# Noether's Theorem and Liouville's Theorem

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?

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

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.