Hamiltonian flow - Imported Knowlpedia Content

Definition

Let $H$ be a Hamiltonian function on a symplectic manifold $(M,\omega)$ , with Hamiltonian vector field $X_H$ . The Hamiltonian flow is the maximal local flow $\Phi:D\to M$ , defined on an open set $D\subseteq\mathbb R\times M$ containing $\{0\}\times M$ , such that

\Phi_0(x)=x,\qquad \frac{d}{dt}\Phi_t(x)=X_H(\Phi_t(x)).

Where both sides are defined, $\Phi_{t+s}=\Phi_t\circ\Phi_s$ . If $X_H$ is complete, then $D=\mathbb R\times M$ , and the flow is a global one-parameter group of diffeomorphisms.

Preserved structures

Each flow map is locally a symplectomorphism because

\frac{d}{dt}\Phi_t^*\omega=\Phi_t^*(\mathcal L_{X_H}\omega)=0.

For an autonomous Hamiltonian, it also preserves energy:

H\circ\Phi_t=H.

Consequently, every trajectory lies in a level set of $H$ . These conclusions follow from the defining equation and are recorded in Cannas da Silva, Lecture 18.1.

Examples and completeness

On $\mathbb R^2$ with $\omega=dq\wedge dp$ and $H=(q^2+p^2)/2$ , the flow rotates each circle centered at the origin and is defined for all time. A constant Hamiltonian has $X_H=0$ , so its flow is the identity. By contrast, a smooth Hamiltonian vector field on a noncompact manifold can escape to infinity in finite time; smoothness alone does not imply completeness.

Autonomous and time-dependent flows

References

Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: Lecture 18.1, Hamiltonian vector fields and their flows.
V. I. Arnol'd, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989. DOI record. Relevant: Chapter 8, Hamiltonian mechanics.