Hamiltonian system - Imported Knowlpedia Content

Definition

An autonomous Hamiltonian system is a triple $(M,\omega,H)$ , where $(M,\omega)$ is a classical phase space and $H:M\to\mathbb R$ is a smooth Hamiltonian function. With the convention

\iota_{X_H}\omega=dH,

the system's trajectories are the maximal curves $\gamma$ satisfying

\dot\gamma(t)=X_H(\gamma(t)).

Thus $M$ is the state space, $\omega$ converts the differential of $H$ into the Hamiltonian vector field, and $H$ specifies the dynamics. The system is more data than any one of these ingredients separately.

Evolution and observables

The Hamiltonian flow preserves both $\omega$ and $H$ . For an observable $f\in C^\infty(M)$ ,

\frac{d}{dt}f(\gamma(t))=\{f,H\},

using the Poisson-bracket convention associated with $\iota_{X_H}\omega=dH$ . In particular, $f$ is a first integral exactly when $\{f,H\}=0$ . See Cannas da Silva, Lecture 18.4, Definitions 18.8 and 18.10 for Hamiltonian and completely integrable systems.

Example

The harmonic oscillator on $\mathbb R^2$ has

\omega=dq\wedge dp,\qquad H(q,p)=\frac{p^2}{2m}+\frac{m\Omega^2q^2}{2}.

Its Hamilton equations are $\dot q=p/m$ and $\dot p=-m\Omega^2q$ , and its trajectories lie on the ellipses $H=\text{constant}$ . The symplectic plane without the choice of $H$ is only a phase space, not this Hamiltonian system.

Scope

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: Lecture 18.4, Definition 18.8 and the discussion of integrals of motion.
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 8.