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I'm trying to follow a professor's notes for finding Christoffel symbols for a twosphere. He gives the following two equations:
The Lagrangian for a two sphere:[tex]L = \left( \frac{d\theta}{ds} \right)^2 + sin^2\theta \left( \frac{d\phi}{ds} \right)^2[/tex]
The Euler Lagrange equation:[tex] \frac{d}{ds} \left( \frac{\partial L}{\partial (dx^\mu/ds)} \right )  \frac{\partial L}{\partial x^\mu} = 0[/tex]
Using these, the professor magically gets:
for [tex]x^\mu = \theta[/tex]:[tex]2\frac{d^2\theta}{ds^2}  2 sin\theta cos\theta \left(\frac{d\phi}{ds^2}\right)[/tex]
for [tex]x^\mu = \phi[/tex]:[tex]2sin^2 \theta \frac{d^2\phi}{ds^2} + 4 sin\theta cos\theta \left(\frac{d\theta}{ds}\frac{d\phi}{ds}\right)[/tex]
And the Christoffel symbols can be found with minimal effort. The problem is that I can't follow the derivation to get the equations. I feel like this is a really simple thing, yet I'm having trouble getting the same answer as he showed.
For example, in the [tex]\theta[/tex] case:
[tex] \frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left( \left( \frac{d\theta}{ds} \right)^2\right)\right )
+
\frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right)\right )

\frac{\partial}{\partial \theta}\left(\left( \frac{d\theta}{ds} \right)^2\right)

\frac{\partial}{\partial \theta}\left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right) = 0[/tex]
[tex] 2 \frac{d^2\theta}{ds^2}
+
\frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right)\right )

\frac{\partial}{\partial \theta}\left(\left( \frac{d\theta}{ds} \right)^2\right)

2 sin\theta cos\theta \left(
\frac{d\phi}{ds}
\right)^2
= 0[/tex]
But I don't see how to drive the other terms to zero.
The [tex]\phi[/tex] case is even worse:
[tex]
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)
\right )
+
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
\right )

\frac{\partial}{\partial \phi}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)

\frac{\partial}{\partial \phi}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
= 0[/tex]
[tex]
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)
\right )
+
\frac{d}{ds}
\left(
2 sin^2\theta \left( \frac{d\phi}{ds} \right)
\right )

\frac{\partial}{\partial \phi}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)

\frac{\partial}{\partial \phi}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
= 0[/tex]
I'm not sure how to simplify it from there, since each time I try a method I get the wrong answer. I especially don't see where his [tex]+4sin\theta cos\theta[/tex] came from.
All my attempts have failed. It seems like such a trivial thing, since the professor left it out. And obviously the results are correct since they give the correct Christoffel terms. I would be eternally grateful if someone with a working knowledge of how to combine ordinary and partial derivatives could give me a step by step.
The Lagrangian for a two sphere:[tex]L = \left( \frac{d\theta}{ds} \right)^2 + sin^2\theta \left( \frac{d\phi}{ds} \right)^2[/tex]
The Euler Lagrange equation:[tex] \frac{d}{ds} \left( \frac{\partial L}{\partial (dx^\mu/ds)} \right )  \frac{\partial L}{\partial x^\mu} = 0[/tex]
Using these, the professor magically gets:
for [tex]x^\mu = \theta[/tex]:[tex]2\frac{d^2\theta}{ds^2}  2 sin\theta cos\theta \left(\frac{d\phi}{ds^2}\right)[/tex]
for [tex]x^\mu = \phi[/tex]:[tex]2sin^2 \theta \frac{d^2\phi}{ds^2} + 4 sin\theta cos\theta \left(\frac{d\theta}{ds}\frac{d\phi}{ds}\right)[/tex]
And the Christoffel symbols can be found with minimal effort. The problem is that I can't follow the derivation to get the equations. I feel like this is a really simple thing, yet I'm having trouble getting the same answer as he showed.
For example, in the [tex]\theta[/tex] case:
[tex] \frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left( \left( \frac{d\theta}{ds} \right)^2\right)\right )
+
\frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right)\right )

\frac{\partial}{\partial \theta}\left(\left( \frac{d\theta}{ds} \right)^2\right)

\frac{\partial}{\partial \theta}\left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right) = 0[/tex]
[tex] 2 \frac{d^2\theta}{ds^2}
+
\frac{d}{ds} \left( \frac{\partial}{\partial (d\theta/ds)} \left(sin^2\theta \left( \frac{d\phi}{ds} \right)^2\right)\right )

\frac{\partial}{\partial \theta}\left(\left( \frac{d\theta}{ds} \right)^2\right)

2 sin\theta cos\theta \left(
\frac{d\phi}{ds}
\right)^2
= 0[/tex]
But I don't see how to drive the other terms to zero.
The [tex]\phi[/tex] case is even worse:
[tex]
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)
\right )
+
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
\right )

\frac{\partial}{\partial \phi}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)

\frac{\partial}{\partial \phi}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
= 0[/tex]
[tex]
\frac{d}{ds}
\left(
\frac{\partial}{\partial (d\phi/ds)}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)
\right )
+
\frac{d}{ds}
\left(
2 sin^2\theta \left( \frac{d\phi}{ds} \right)
\right )

\frac{\partial}{\partial \phi}
\left(
\left( \frac{d\theta}{ds} \right)^2
\right)

\frac{\partial}{\partial \phi}
\left(
sin^2\theta \left( \frac{d\phi}{ds} \right)^2
\right)
= 0[/tex]
I'm not sure how to simplify it from there, since each time I try a method I get the wrong answer. I especially don't see where his [tex]+4sin\theta cos\theta[/tex] came from.
All my attempts have failed. It seems like such a trivial thing, since the professor left it out. And obviously the results are correct since they give the correct Christoffel terms. I would be eternally grateful if someone with a working knowledge of how to combine ordinary and partial derivatives could give me a step by step.