Proving basic properties of Poisson brackets on symplectic manifolds

[sarason]

Dear all,
Please help me to solve the following problems
about Poisson brackets.
Let M be a 2n-manifold and w is a closed non-degenerate di®eren-
tial 2-form. (Locally we write w = w_ij dx^i ^ dx^j with [w_ij ] being a
non-degenerate anti-symmetric real matrix-valued local function on M)
Let f, g be two smooth functions on M. Define the Poisson bracket
{f, g} as follows: locally,
{f, g} = w^ij \partial_i f \partial_j g
where [w^ij ] is the inverse of [w_ij ].
SHOW that
1) The Poisson bracket is well-defined, i.e., on the intersection of two
coordinate patches, the two definitions, one written in each local co-
ordinate system, actually always agree.
2) {f, g} = -{g, f} and
{f, gh} = {f, g}h + g{f, h}
for any three smooth functions f, g, h.
3) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0
for any three smooth functions f, g, h.

Thank you very much for taking time to consider these problems.

Sarason

 

[matt grime]

Erm, the answer is to "just do it".

calculate {f,g} on the overlap of two elements in the atlas and use the transformation rules for f and g (it's a smooth function) to show they are equal.

2 and 3, it now suffices to work locally, so do so.

it's not pleasant but that's the way it works I'm afraid.

 

[sarason]

Thanks for your reply. Would you please cite some
references so that I may consult the transformation rules
or the information related to my questions.

Sarason

 

[matt grime]

Eh? any book on differential geometry or differential manifolds, or even differentiable manifolds will tell you what it means for a function to be smooth on the manifold. If you've not seen this then trying to do poisson brackets is a little adventurous.