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?


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 $1/\|\nabla H\|$.

  • $\begingroup$ 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

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