Euler-Lagrange Equations for geodesics

[Pentaquark5]

Homework Statement

The Lagrange Function corresponding to a geodesic is $$\mathcal{L}(x^\mu,\dot{x}^\nu)=\frac{1}{2}g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$

Calculate the Euler-Lagrange equations

Homework Equations

The Euler Lagrange equations are $$\frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^\mu} \right) =\frac{\partial \mathcal{L}}{\partial x^\mu}$$

The solution should be
$$\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}s^2}+\Gamma^\mu_{\;\;\alpha \beta} \frac{\mathrm{d}x^\alpha}{\mathrm{d}s} \frac{\mathrm{d}x^\beta}{\mathrm{d}s}=0$$

The Attempt at a Solution

Calculate LHS:

$$\frac{\partial \mathcal{L}}{\partial \dot{x}^\mu}=\frac{1}{2}\dot{x}^\beta (g_{\mu\beta}+g_{\beta\mu})=\dot{x}^\beta g_{\mu \beta}$$

$$\frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^\mu} \right)=\frac{\mathrm{d}}{\mathrm{d}s} \left(\dot{x}^\beta g_{\mu \beta}\right)=\ddot{x}^\beta g_{\mu\beta}$$

Calculate RHS:

$$\frac{\partial \mathcal{L}}{\partial x^\mu}=\frac{1}{2} \partial_m g_{\alpha \beta}(x^\mu)\dot{x}^\alpha\dot{x}^\beta$$Equating and multiplying with $g^{kl}$:

$$ \ddot{x}^\beta=\frac{1}{2} g^{\mu \beta} \partial_m g_{\alpha \beta}(x^\mu)\dot{x}^\alpha\dot{x}^\beta $$This kind of looks like the definition of the Christoffel Symbol that I need, however two derivations of the metric are missing. Where did I go wrong?

Thanks

 

[Orodruin]

You did not go wrong. I suggest you check what the missing terms are given that they are contracted with the $\dot x^\mu$.

Edit: Note that what you compute with the EL is the affinely parametrised curve that minimises the distance. This is the same as the geodesic only if the connection is the Levi-Civita connection.

 

[Pentaquark5]

Ok, I just realized my indecess are off.

$$\ddot{x}^\beta \underbrace{g^{kl}g_{\mu\beta}}_{\delta^k_{\;\mu} \delta^l_{\; \beta}}=\frac{1}{2} g^{kl} \partial_\mu g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$
$$ \ddot{x}^l = \frac{1}{2} g^{kl} \partial_\mu g_{\alpha \beta}(x^\mu)\dot{x}^\alpha \dot{x}^\beta$$

However, that is a problem, as I would need the $k$ index to appear in the $\partial_\mu g_{\alpha \beta}$ term, wouldn't I?

Then, I'm still missing some terms along the lines of $\partial_\alpha g_{k \mu} - \partial_k g_{\mu \alpha}$, and the signs are all wrong.

I really don't see how to get there!

 

[Orodruin]

Ignore what I said in #2. I had some more time to read your post more carefully.

I suggest you drop the argument of the metric because it seems that its index is confusing you (it is not an actual index in the equation, it is just telling you that the metric depends on the position, which you should already know). In addition, this is not correct:

$$\underbrace{g^{kl}g_{\mu\beta}}_{\delta^k_{\;\mu} \delta^l_{\; \beta}}$$

What you are looking for is to contract one of the indices of the inverse metric with the free index of the metric, i.e.,
$$
g^{k\beta} g_{\mu\beta} = \delta^k_\beta.
$$

Then look at the Christoffel symbols of the Levi-Civita connection
$$
\Gamma_{\mu\nu}^\lambda = \frac{1}{2} g^{\lambda\rho} (\partial_\mu g_{\rho\nu} + \partial_\nu g_{\mu\rho} - \partial_{\rho}g_{\mu\nu}).
$$
What happens when you insert this into the geodesic equation
$$
(\nabla_{\dot\gamma} \dot\gamma)^\mu = \ddot{x}^\mu + \Gamma_{\alpha\beta}^\mu \dot x^\alpha \dot x^\beta = 0?
$$

Checking it closer, you are actually missing terms on the LHS. Consider this step more carefully:

$$\frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^\mu} \right)=\frac{\mathrm{d}}{\mathrm{d}s} \left(\dot{x}^\beta g_{\mu \beta}\right)=\ddot{x}^\beta g_{\mu\beta}$$

In addition, in this step on the RHS

Equating and multiplying with $g^{kl}$:

$$ \ddot{x}^\beta=\frac{1}{2} g^{\mu \beta} \partial_m g_{\alpha \beta}(x^\mu)\dot{x}^\alpha\dot{x}^\beta $$

suddenly an $m$ and a $\mu$ appeared that are unmatched on the LHS, they should be the same (and again, you would probably do better to drop writing out the index on the argument of the metric, it is just confusing you). Also, your index $\beta$ appears three times in the RHS and only once on the LHS. Probably you should use a different index in the inverse metric that you are multiplying by not to confuse the indices.

 

[Pentaquark5]

Thanks! I think I get it now!

 

[Pentaquark5]

Ok, sorry, but I still don't get it.

When you refer to missing terms, I assume you mean the product rule in the following equation?$$\frac{\mathrm{d}}{\mathrm{d}s} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^\mu} \right)=\frac{\mathrm{d}}{\mathrm{d}s} \left(\dot{x}^\beta g_{\mu \beta}\right)=\ddot{x}^\beta g_{\mu\beta}+\dot{x}^\beta \frac{\mathrm{d}}{\mathrm{d}s} g_{\mu \beta}$$

I really don't know what do do with $\frac{\mathrm{d}}{\mathrm{d}s} g_{\mu \beta}$. Doesn't that bring me even further from the required form?

And I also can't seem to work out how inserting the Christoffel symbol in terms of the metric into the equation helps me.

 

[strangerep]

I really don't know what do do with $\frac{\mathrm{d}}{\mathrm{d}s} g_{\mu \beta}$. Doesn't that bring me even further from the required form?

No, it will bring you closer. Use the chain rule:
$$\frac{d}{ds} X ~=~ X,_\nu \, \dot x^\nu $$