Christoffel symbols from definition or Lagrangian

[noospace]

I asked this question in the tensor analysis formum but did we did not reach a satisfactory conclusion.

Here is the problem:

Let $\mathbf{x} : U \subset\mathbb{R}^2 \to S$ be a local parametrization of a regular surface S. Then the coefficients of the second derivatives of x in the basis of the tangent space are defined to be the Christoffel symbols.

On the other hand, if we write the first fundamental form $ds^2 = E du^2 + 2F du dv + G dv^2$ in differential form we have an extremization problem of the arc-length

Then the coefficients of of the corresponding Euler-Lagrange equations are essentially the Christoffel symbols.

Are there any interesting examples where the Lagrangian method of computing Christoffel symbols breaks down?

 

[kurt.physics]

The difficulty with this question is that it is not clear what is meant by "Lagrangian method of computing Christoffel symbols". It is possible to compute Christoffel symbols using a variety of methods, such as the direct computation of the derivatives of the coordinates, or the use of the geodesic equation. It is also possible to compute them using the first fundamental form and the Euler-Lagrange equations. However, it is not clear which of these methods is referred to as the "Lagrangian method".