The Lagrangian/Hamiltonian dynamics of a particle moving in a manifold

[vertices]

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!

Homework Statement

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

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

where

$$ds^{2}=g_{ij}(x)dx^{i}dx^{j}$$

is the metric on M.

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

$${\frac{{\partial}L}{{\partial}{x^{i}}}$$

in order to do this.

Homework Equations

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

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

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:

$${\frac{{\partial}L}{{\partial}{x^{i}}}$$

The issue is there is no term in $$x^i$$ (the position) in the Lagrangian - it is only a function of the velocity $$\dot{x}^i$$. And also, why is $$g_{ij} = g_{ij}(x)$$ - 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:

$${\frac{{\partial}L}{{\partial}{x^{i}}}= \frac{m}{2} {\partial}_i g_{jk} \dot{x}^{j}\dot{x}^{k} - ({\partial}_k g_{ij}) \dot{x}^{k}\dot{x}^{j}$$

 

[diazona]

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.