4
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ $V$ is only a function of $t$, and has nothing to do with the phase space coordinates $(q, p)$? $\endgroup$ – rschwieb Feb 1 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 at 14:55
4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.