Canonical transformations as a category

[espen180]

I just realized that given a hamiltonian H, the set generated by its canonical transformations under composition just might be a category.

Just checking the axioms:
Given canonical transformations $f:(H,q,p)\rightarrow (K,Q,P)$ and $g:(K,Q,P)\rightarrow (H^\prime, q^\prime,p^\prime)$, $g\circ f$ is also canonical.

Also, the indentity transformation $\text{id}:(H,q,p)\rightarrow (H,q,p)$ exists and is canonical for any H, and associativity is trivially fulfilled. However, I cannot find any treatment of this category anywhere. It there simply no interest in it or anything to gain from this viewpoint?

 

[Valerie Park]

It is possible that this category has been studied in the literature, although I could not find any specific references. It is certainly an interesting idea, and it might be fruitful to explore it further. One potential application could be to use the category structure to study how different canonical transformations interact with one another. Additionally, the category structure could be used to understand the properties of the Hamiltonian system as a whole.