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[QUOTE="andrewkirk, post: 6244054, member: 265790"] It's not clear what the problem is asking. If the additional constraint has an effect (and I am not yet convinced that it does, given that a geodesic is sometimes described as a 'constant speed trajectory') then the result will not be a geodesic, and so we cannot use the geodesic equation. But then the only constraint we have is the constant speed one, and there are infinitely many curves that have constant speed. So there is no unique solution. For all those curves the particle will travel distance ##k(t_1-t_0)## in the allotted time. Also, I am not sure what is meant above by 'with the smallest distance'. From the context it appears to mean 'smallest distance travelled' which, again, will be ##k(t_1-t_0)##. But it could also mean the smallest (geodesic) distance from start to end point. Is that what was intended? If so, that problem could have a unique solution, but I expect it will be possible for the distance to be zero - ie that it finishes where it starts, and that there will be multiple ways of achieving that result. [/QUOTE]
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