# Problem of brachistochrone

1. Mar 23, 2005

### elessar_telkontar

I am gathering my mechanics notes and I put into it some examples. When I get the Hamilton principle I put a section for some basic variation calculus. There's the problem of brachistochrone, I try to solve it, but I get stuck with a integral:

the integral that I should make minimal is (I'm so sorry, but I don't know how to put it in LaTex):

t=int((sqrt(1+((dy/dx)^2))/sqrt(2gy))dx)

and from the variation calculus, the y must be the one that complies:

df/dy-(df/(dy/dx))/dx=0 where f=(sqrt(1+((dy/dx)^2))/sqrt(2gy)).

then calculating the partial derivatives of f and putting them into the eq:

m(1+((dy/dx)^2))+2y(dy/dx)(dm/dx)=0 or
m(1+(y'^2))+2yy'(dm/dx)

where y'=dy/dx and m=(1/f)(y'/(gy))

The question is how to integrate it? or which change of variable is good to integrate it in order to get the eq of the brachistochrone?

NOTE: I have tried to separate variables, but this is impossible.

2. Mar 23, 2005

### Andrew Mason

Bernoulli posed this problem in the late 17th C and Newton solved it but he took 12 hours to do it. And he invented Calculus. Mind you, he did not have the benefit of the Euler-Lagrange approach. Just so you don't drive yourself crazy, a complete solution can be found here:
http://mathworld.wolfram.com/BrachistochroneProblem.html

AM

3. Mar 23, 2005

### elessar_telkontar

thank you so much.