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There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##, subject to the restriction that the velocity module is a constant ##k##.

The problem without the restriction is well known, and can be solved by the following lagrangian:

##S=\int_{t_{0}}^{t_{1}}\sqrt{g_{ij}v^{i}v^{j}}dt##

Where ##g_{ij}## is the metric tensor, and ##v^{i}## is the velocity vector. By using the Euler-Lagrange equation:

##\frac{\partial \sqrt{g_{ij}v^{i}v^{j}}}{\partial x^{j}}=\frac{d}{dt}\left(\frac{\partial \sqrt{g_{ij}v^{i}v^{j}}}{\partial v^{j}}\right)##

However, the restriction being precisely ##\sqrt{g_{ij}v^{i}v^{j}}=k##, doesn't allow to set up the problem as before (or that's what I think). Would you know the correct way to solve it?

Thanks in advance.