What is the physical significance of Poisson brackets?

AI Thread Summary
The discussion centers on the physical interpretation of the Poisson bracket, particularly the relationship between position and momentum, expressed as {q_i, p_j} = δ_ij. It raises questions about whether this indicates that momentum generates changes in position or has a different meaning. Participants express confusion over the non-zero nature of the Poisson bracket in classical mechanics, contrasting it with the quantum commutator, which leads to misunderstandings. The motivation behind the Poisson bracket is also questioned, whether it arises naturally or is derived. A geometrical interpretation involving the symplectic form in phase space is mentioned but not fully explored.
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I know the definition of the Poisson bracket and how to derive elementary results from it, but I'm struggling to understand intuitively what they are describing physically? For example, the Poisson bracket between position q_{i} and momentum coordinates p_{j} is given by \lbrace q_{i},p_{j}\rbrace = \delta_{ij}
but what is this describing physically? is it that momentum generates a change in position or is it something else?

Also, how is the Poisson bracket motivated? Is it a quantity that arises naturally which we define to be the Poisson bracket, or is it derived in some manner?
 
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Thanks for the link. I've actually already had a look at that thread and it didn't really help unfortunately.

I think part of the issue is how to interpret the fact that the Poisson bracket for position and momentum is non-zero, but these commute in classical mechanics (the confusion arising from the fact that people often seem to describe the Poisson bracket by comparing it to the quantum commutator)?!
 
This might help...
(it's been on my to-read list)
http://projecteuclid.org/euclid.cmp/1103907394
On the relation between classical and quantum observables (Abhay Ashtekar)

There's a geometrical interpretation involving the symplectic form in phase space... but I haven't absorbed it well enough to explain.
 
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