Hamiltonian flow

Hamiltonian flow is the phase-space evolution generated by Hamilton's equations for a fixed Hamiltonian.

For $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$, the displayed linear Hamiltonian flow solves $\dot{X}=V$, $\dot{V}=-{\omega }^{2}X$. Its inverse is $S_{-t}$, its determinant is one and it preserves $H_{\omega }$. The zero-frequency limit is the shear $(x,v)\mapsto (x+tv,v)$.

For homogeneous Hamiltonian Liouville equation transport with autonomous $H$, compact initial support remains in the fixed energy sublevel determined by its maximum initial energy. If these sublevels are compact, this proves uniform compact support without solving the flow. The harmonic energy $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$ has this property exactly when $\omega \ne 0$; at zero frequency spatial free streaming can escape every compact set.

If $H$ is $C^2$ and its spatial Hessian matrix is uniformly bounded on every finite time interval, its Hamiltonian vector field is globally Lipschitz continuous in phase space with at most linear growth. The Picard-Lindelöf theorem and Gronwall inequality give a unique Hamiltonian flow for all finite forward and backward times. Bounds on ${\nabla }_{z}H(t,0)$ follow from time continuity on compact intervals.

With the displayed signed Lie-Poisson bracket and evolution $\dot{F}=\{F,H\}$, the equations are ${\dot{x}}_1=2(C-B)x_2{x}_3$, ${\dot{x}}_2=2(A-C)x_3{x}_1$, and ${\dot{x}}_3=2(B-A)x_1{x}_2$. The energy and the Casimir function of a Poisson manifold $x_1^2+x_2^2+x_3^2$ are conserved. For $A=1/(2I_1)$, $B=1/(2I_2)$, $C=1/(2I_3)$ with positive inertias, these are the Euler equations for a torque-free rigid body. Reversing the bracket while fixing $H$ reverses time orientation.