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How does one show the uniqueness of the solution to the brachistochrone problem? Doesn't the fact that the solution is of the form $x=a-c(2t+\sin2t)$ and $y=c(1+\cos2t)$ naturally guarantee uniqueness given the 2 endpoints of the path -- 2 unknowns $(a,c)$ and 2 restraints (the 2 endpoints)?

Thanks!

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    Sure, you have those conditions. Now, how sure are you an *algebraic* curve doesn't suit the bill? That's where solving the associated differential equations comes in...2011-07-27
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    Hi, J.M., I'm afraid I don't quite understand... Would you mind elaborating a bit? Thanks!2011-07-27
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    I can for instance construct a parametric *cubic* which matches the boundary conditions, but does not satisfy the brachistochrone condition...2011-07-27

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