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Derivation of geodesic equation from hamiltonian (lagrangian) equations

  1. Jul 31, 2011 #1
    1. The problem statement, all variables and given/known data
    Hello, I would like to derive geodesics equations from hamiltonian
    [itex]H=\frac{1}{2}g^{\mu\nu}p_{\mu}p_{\nu}[/itex]
    using hamiltonian equations.

    A similar case are lagrangian equations. With the definition

    [itex]L=g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu[/itex]

    I tried to solve the Euler-Lagrange equation

    [itex]\frac{d}{d\lambda}(\frac{\partial{}L}{\partial\dot{x}^\alpha})-\frac{\partial{}L}{\partial{}x^\alpha}=0[/itex].

    2. Relevant equations

    I'm stuck. Apparantely I got lost in the indices. Could anyone please help?

    3. The attempt at a solution
    So far I have

    [itex]\dot{x}^\alpha=\frac{\partial{}H}{\partial{}p_{\alpha}}=g^{\alpha\beta}p_\beta[/itex]
    [itex]\dot{p}_\alpha=-\frac{\partial{}H}{\partial{}x^\alpha}=-\frac{1}{2}(g^{\beta\gamma})_{,\alpha}p_\beta{}p_{\gamma}[/itex]

    where dot means derivation with respect to a parameter [itex]\lambda[/itex].

    I know I should substitute one into another. So to get geodesic equation I write

    [itex]\ddot{x}^\alpha=\frac{d}{d\lambda}(g^{\alpha\beta}p_\beta)=(g^{\alpha\beta})_{,\mu}p^{\mu}p_{\beta}-\frac{1}{2}g^{\alpha\beta}(g^{\nu\gamma})_{,\beta}p_\nu{}p_\gamma[/itex]

    but what do I do now?

    For lagrangian problem:

    [itex]\frac{d}{d\lambda}(g_{\alpha\nu}\dot{x}^\nu+g_{\mu\alpha}\dot{x}^\alpha)-g_{\mu\nu,\alpha}\dot{x}^\mu\dot{x}^\nu=0[/itex]

    [itex]g_{\alpha\nu,\beta}\dot{x}^\beta\dot{x}^\nu+g_{\alpha\nu}\ddot{x}^\nu+g_{\mu\alpha,\kappa}\dot{x}^{\kappa}\dot{x}^{\alpha}+g_{\mu\alpha}\ddot{x}^\alpha-g_{\mu\nu,\alpha}\dot{x}^\mu\dot{x}^\nu=0[/itex]
     
  2. jcsd
  3. Aug 4, 2011 #2
    Your latexing has messed up for your lagrangian one but i think you're almost there.

    I find that your 2nd line should be

    [itex]g_{\alpha \nu, \mu} \dot{x}^\mu \dot{x}^\nu + g_{\alpha \nu} \ddot{x}^\nu + g_{\mu \alpha , \nu} \dot{x}^\mu \dot{x}^\nu + g_{\mu \alpha} \ddot{x}^\mu - g_{\mu \nu , \alpha} \dot{x}^\mu \dot{x}^\nu=0[/itex]

    Now the [itex]\ddot{x}[/itex] terms can be combined by relabelling dummy indices. then divide the whole thing by 2 and multiply by an inverse metric so as to "isolate" the [itex]\ddot{x}[/itex] term from the metric it's currently paired with. You should now be able to identify the Christoffel symbol term and rewrite your answer as the familiar

    [itex]\ddot{x}^\mu + \Gamma^\mu{}_{\nu \alpha} \dot{x}^\nu \dot{x}^\alpha =0[/itex]

    I haven't been through your hamiltonian one explicitly but i think you're on the right lines. remember if you need an expression for [itex]p_\nu[\itex] you can always multiply [itex]\dot{x}^\alpha = g^{\mu \alpha} p_\mu[\itex] by [itex]g_{\nu \alpha}[\itex]
     
    Last edited: Aug 4, 2011
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