- #1

Grand

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## Homework Statement

Prove that a particle constrained to move on a surface [tex]f(x,y,z)=0[/tex] 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:

[tex]\int dt=\int\frac{ds}{v}=\int\frac{\sqrt{dx^2+dy^2+dz^2}}{v}[/tex]

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:

[tex]\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}=\frac{\partial L}{\partial x}[/tex]

etc.

But where from now on??