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thepopasmurf
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I'm going through Landau/Lifgarbagez's book II of theoretical physics. In it they have a derivation of the equation of motion from the principle of least action, however I don't understand one step.
Derive the equation of motion:
[itex]\frac{d^2x^i}{ds^2}+\Gamma^i_{kj} \frac{dx^k}{ds} \frac{dx^j}{ds}=0[/itex]
Using the principle of least action:
[itex]\delta S=-mc\delta\int ds=0[/itex]
[itex]\Gamma_{i,kj}=\frac{1}{2}\left(\frac{\partial g_{ik}}{\partial x^j}+\frac{\partial g_{ij}}{\partial x^k}-\frac{\partial g_{kj}}{\partial x^i}\right)[/itex]
[itex]\delta ds^2=2ds\delta ds = \delta(g_{ik}dx^i dx^k)=dx^i dx^k \frac{\partial g_{ik}}{\partial x^j}\delta x^j + 2g_{ik}dx^i d\delta x^k[/itex]
Therefore
[itex]\delta S = -mc\int\left\{\frac{1}{2}\frac{dx^i}{ds}\frac{dx^k}{ds}\frac{\partial g_{ik}}{\partial x^j}\delta x^j + g_{ik}\frac{dx^i}{ds}\frac{d\delta x^k}{ds}\right\}[/itex]
which equals
[itex]\delta S = -mc \int \left\{\frac{1}{2} \frac{dx^i}{ds} \frac{dx^k}{ds} \frac{\partial g_{ik}}{\partial x^j}\delta x^j - \frac{d}{ds}\left\{ g_{ik} \frac{x^i}{ds}\right\} \delta x^k \right\} ds[/itex]
The step I don't understand is going from the second last line to the last line.
Thanks
Homework Statement
Derive the equation of motion:
[itex]\frac{d^2x^i}{ds^2}+\Gamma^i_{kj} \frac{dx^k}{ds} \frac{dx^j}{ds}=0[/itex]
Using the principle of least action:
[itex]\delta S=-mc\delta\int ds=0[/itex]
Homework Equations
[itex]\Gamma_{i,kj}=\frac{1}{2}\left(\frac{\partial g_{ik}}{\partial x^j}+\frac{\partial g_{ij}}{\partial x^k}-\frac{\partial g_{kj}}{\partial x^i}\right)[/itex]
The Attempt at a Solution
[itex]\delta ds^2=2ds\delta ds = \delta(g_{ik}dx^i dx^k)=dx^i dx^k \frac{\partial g_{ik}}{\partial x^j}\delta x^j + 2g_{ik}dx^i d\delta x^k[/itex]
Therefore
[itex]\delta S = -mc\int\left\{\frac{1}{2}\frac{dx^i}{ds}\frac{dx^k}{ds}\frac{\partial g_{ik}}{\partial x^j}\delta x^j + g_{ik}\frac{dx^i}{ds}\frac{d\delta x^k}{ds}\right\}[/itex]
which equals
[itex]\delta S = -mc \int \left\{\frac{1}{2} \frac{dx^i}{ds} \frac{dx^k}{ds} \frac{\partial g_{ik}}{\partial x^j}\delta x^j - \frac{d}{ds}\left\{ g_{ik} \frac{x^i}{ds}\right\} \delta x^k \right\} ds[/itex]
The step I don't understand is going from the second last line to the last line.
Thanks