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: 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:

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

    where

    [tex]
    ds^{2}=g_{ij}(x)dx^{i}dx^{j}
    [/tex]

    is the metric on M.

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

    [tex]
    {\frac{{\partial}L}{{\partial}{x^{i}}}
    [/tex]

    in order to do this.

    2. Relevant equations

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

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

    3. The attempt at a solution

    It took me a while but I understand why:

    [tex]
    \frac{{\partial}L}{{\partial}\dot{x^{i}}}=mg_ {ij}x^{j}
    [/tex]

    So that's fine.

    Now I need to evaluate:

    [tex]
    {\frac{{\partial}L}{{\partial}{x^{i}}}
    [/tex]

    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:

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

    diazona

    User Avatar
    Homework Helper

    I think the Powers That Be on this forum would rather you keep it there:
    https://www.physicsforums.com/showthread.php?t=366151
    (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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook