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Differential Geometry
Geodesics subject to a restriction
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[QUOTE="MrTictac, post: 6243982, member: 655908"] Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. 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. [/QUOTE]
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