Symplectomorphism
A symplectomorphism is a diffeomorphism
Relation to Lagrangian Submanifolds
Given a map
is a Lagrangian submanifold of
Proof
Consider the map
The image
Therefore,
Twisting Trick
Consider the case with cotangent bundles
Sometimes, you are given a Lagrangian submanifold of
In this case, in order to use the relationship above with the twisted product form, we can use the involution on
Using this,
Then, we can define the function
Using this symplectomorphism, for a Lagrangian submanifold