generating function

Definition/ Construction

Let and be smooth manifolds, and a smooth function. Therefore, is a closed 1-form on , so it corresponds to a Lagrangian submanifold of . This submanifold is called the Lagrangian submanifold generated by . Using coordinates for and for , then we can denote as

Therefore, we can take this manifold and use the symplectomorphism twisting trick to obtain the submanifold

If this manifold is the graph of a symplectomorphism , then we call the symplectomorphism generated by , and the generating function.

Relation to Hamilton’s equations

The goal of this section is to understand under what conditions the submanifold is the graph of a symplectomorphism Let and . If these elements are in the graph of the symplectomorphism generated by (that is ) then

Broken into component functions, we have

These give us the Hamilton’s equations, so for a point , to find the point that is mapped to under (with the goal of constructing the symplectomorphism ) we must solve

Note that the coordinates are given, so we must solve the first function using the Implicit function theorem to get coordinates . We can then use this in the second equation to get . Thus, the symplectomorphism is constructed as

In order to use the implicit function theorem we need the local condition