

Hamiltonian becomes Thus it is in action-angle form with A more general, anharmonic

oscillator, with a natural Hamiltonian of the form (3) with a potential energy with a unique minimum at

has a Hamiltonian that depends in a nonlinear way upon the action, but which nevertheless can be

reduced to action-angle form.

Hamiltonian systems with two or more degrees of freedom cannot always be reduced to action-angle form,

giving rise to chaotic motion.

Liouville Integrability

Liouville and Arnold showed that the motion in a larger class of Hamiltonian systems is as simple as that of (11).

Suppose that an degree-of-freedom Hamiltonian system (2) has a set of invariants that are almost

everywhere independent (their gradients span an -dimensional space except on sets of zero measure) and that

are in involution, that is, their Poisson brackets vanish:

Then if a regular level set of the invariants is compact it must be a torus .

Moreover, there is a neighborhood of in which there exist action-angle coordinates such that the equations

of motion reduce to (11). See (Arnold, 1978).

For example, every one degree-of-freedom, autonomous Hamiltonian system is Liouville integrable. However,

the action-angle coordinates may not be globally defined. In the case of the pendulum (4), there are action-

variables away from the separatrix.

Generically the dynamics on an invariant torus are quasiperiodic.

KAM Theory

Andrey Kolmogorov discovered a general method for the study of perturbed, integrable Hamiltonian systems.

The method lead to theorems by Vladimir Arnold for analytic Hamiltonian systems (Arnold, 1963) and by

Jurgen Moser for smooth enough area-preserving mappings (Moser 1962), and the ideas have become known as

KAM theory.

Roughly speaking, KAM theory implies that a Hamiltonian system of the form

which is integrable at still has a large set of invariant tori if is small enough (a set whose measure

approaches the total measure as ). In order that KAM theory apply, the Hamiltonian must be sufficiently

smooth, and (for the simplest version of the theorem) the unperturbed Hamiltonian must satisfy a

nondegeneracy or twist condition, that is nonsingular.

For more details see Kolmogorov-Arnold-Moser Theory.

(q, p) = ( sin θ, cos θ) .2I−−√ 2I

−−√dq ∧ dp = dθ ∧ dI

H(θ, I) = I . Ω = 1 .V (q)

q = 0

n n Fi

n

{ , } ≡ ω(∇ , ∇ ) = 0 .Fi Fj Fi Fj

= { (q, p) = : i = 1, … n}Lc Fi ci

Lc

H(θ, I) = (I) + ϵ (θ, I) ,H0 H1

ϵ = 0 , ϵ

ϵ → 0

(I)D2H0

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Figure 4: Poincaré section at of the two-wave

Hamiltonian (12) for and Resonant

island chains with rotation numbers and

are shown.

Hamiltonian Chaos

Though many invariant tori of an

integrable system typically persist

upon a perturbation, tori that

commensurate or nearly

commensurate are typically

destroyed. Chaotic dynamics often

occurs in the neighborhood of these

destroyed tori.

An invariant torus is characterized by

its frequency vector It is

commensurate if there exists an

integer vector such that

Commensurate tori of an integrable

system are generically destroyed by

any perturbation.

For example, consider the 1.5 degree-

of-freedom system

that represents the motion of (for example) a charge particle in the field of two electrostatic waves. Here the

phase space can be taken to be since is a periodic function of and For the momentum is

constant and the orbits lie on two-dimensional tori with the frequency vector Consequently every

torus with a rational value of is commensurate--indeed such orbits are periodic in this case.

KAM theory implies that if is "sufficiently" irrational, then the torus is preserved for However

commensurate tori and nearby irrational tori are destroyed. For small the destroyed tori are replaced by

chains of islands formed from a pair of periodic orbits, one a saddle and the other elliptic (see Stability of

Hamiltonian Flows). Surrounding the elliptic orbit are a family of two-dimensional tori with a new topology (not

homotopic to see (Figure 4). Moreover, the stable and unstable manifolds of the saddle

typically intersect transversely, giving rise to a Smale horseshoe and chaotic motion (albeit chaos that is limited

to a narrow layer about the separatrix. As grows these chaotic layers also grow, and they can envelope larger

regions of phase space, see (Figure 5).

References

Abraham, R. and J. E. Marsden (1978). Foundations of Mechanics. Reading, Benjamin.

t = 2πka = 4 , b = 6 ϵ = 0.1 .

0/1, 1/2, 2/3, 4/5 1/1

Ω .

m ∈ Zn

m ⋅ Ω = 0

H(q, p) = + ϵ(acos(2πq) + bcos(2π(q − t))12

p2 (12)

× RT2 H q t ϵ = 0 ,Ω = (p, 1 .)T

p

p |ϵ| ≪ 1 .ϵ

p = constant ,

ϵ

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Figure 5: Poincaré section of the two-wave Hamiltonian (12) for

Arnold, V. I. (1963). “Proof of a Theorem of A.N. Kolmogorov on the Invariance of Quasiperiodic Motions

Under Small Perturbations of the Hamiltonian.” Russ. Math. Surveys 18:5: 9-36.

Arnold, V. I. (1978). Mathematical Methods of Classical Mechanics. New York, Springer.

MacKay, R. S. and J. D. Meiss, Eds.

(1987). Hamiltonian Dynamical

Systems: a reprint selection. London,

Adam-Hilgar Press.

McDuff, D. and D. Salamon (1995).

Introduction to Symplectic Topology.

Oxford, Clarendon Press.

Meyer, K. R. and G. R. Hall (1992).

Introduction to the Theory of

Hamiltonian Systems. New York,

Springer-Verlag.

Moser, J. K. (1962). “On Invariant

Curves of Area-Preserving Mappings

of an Annulus.” Nachr. Akad. Wiss.

Göttingen, II Math. Phys. 1: 1-20.

Siegel, C. L. and J. K. Moser (1971).

Lectures on Celestial Mechanics. New

York, Springer-Verlag.

Internal references

Paul M.B. Vitanyi (2007) Andrey Nikolaevich Kolmogorov. Scholarpedia, 2(2):2798.

Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.

James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.

Ferdinand Verhulst (2007) Hamiltonian normal forms. Scholarpedia, 2(8):2101.

Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.

Anatoly M. Samoilenko (2007) Quasiperiodic oscillations. Scholarpedia, 2(5):1783.

Steve Smale and Michael Shub (2007) Smale horseshoe. Scholarpedia, 2(11):3012.

Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

Alain Chenciner (2007) Three body problem. Scholarpedia, 2(10):2111.

External Links

James Meiss's website (http://amath.colorado.edu/faculty/jdm/)

See Also

ϵ = 0.2 .

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