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Velocities as function of canonical momenta

  1. Jul 5, 2008 #1
    Assuming I've understood some claims correctly, having defined the canonical momenta with equation

    [tex]
    p_k = \frac{\partial L}{\partial \dot{q}_k},
    [/tex]

    we can solve the velocities as functions

    [tex]
    \dot{q}_k(q_1,\ldots,q_n,p_1,\ldots, p_n)
    [/tex]

    precisely when the determinant

    [tex]
    \textrm{det}\Big(\Big(\frac{\partial^2 L}{\partial \dot{q}_k\partial\dot{q}_{k'}}\Big)_{k,k'\in\{1,\ldots,n\}}\Big)
    [/tex]

    is non-zero. Why is this the case? The result looks reasonable, but I have difficulty seeing where this is coming from.
     
  2. jcsd
  3. Jul 7, 2008 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Basically, to solve the velocities, you will want to use the implicit function theorem, from which it follows that the Hessian must be invertible. Also see the exact statement in the link.
     
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