Symplectomorphism

A symplectomorphism is a diffeomorphism from symplectic manifolds that preserves the symplectic structure of both. Symbolically, this means

Relation to Lagrangian Submanifolds

Given a map of symplectic manifolds, is a symplectomorphism if and only if

is a Lagrangian submanifold of using the twisted product form (where and are the canonical projections).

Proof

Consider the map

The image , so is Lagrangian if and only if .

Therefore,

Twisting Trick

Consider the case with cotangent bundles and and the symplectic manifold

Sometimes, you are given a Lagrangian submanifold of with the canonical symplectic form

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 Let be the twisted product from above, then

Using this symplectomorphism, for a Lagrangian submanifold using , then is a Lagrangian submanifold using the twisted product form.