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Geodesics on a surface

  • Thread starter Grand
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  • #1
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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??
 

Answers and Replies

  • #2
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Anyone with an idea?
 

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