Liouville's theorem for the submanifold of given conserved quantities? Liouville's theorem states that phase space volume is conserved over time with respect to the dynamical system generated by the Hamiltonian and Hamilton's equations.
However, any given point in phase space will evolve within a submanifold characterized by certain values of the conserved quantities (energy, momentum,...).
It's not obvious to me that the "phase volume" within this submanifold is also conserved over time, since it is a volume of lower dimension than that of tbe phase space.
Is there a result here that you could point me to?
 A: In order to ask if phase volume on the submanifold is conserved, we first need to define phase volume on the submanifold. It's not obvious how to do this - the symplectic form might vanish on the submanifold, or the submanifold might even by odd-dimensional, so we aren't guaranteed to get a natural volume measure from the symplectic form. A better question is "can we define phase volume on a submanifold such that Liouville's theorem holds?
Defining a volume measure over a submanifold is equivalent to defining integration over that submanifold.
For Riemannian manifolds, we usually do this by integrating over an $\epsilon$-thickening of the submanifold, then taking the limit as $\epsilon \rightarrow 0^+$. For a sympletic manifold, an $\epsilon$-thickening doesn't make sense, since there's no notion of distance. However, we can sometimes do something similar using orbits. Fortunately, we don't care about defining volume on an arbitrary submanifold. We care about defining volume on the orbit of of some initial point under the Hamiltonian flow.
Let $p$ be the initial point we care about, and let $M$ be the original manifold.
Let $U \subset M$ be a neighborhood of $p$. $\dim U = \dim M$, so we know how to integrate over $U$. We also know how to integrate over the orbit of $U$. To integrate over the orbit of $p$, we can integrate over the orbit of $U$, then divide by $\int 1 $ and take the limit as $U$ shrinks to $p$. This integration gives a well-defined volume measure on the orbit of $p$. With respect to this volume measure, Liouville's theorem is satisfied.
Exercises for the reader:

*

*Show that the volume measure really is well-defined (i.e. the limit exists)

*Show that it satisfies Liouville's theorem

*On further thought, it's not actually obvious to me that the orbit of $U$ always has a well-defined dimension. Are there Hamiltonian systems with fractal orbits?

*If we have two different Hamiltonians on $M$ with the same orbits, will the associated volume measures be the same? I don't know the answer to this one either.

