- #1
vertices
- 62
- 0
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!
There is a particle of mass 'm' moving in a manifold with the following Lagrangian:
<br /> L={\frac{m}{2}}{g_{ij}(x)}.{\dot{x^{i}}{\dot{x^{j} }<br />
where
<br /> ds^{2}=g_{ij}(x)dx^{i}dx^{j}<br />
is the metric on M.
The question is to find the Equation of Motion - I need to work out:
<br /> {\frac{{\partial}L}{{\partial}{x^{i}}}<br />
in order to do this.
To find EOM we need to set the the follow expression to 0:
<br /> {\delta}A=\int{dt[{\frac{{\partial}L}{{\partial}{x^{i}}}dx^{i}+\frac {{\partial}L}{{\partial}\dot{x^{i}}}d{\dot{x^{i}}]}=0<br />
It took me a while but I understand why:
<br /> \frac{{\partial}L}{{\partial}\dot{x^{i}}}=mg_ {ij}x^{j}<br />
So that's fine.
Now I need to evaluate:
<br /> {\frac{{\partial}L}{{\partial}{x^{i}}}<br />
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}}}= <br /> \frac{m}{2} {\partial}_i g_{jk} \dot{x}^{j}\dot{x}^{k} - ({\partial}_k g_{ij}) \dot{x}^{k}\dot{x}^{j} <br />
Homework Statement
There is a particle of mass 'm' moving in a manifold with the following Lagrangian:
<br /> L={\frac{m}{2}}{g_{ij}(x)}.{\dot{x^{i}}{\dot{x^{j} }<br />
where
<br /> ds^{2}=g_{ij}(x)dx^{i}dx^{j}<br />
is the metric on M.
The question is to find the Equation of Motion - I need to work out:
<br /> {\frac{{\partial}L}{{\partial}{x^{i}}}<br />
in order to do this.
Homework Equations
To find EOM we need to set the the follow expression to 0:
<br /> {\delta}A=\int{dt[{\frac{{\partial}L}{{\partial}{x^{i}}}dx^{i}+\frac {{\partial}L}{{\partial}\dot{x^{i}}}d{\dot{x^{i}}]}=0<br />
The Attempt at a Solution
It took me a while but I understand why:
<br /> \frac{{\partial}L}{{\partial}\dot{x^{i}}}=mg_ {ij}x^{j}<br />
So that's fine.
Now I need to evaluate:
<br /> {\frac{{\partial}L}{{\partial}{x^{i}}}<br />
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}}}= <br /> \frac{m}{2} {\partial}_i g_{jk} \dot{x}^{j}\dot{x}^{k} - ({\partial}_k g_{ij}) \dot{x}^{k}\dot{x}^{j} <br />
Last edited:
