Neumann’s dynamical system

  • David Mumford
Part of the Modern Birkhäuser Classics book series (MBC)


In classical mechanics, one encounters the class of problems: M = real 2n-dimensional manifold, with a closed non-degenerate differential 2-form ω
$$ \hat \omega = {\text{dual skew - symmetric form on T}}_{\text{M}}^{\text{*}} $$
$$ {\rm H} = e^\infty - {\text{function on M, called the Hamiltonian}}{\text{.}} $$
$$ {\rm X}_{\rm H} = \left\{ {\begin{array}{*{20}c} {the vector field on M defined by} \\ {\omega \left( {{\rm X}_{\rm H} ,Y} \right) = \left\langle {Y,dH} \right\rangle for all vectors Y} \\ {or} \\ {\hat \omega \left( {dH,\alpha } \right) = \left\langle {{\rm X}_{\rm H} ,\alpha } \right\rangle for all 1 - forms \alpha } \\ \end{array} } \right\} $$


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Copyright information

© Birkhäuser Boston 2007

Authors and Affiliations

  • David Mumford
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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