Question:

When to set a quantity to constant in a Lagrangian (of geodesic equation)?

Everly: 3 days ago

I've been wrestling with this problem for quite a while now, and I can't seem to understand what I am allowed to do or not.

Let us consider the following action : $$S = \int \sqrt{-g_{\mu\nu}\frac{dx^{\nu}}{|ds|}\frac{dx^{\mu}}{|ds|}}|ds|$$

Now, I want to show that this gives the famous geodesic equation if we choose $|ds| = d\tau$, the proper time. I know how the proof goes with the variation of S, but I wanted to prove it with the Euler-Lagrange expression, and that is how I arrive to the root of my problem.

I know that if we choose the parameter as stated before, we must have that $$g_{\mu\nu}\frac{dx^{\nu}}{|ds|}\frac{dx^{\mu}}{|ds|} = -1.$$ Obviously, we can't replace that directly in the initial equation for S, otherwise we will have a constant lagrangian. My question is therefore the following : when can I effectively set $g_{\mu\nu}\frac{dx^{\nu}}{|ds|}\frac{dx^{\mu}}{|ds|}$ to be $-1$, without coming up with a wrong result ? Indeed, let me write the steps for the Euler-Lagrange equation :

$$\frac{d}{d\tau}\left(\frac{\partial}{\partial U^{\alpha}}(\sqrt{-g_{\mu\nu}U^{\nu}U^{\mu}}) \right) = \frac{\partial}{\partial x^\alpha}(\sqrt{-g_{\mu\nu}U^{\nu}U^{\mu}})$$

$$\frac{d}{d\tau}\left(\frac{g_{\alpha \nu}U^{\nu}}{\sqrt{M}} \right) = \frac{\partial_{\alpha}g_{\mu\nu}}{2\sqrt{M}}U^{\mu}U^{\nu}$$

With $$U^{\nu} = \frac{dx^{\nu}}{|ds|}$$ and $$M = -g_{\mu\nu}U^{\nu}U^{\mu}.$$

Now, in the above equation, if I set $M = 1$, then I do find the geodesic equation. Let me rephrase the question : why can I set $M = 1$ now? If I had set $M = 1$ before doing any of the $\frac{\partial}{\partial U^\nu}$ or $\frac{\partial}{\partial x^\nu}$ derivatives, I would have gotten a wrong result. Why can I set $M=1$ inside the $\frac{d}{|ds}$ derivative, but not inside the others? I hope I made myself clear!

Answer:
Claire: 3 days ago

  1. Because the square root action is reparametrization invariant, the solutions to Euler-Lagrange (EL) equation (http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation) are geodesics (https://en.wikipedia.org/wiki/Geodesic) with arbitrary parametrization.

  2. As you already noted, it would be inconsistent to choose $$M~=~{\rm constant}$$ before performing the variation and before doing all the partial differentiations in the EL equation.

  3. By restricting to $$M~=~{\rm constant}$$ after the partial differentiations in the EL equation, you are restricting your parametrized geodesics solutions to only those which are affinely parametrized. (Note that the non-affinely parametrized geodesic equation has an additional term.)

  4. You are allowed to set $$M~=~{\rm constant}$$ before the final total parameter differentiation in the EL equation because this differentiation is along the very same curve rather than, say, a differentiation comparing neighboring curves in a variational process.

  5. See also my related Phys.SE answer here (https://physics.stackexchange.com/a/149204/2451) where all of this is explained in more details.