# Canonical transformations as a category

• espen180
In summary, the idea of considering the set of canonical transformations under composition as a category is an interesting one that could have potential applications in studying Hamiltonian systems.
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?

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.

## What are canonical transformations?

Canonical transformations are mathematical transformations that preserve the fundamental structure of a system, such as the Hamiltonian equations of motion. They are used to transform coordinates and momenta in a system while still preserving the overall dynamics.

## How are canonical transformations different from other transformations?

Canonical transformations are unique in that they are symplectic, meaning they preserve the symplectic structure of a system. This structure is related to the conserved quantities in a system, making canonical transformations crucial in understanding the dynamics of a system.

## What are the applications of canonical transformations?

Canonical transformations are used extensively in classical mechanics and Hamiltonian dynamics, as well as in other areas of physics such as quantum mechanics and statistical mechanics. They are also useful in solving problems involving constraints, as they can simplify the equations of motion.

## How are canonical transformations represented mathematically?

Canonical transformations are typically represented by a generating function, which is a function of the old and new coordinates and momenta. The transformation equations can then be derived from this generating function using the Poisson bracket operation.

## What is the role of canonical transformations in Hamiltonian mechanics?

Canonical transformations are essential in Hamiltonian mechanics as they preserve the Hamiltonian structure of a system. This means that the equations of motion and the conserved quantities remain the same under a canonical transformation, making it a powerful tool in analyzing and solving problems in this field.

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