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!!(adsbygoogle = window.adsbygoogle || []).push({});

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]

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

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