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Poisson-brackets: oh look, we have a group!
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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