§ Symplectic version of classical mechanics
§ Basics, symplectic mechanics as inverting :
I've never seen this kind of "inverting " perspective written down
anywhere. Most of them start by using the inteior product without
ever showing where the thing came from. This is my personal interpretation of
how the symplectic version of classical mecanics comes to be.
If we have a non-degenerate, closed
Now, given a hamiltonian , we can construct a
vector field under the definition:
This way, given a hamiltonian , we can construct
an associated vector field , in a pretty natural way.
We can also go the other way. Given the , we can build the
under the equivalence:
This needs some demands, like the one-form being integrable. But this
works, and gives us a bijection between and as we wanted.
We can also analyze the definition we got from the previous manipulation:
We can take this as a relationship between and . Exploiting
this, we can notice that . That is, moving along does
not modify :
§ Preservation of
We wish to show that . That is, pushing forward
along the vector field preserves .
§ Moment Map
Now that we have a method of going from a vector field to a Hamiltonian
, we can go crazier with this. We can generate vector fields using
Lie group actions on the manifold, and then look for hamiltonians corresponding
to this lie group. This lets us perform "inverse Noether", where for a given
choice of symmetry, we can find the Hamiltonian that possesses this symmetry.
We can create a map from the Lie algebra to
a vector field , performing:
We can then attempt to recover a hamiltonian from
. If we get a hamiltonian from this process, then it
will have the right symmetries.