TheoremProved

Liouville's theorem (Hamiltonian mechanics)

Statement

Let a Hamiltonian system with phase space coordinates $(q, p)$ evolve under Hamilton's equations $\dot q = \partial H/\partial p$ , $\dot p = -\partial H/\partial q$ , and let $\varphi_t$ denote the flow that advances each phase point by time $t$ . Then for any region $R$ of phase space, the volume of $\varphi_t(R)$ (computed with the standard Lebesgue measure $dq\, dp$ ) equals the volume of $R$ for every $t$ : phase-space volume is conserved under Hamiltonian flow.

Why is it true?

Picture a cloud of possible initial states of a system, filling some blob of phase space. As time runs, each point of the blob moves along its own trajectory, so the blob stretches, twists, and deforms — but Liouville's theorem says it can never shrink or swell in total volume, like an incompressible fluid. Even though the system's uncertainty in position or momentum individually can grow (the blob can get thin and stretched, spreading over a huge region), the total 'amount of phase space' it occupies never changes.

Proof sketch

The rate of change of a volume element transported by a flow equals the volume element times the divergence of the flow's velocity field. For Hamilton's equations, the velocity field is $(\dot q, \dot p) = (\partial H/\partial p, -\partial H/\partial q)$ , whose divergence is $\frac{\partial}{\partial q}\left(\frac{\partial H}{\partial p}\right) + \frac{\partial}{\partial p}\left(-\frac{\partial H}{\partial q}\right) = \frac{\partial^2 H}{\partial q\, \partial p} - \frac{\partial^2 H}{\partial p\, \partial q} = 0$ by equality of mixed partials. A divergence-free velocity field is exactly the condition (via the continuity equation / Liouville's equation for the phase-space density) for the flow to preserve volume, which proves the theorem.

Topics that use this theorem

Lagrangian and Hamiltonian mechanics

Related theorems

Noether's theorem Euler–Lagrange equation

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

V. I. Arnold (1989). Mathematical Methods of Classical Mechanics
Herbert Goldstein, Charles P. Poole, John L. Safko (2002). Classical Mechanics