What is the physical significance of Poisson brackets?

["Don't panic!"]

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?

 

[jedishrfu]

You might find some discussion here on its significance:

https://www.physicsforums.com/threads/poisson-bracket-significance-classical-mechanics.169573/

 

["Don't panic!"]

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)?!

 

[robphy]

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.