Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

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

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# About the basics of Poisson bracket

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