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Homework Help: The Lagrangian/Hamiltonian dynamics of a particle moving in a manifold

  1. Jan 3, 2010 #1
    I posted this problem on the Classical Mechanics Subforum last week but have not received many responses - hopefully someone can help here as I've spent hours racking my brain, trying to work this out!!

    1. The problem statement, all variables and given/known data

    There is a particle of mass 'm' moving in a manifold with the following Lagrangian:

    L={\frac{m}{2}}{g_{ij}(x)}.{\dot{x^{i}}{\dot{x^{j} }



    is the metric on M.

    The question is to find the Equation of Motion - I need to work out:


    in order to do this.

    2. Relevant equations

    To find EOM we need to set the the follow expression to 0:

    {\delta}A=\int{dt[{\frac{{\partial}L}{{\partial}{x^{i}}}dx^{i}+\frac {{\partial}L}{{\partial}\dot{x^{i}}}d{\dot{x^{i}}]}=0

    3. The attempt at a solution

    It took me a while but I understand why:

    \frac{{\partial}L}{{\partial}\dot{x^{i}}}=mg_ {ij}x^{j}

    So that's fine.

    Now I need to evaluate:


    The issue is there is no term in [tex]x^i[/tex] (the position) in the Lagrangian - it is only a function of the velocity [tex] \dot{x}^i[/tex]. And also, why is [tex]g_{ij} = g_{ij}(x)[/tex] - ie. a function of x - as I understand it, it is just a matrix of dim M with diag(-1,1,1,1,1,...).

    FYI - the answer to my question should be:

    \frac{m}{2} {\partial}_i g_{jk} \dot{x}^{j}\dot{x}^{k} - ({\partial}_k g_{ij}) \dot{x}^{k}\dot{x}^{j}
    Last edited: Jan 3, 2010
  2. jcsd
  3. Jan 3, 2010 #2


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    Homework Helper

    I think the Powers That Be on this forum would rather you keep it there:
    (unless it has been a homework problem all along, then it should have been posted here in the first place...) I'll take it up on that other thread.
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