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What are the hamilton equations of motion for homogeneous lagrangians?

  1. Feb 21, 2013 #1
    For a Lagrangian [itex]L(x^k,\dot{x}^k)[/itex] which is homogeneous in the [itex]\dot{x}^k[/itex] in the first degree, the usual Hamiltonian vanishes identically. Instead an alternative conjugate momenta is defined as

    [itex]y_j=L\frac{\partial L}{\partial \dot{x}^j}[/itex]

    which can then be inverted to give the velocities as a function of the position and momenta

    [itex]\dot{x}^i=\phi^{i}(x^k,y_k)[/itex]

    The Hamiltonian is then equal to the Lagrangian with the velocities replaced with this function

    [itex]H(x^k,y_k)=L(x^k,\phi^{k}(x^l,y_l))[/itex]

    We then find that

    [itex]\dot{x}^i=H\frac{\partial H}{\partial y_i}[/itex]

    which is one half of the Hamilton equations of motion. But what about [itex]\dot{y}_i[/itex]?

    I am following Hanno Rund The Hamilton-Jacobi equation in the Calculus of Variations. But Rund moves on from this point to the H-J equation, leaving me wondering about this question.
     
  2. jcsd
  3. Feb 22, 2013 #2
    Re: what are the hamilton equations of motion for homogeneous lagrangi

    I found the answer to this
     
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