In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical transformations comply with a group structure. But what should be the group operation? In other words, is there a notion of "product" under which it is closed and associative?
- Existence of identity element: Identity transformation is canonical because the coordinates are mapped to themselves.
- Existence of inverse: Inverse of a canonical transformation is canonical since the Poisson brackets are invariant.