Integrating the Brachistochrone Problem: Solving for the Optimal Path

[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.

 

[Andrew Mason]

[tex]t=\int\frac{\sqrt{1+(dy/dx)^2}}{\sqrt{2gy}}dx[/tex]

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?

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

 

[elessar_telkontar]

thank you so much.