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Hamilton-Jacobi equation in spherical coordinates

  1. Dec 19, 2013 #1

    DrClaude

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    Staff: Mentor

    I was looking at the Wikipedia entry on the Hamilton-Jacobi equation, and was confounded by the equation at the beginning of the section on spherical coordinates:

    http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation#Spherical_coordinates

    Shouldn't the Hamiltonian simply be
    $$
    H = \frac{1}{2m} \left[ p_r^2 + p_\theta^2 + p_\phi^2 \right] + U(r, \theta, \phi)
    $$
    ?
     
  2. jcsd
  3. Dec 19, 2013 #2
    In spherical coordinates, kinetic energy is $$ T = {m \over 2} \left( v_{r}^2 + v_{\theta}^2 + v_{\phi}^2 \right) = {m \over 2} \left( \dot r ^2 + (r \dot \theta)^2 + (r \sin \theta \ \dot \phi)^2 \right) $$ By definition, $$ p_r = {\partial T \over \partial r} = m \dot r \\ p_{\theta} = {\partial T \over \partial \theta } = m r^2 \dot \theta \\ p_{\phi} = {\partial T \over \partial \phi} = m r^2 \sin^2 \theta \ \dot \phi $$
     
  4. Dec 19, 2013 #3

    DrClaude

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    Staff: Mentor

    Thanks a lot! I now realize I missed a square in my derivation :redface:
     
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