Earlier today I was struggling to see why I couldn’t derive the general geodesic equation (*)
where a general nonzero function. I had been trying to do this by varying the action
with Lagrangian .
The standard technique is use as a Lagrangian instead, and claim that this produces the same results as we would have got with . I’d always accepted this as gospel, but a simple calculation shows that we need an additional assumption for this to be true.
Indeed the Euler-Lagrange equations for give
To regain the Euler-Lagrange equations for we want to cancel our factors. Clearly a sufficient condition for this is that is constant. But by the definition of this means that is constant, which precisely says that is an affine parameter.
Hence, by the use of this method we lose the generality needed to obtain equation (*).
Nevertheless it is easy to derive (*) from the affine geodesic equation (**)
Indeed let be a general parameter and write . Then rewriting the geodesic equation (**) in terms of using the chain rule yields (*) with (try it).
Alternatively you can reach a manifestly reparameterisation invariant version of (*) by directly varying our original action . It’s not pretty though!