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!