About the basics of Poisson bracket

1. Nov 20, 2004

sarason

Dear all,
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

2. Nov 21, 2004

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.

3. Nov 22, 2004