Geodesics on Surfaces: Proving the Relationship to Particle Motion

[Grand]

Homework Statement

Prove that a particle constrained to move on a surface $$f(x,y,z)=0$$ and subject to no forces, moves along the geodesic of the surface.

Homework Equations

The Attempt at a Solution

OK, we should prove that the path the particle takes and the geodesic are given by the same expression.

For the geodesic:
$$\int dt=\int\frac{ds}{v}=\int\frac{\sqrt{dx^2+dy^2+dz^2}}{v}$$
v must be constant since there are no forces - components of v may change along the path, but the speed will remain the same.

Now for the path:
$$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x}$$
etc.

But where from now on??

 

[Grand]

Anyone with an idea?