Connectedness & Canonical Transformations

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SUMMARY

Connectedness and simply-connectedness are preserved under canonical transformations in phase space. Canonical transformations, which are continuous mappings between sets of canonical variables, maintain the topological properties of the original phase space. Therefore, if an area in phase space is connected or simply connected, it will remain so after the transformation to new canonical variables.

PREREQUISITES
  • Understanding of phase space concepts in Hamiltonian mechanics
  • Familiarity with canonical transformations in classical mechanics
  • Basic knowledge of topology, specifically connectedness and simply-connectedness
  • Proficiency in mathematical analysis related to continuous functions
NEXT STEPS
  • Study the properties of canonical transformations in classical mechanics
  • Explore topological concepts such as connectedness and simply-connectedness
  • Investigate Hamiltonian mechanics and its applications in physics
  • Learn about continuous mappings and their implications in mathematical analysis
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Physicists, mathematicians, and students of classical mechanics interested in the implications of canonical transformations on topological properties in phase space.

kakarukeys
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This is a question I found no answer from books.

Is Connectedness and Simply-connectedness preserved by Canonical Transformations?

If an area in phase space is connected (simply connected), will it still connected (simply connected) in the new phase space of new canonical variables?
 
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Isn't a canonical transformation continuous?
 

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