Differential geometry and hamiltonian dynamics

[luisgml_2000]

Hello everybody!

I'm currently attending lectures on Hamiltonian dynamics from a very mathematical viewpoint and I'm having trouble understanding two facts:

1. An inner product defined in every tangent space and a symplectic form both establish a natural isomorphism between tangent and contanget spaces. My question is: what is the nature of this isomorphism?

2. The relationship between the invariance of the symplectic form under a hamiltonian flux and the Liouville theorem.

Can someone help me out? Thanks a lot.

 

[Ben Niehoff]

1. An inner product defined in every tangent space and a symplectic form both establish a natural isomorphism between tangent and contanget spaces. My question is: what is the nature of this isomorphism?

This is easiest to show using the inner product: Let [itex]\alpha[/itex] be an element of the cotangent space, and v be an element of the tangent space. Say the tangent space has some inner product [itex]g(\cdot,\cdot)[/itex]. Then we can associate [itex]\alpha[/itex] to some element A of the tangent space via

[tex]\alpha(u) = g(A,u)[/tex]

for all vectors u. Using the symplectic form is similar; we just put

[tex]\alpha(u) = \omega(A,u)[/tex]

for all vectors u.

2. The relationship between the invariance of the symplectic form under a hamiltonian flux and the Liouville theorem.

The symplectic form is simply the volume element in phase space. The invariance of the symplectic form under the action of the Hamiltonian is equivalent to saying the phase space "fluid" is incompressible; i.e., the infinitesimal bits of volume do not change in size. I can't say much more without knowing which variant of the Liouville theorem you're familiar with; sometimes, the Liouville theorem simply IS the statement that the symplectic form is invariant under Hamiltonian flows.

 

[luisgml_2000]

I can't say much more without knowing which variant of the Liouville theorem you're familiar with.

The version of Liouville's theorem I'm familiar with is: a hamiltonian flux preserves volume in phase space.

One more thing: why a symplectic form is the volume in phase space?

 

[Ben Niehoff]

I was slightly wrong; the symplectic form is not identical to the volume form. Rather, in a system of N coordinates and N momenta, the Nth exterior power of the symplectic form is proportional to the volume form. Hence if the symplectic form is invariant, the phase space volume must also be invariant.