Most General Canonical Transformation?

[strangerep]

In classical Hamiltonian mechanics, the concept of a canonical transformation ("CT")
preserving the form of Hamilton's eqns is well known. Textbooks (e.g., Goldstein)
distinguish "restricted" CTs that just mix the q's and p's (generalized coordinates and
generalized momenta respectively). These form the usual group of symplectomorphisms
on Poisson manifolds discussed in the more high-brow books.
Textbooks also mention "extended" CTs that involve the time parameter.

But I can also imagine transformations that only mix H and t, but still preserve the
form of Hamilton's eqns (assuming the various functions and derivatives are
sufficiently well-behaved).

So... what are the most general CTs, and what group is formed from them?

TIA.

 

[Aaron Bex]

I'm not an expert in classical Hamiltonian mechanics, but from what I understand, the most general CTs are those which mix both generalized coordinates and momenta, as well as the time parameter. These transformations form the group of symplectomorphisms on Poisson manifolds, as discussed in more advanced texts. Hope this helps!

 

[charng]

The concept of canonical transformations is an important one in classical Hamiltonian mechanics, as it allows us to transform a system's coordinates and momenta while preserving the form of Hamilton's equations. In textbooks, there is often a distinction made between restricted canonical transformations, which only mix the coordinates and momenta, and extended canonical transformations, which also involve the time parameter. However, it is possible to imagine transformations that only mix the Hamiltonian and time, while still preserving the form of Hamilton's equations.

To answer your question, the most general canonical transformations are those that involve both the coordinates and momenta, as well as the Hamiltonian and time. These transformations form a group known as the symplectomorphisms on Poisson manifolds, which is discussed in more advanced texts on the subject.

It is important to note that the functions and derivatives involved in these transformations must be well-behaved in order for them to preserve the form of Hamilton's equations. This is necessary to ensure the physical consistency of the system.

In summary, the most general canonical transformations are those that involve both the coordinates and momenta, as well as the Hamiltonian and time, forming the symplectomorphisms group on Poisson manifolds. These transformations play a crucial role in classical Hamiltonian mechanics and allow us to analyze and manipulate systems in a mathematically rigorous manner.