# Do canonical transformations form a group?

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?

1. Existence of identity element: Identity transformation is canonical because the coordinates are mapped to themselves.
2. Existence of inverse: Inverse of a canonical transformation is canonical since the Poisson brackets are invariant.
• You seem to have answered your own question? – jacob1729 May 21 '19 at 19:49

There are, confusingly, different definitions of what exactly a "canonical transformation" is. See this answer by Qmechanic for an extended discussion and more useful links. The definition of a transformation in terms of canonical coordinates $$q,p$$ (a fixed "Darboux chart") does not suffice to talk about these transformations in full generality since the phase space of a generic system is not guaranteed to be covered by a single such chart.

If we either take the viewpoint that a canonical transformation is a symplectomorphism or an infinitesimal symplectomorphism as in this answer by Qmechanic, then the group operation is simply "composition" - carry out the two transformations in succession. Symplectomorphisms are a group under composition: Diffeomorphisms are a group under composition and it's easy to show that the condition for being a symplectomorphism is also fulfilled for the composition of two symplectomorphisms.

• +1. When will the group be abelian? Could you please tell me a resource to look more on this? – Abhay Hegde May 22 '19 at 5:18
• @expikx It is never Abelian (possible proof: The real symplectic group is always a subgroup and it is never Abelian). I'm not really sure what you're looking for in a reference, but one standard reference for symplectic geometry in classical mechanics would be Abraham's and Marsden's Foundations of Mechanics. – ACuriousMind May 22 '19 at 17:00