Suppose you’ve got a Lagrangian on the Tangent bundle of your configuration space. Fiber derivatives of L give you momenta,
Which transform like the components of a covector. There is the remaining question (this is probably obvious but I’m missing it at the moment) as to how we know these are the components of a 1-form on M (so a point in T*M). But let’s suppose that’s the case. Then we can pull that form back to a form in T*(T*M), so a form on phase space, which looks the same:
Then we can take the exterior derivative of this 1-form to get the sympletic form on T*M:
which can be wedged with itself to generate volume elements. But hold on right there. It can also be used to write Hamilton’s equations in a coordinate free manner. I’ll go into this at some point, but for now it’s sufficient to say (or tease) that we have:
Where is the Hamiltonian vector field$ and H is the Hamiltonian itself. The little “i” is just telling us to insert the vector field into the 2-form, contracting it to a 1-form.
The components of also determine the Poisson bracket:
Yeah, the fellow shows up a lot! Looking at the coordinate free Hamilton’s equations we can see that it even acts like a metric tensor by creating an association between vectors and covectors. Though I haven’t looked into that direction yet.
One last thing. Since we can make volume elements out of the ‘s and since the microcanonical distribution of statistical mechanics is built out of ratios of hypervolumes in phase space we’d be interested in knowing how these volumes change in time. We’ll do it out here in the near future, but its sufficient to end by saying that the Lie derivative of is zero. All of this from the fiber derivative.