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About the basics of Poisson bracket

  1. Nov 20, 2004 #1
    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
     
  2. jcsd
  3. Nov 21, 2004 #2

    matt grime

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    Homework Helper

    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.
     
  4. Nov 22, 2004 #3
    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
     
  5. Nov 22, 2004 #4

    matt grime

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    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.
     
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