# Brachistochrone problem

I am trying to solve the Brachistochrone problem.

First of all I was asking myself whether there is any good reason that the curve $x$ we are looking for is in $C^2[0,t_\text{end}]$. Then most textbook argue, that one could bring a curve of the form $x(t)=(x_1(t),x_2(t))$ to the form $x(t)=(x_1(t), f(x_1(t)))$ in order to obtain the integral that we want to examine. As far as I know, one would need that $x_1(t)$ is continuously differentiable and that $\forall t \in (0,t_\text{end}]: \dot{x_1}(t) >0$ to do this but how do I know that there are no curves that are much faster that do not fulfill this property. for instance the curve where the ball falls free for several seconds is not included in this set, as this curve cannot be written as a function. but what is the right argument that one does not have to consider these curves? i have looked through so many textbooks but they are all fairly inaccurate about these things.

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As far as I know the most common way to solve the Brachistochrone problem is using Hamilton's variational principle. Maybe you could not find anything because you didn't look in the right textbooks: try in texts on classical mechanics. If I'll find where I'll add a link. – Alessandro Mar 15 '13 at 18:29
EDIT: Found! Look in Goldstein, "Classical mechanics" second edition, chapter 2 – Alessandro Mar 15 '13 at 18:41

You know that $x_1'(t)\ge 0$. If for some $t$ you had $x_1'(t)=0$ when looking for a solution among functions $x_2=f(x_1)$ you would not find a solution. In fact any such curve (with vertical parts) could be approximated by graphs.

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thanks.what is the exact mathematical theorem, that tells me, that i would not find a solution when i would look among functions $x_2=f(x_1)$? It is obviously consequence by the approximation theorem you mentioned,but I do not know,although it sounds fairly plausible, the proper name for it either?by the way:i do not know,where it comes from,that so many textbook say that this curve has to be continously differentiable.is this assumption just made as it is the only way to the apparatus of mathematics or can it be shown that there are no "faster solutions" if we say that our curve is continous? – user66906 Mar 15 '13 at 18:01
If the curve is not differentiable how can you define the motion of the point? The velocity is constrained on the tangent line... – Emanuele Paolini Mar 15 '13 at 18:26
and if you now tell me what the name of this approximation theorem is, then i will sleep well and deep :-) – user66906 Mar 15 '13 at 19:16
i would highly appreciate it if you could give me a few more hints about this approximation thing. unfortunately, there are so few text that threat this problem with the mathematical rigour that would be neceesary. – user66906 Mar 16 '13 at 1:48
First of all: how do you prove that your solution is the minimum among functions? – Emanuele Paolini Mar 16 '13 at 9:58