1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
    using hamiltonian equations.

    A similar case are lagrangian equations. With the definition


    I tried to solve the Euler-Lagrange equation


    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


    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


    but what do I do now?

    For lagrangian problem:


  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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook