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Liouville's theorem states that for Hamiltonian systems the phase space volume $V(t)$ is a conserved quantity, i.e., $\frac{d}{dt}V(t)=0$. This is related to the fact that trajectories in phase space do not cross and a point in phase space has a unique time evolution.

Noether's theorem tells us that conserved quantities correspond to continuous symmetries/cyclic coordinates, and vice versa.

My question is: what is the continuous symmetry/cyclic coordinate corresponding to the conservation of phase space volume?

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  • $\begingroup$ $V$ is only a function of $t$, and has nothing to do with the phase space coordinates $(q, p)$? $\endgroup$
    – rschwieb
    Feb 1, 2021 at 14:51
  • $\begingroup$ @rschwieb It's the volume of a submanifold of the whole phase space, defined as an integral over phase space coordinates. $\endgroup$
    – FractalScout
    Feb 1, 2021 at 14:55

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There are several versions of Liouville's theorem. One version states that a Hamiltonian vector field (HVF) $X_H=\{H,\cdot\}_{PB}$ on a symplectic manifold $(M,\omega)$ is divergence-free $$ {\rm div} X_{H}~=~0.$$ One may view the above HVF as the underlying symmetry.

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