3
$\begingroup$

This is a reference request, as I can't for the life of me find anything that answers my question in the literature.

If $(M,\omega,H)$ is a Hamiltonian system, we know from Liouvile's theorem that its level sets $H^{-1}(c)$ for $c$ some regular value are tori. Several of the texts I am reading mention that there is a flow-invariant measure on these level sets called the Liouville measure.

As I don't think this is trivial, but don't know how to construct it, how is the Liouville measure defined for arbitrary symplectic manifolds, and why is it given by $$\frac{dS}{||\nabla H||},$$ where $S$ is "surface area," for $M=\mathbb{R}^{2n}$ with the canonical symplectic form?

$\endgroup$
1
$\begingroup$

If you don't care that the answer is not rigorous, here is the idea. Think of Liouville measure $dS$ on $H^{-1}([c,c+\delta c])$ for $\delta c$ small. Then the "thickness" of this "shell" will be proportional to $1/\|\nabla H\|$.

$\endgroup$
  • $\begingroup$ Thanks, this does make sense. I think this means the Liouville measure is a sort of volume form on the level sets. If so, is it just something like $\omega^n$ for an arbitrary symplectic manifold $(M,\omega)$? $\endgroup$ – user117824 Dec 30 '13 at 5:53
  • $\begingroup$ Yes, that is the idea. $\endgroup$ – Stephen Montgomery-Smith Dec 30 '13 at 5:55
  • $\begingroup$ Also, did you check this book: L.D. Landau & E.M. Lifshitz Mechanics ( Volume 1 of A Course of Theoretical Physics ) Pergamon Press 1969. I must confess I haven't looked through it, but people tell me it deals with this stuff rather well (aside from comments in Arnold's book that criticizes some of the math details.) $\endgroup$ – Stephen Montgomery-Smith Dec 30 '13 at 6:01

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.