Poisson-brackets: oh look, we have a group

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SUMMARY

This discussion centers on the specification of the structure of groups, particularly the Galilei and Poincare groups, using canonical coordinates and momenta defined as q[0], q[1], q[2], q[3] and p[0], p[1], p[2], p[3]. The key point raised is whether providing Poisson-bracket relations for these coordinates and momenta is sufficient to define the group's structure. The conversation aims to enhance understanding of group dynamics, specifically through the lens of classical mechanics and the author's professor's class notes.

PREREQUISITES
  • Understanding of canonical coordinates and momenta in classical mechanics
  • Familiarity with Poisson-bracket relations
  • Knowledge of group theory, specifically the Galilei and Poincare groups
  • Basic grasp of system dynamics, including translations and rotations
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  • Research the properties and applications of Poisson brackets in classical mechanics
  • Study the structure and representations of the Galilei group
  • Explore the Poincare group and its significance in relativistic physics
  • Investigate the role of generators in system dynamics and their mathematical formulations
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Students of classical mechanics, physicists interested in group theory, and educators looking to enhance their understanding of group structures in dynamics.

bjnartowt
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Suppose we had canonical-coordinates and momenta, q[0], q[1], q[2], q[3], and p[0], p[1], p[2], p[3], and we said "Oh, and also: here are some *given* Poisson-bracket relations for the canonical coordinates and momenta that we take as Given ("lemma"?)."

Is that all that's needed to specify the so-called "structure" of a group? Namely: the Galilei group or Poincare group, once the appropriate elements are specified? (By "elements" of the aforementioned groups, I think I mean: generators of system-dynamics, like translations, rotations, etc.)

The reason I ask: my classical mechanics prof wrote his own class-notes, and I'm trying to provoke discussion that would supplement his perspectives (and my perspectives on his perspectives).
 
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This would be more appropriate in the homework help section, not the academic section.
 

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