Most General Canonical Transformation?

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SUMMARY

The discussion centers on the concept of canonical transformations (CTs) in classical Hamiltonian mechanics, specifically distinguishing between "restricted" CTs, which mix generalized coordinates (q's) and generalized momenta (p's), and "extended" CTs that incorporate the time parameter. The forum participants explore the possibility of transformations that only mix the Hamiltonian (H) and time (t) while still preserving the form of Hamilton's equations. The inquiry seeks to identify the most general CTs and the group they form, emphasizing the need for well-behaved functions and derivatives in these transformations.

PREREQUISITES
  • Understanding of classical Hamiltonian mechanics
  • Familiarity with canonical transformations and their properties
  • Knowledge of symplectomorphisms on Poisson manifolds
  • Basic grasp of Hamilton's equations
NEXT STEPS
  • Research the properties of symplectomorphisms in more advanced texts
  • Explore the implications of extended canonical transformations
  • Study the mathematical framework of Poisson manifolds
  • Investigate the role of well-behaved functions in Hamiltonian mechanics
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Physicists, particularly those specializing in classical mechanics, mathematicians interested in symplectic geometry, and students seeking a deeper understanding of canonical transformations in Hamiltonian systems.

strangerep
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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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