Calculus of variations

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when inducing that the cycloid is the least time-taking course between the two points in the two dimension, we have to use calculus of variations.

Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?
 

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blue_leaf77
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Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?
In principle yes, in fact the cycloid solution applies to a 3D curve when the two points lie in a plane that also contains the gravitational force vector. Since such plane always exists for any given pair of points, the trajectory will always be a cycloid.
 
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