Most General Canonical Transformation?

Thread starter strangerep
Start date Feb 1, 2008
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Canonical transformation General Transformation

AI Thread Summary

Canonical transformations (CTs) in classical Hamiltonian mechanics are known to preserve the structure of Hamilton's equations, with textbooks typically distinguishing between "restricted" CTs that mix generalized coordinates and momenta, and "extended" CTs that also involve time. The discussion raises the possibility of transformations that exclusively mix the Hamiltonian (H) and time (t) while maintaining the form of Hamilton's equations, suggesting a broader category of CTs. The inquiry seeks to define the most general CTs and the group they form. Understanding these transformations could deepen insights into the symplectic structure of Poisson manifolds. The exploration of this topic may reveal new avenues in the study of Hamiltonian systems.

Feb 1, 2008

strangerep

Science Advisor

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.

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