Brachistochrone Problem/Calculus of Variations

1. Oct 11, 2006

Megatron

I'm working on the Brachistochrone probem and I've gotten to the equation:

(dy/dx)^2 = (c^2*y)/(1-c^2*y)

the 'hint' given is to use y = sin^2(theta)/c^2 to solve the integral. I haven't done any math for 5 months and i haven't been in a pure math class for over a year, so I'm drawing a complete blank on how to solve this. If someone could point me in the right direction that would help.

Also, I'm having trouble approaching a problem in which I'm asked to find the geodesics on a sphere of unit radius using calculus of variations. I'm asked to express phi as a function of theta but that isn't making sense to me...again, some direction would help me get on track.
(^ is supposed to denote a power)

Thanks

Last edited: Oct 11, 2006
2. Oct 11, 2006

dicerandom

If you make the substitution
$$y = \sin^2(\theta)/c^2$$
then you also need to substitute for $dy$ the value
$$dy = 2/c^2 \sin(\theta) \cos(\theta) d\theta$$.
Grouping $\theta$ terms and x terms on opposite sides of the equation should give you something you can integrate.

Look familiar? Think back to your integral calculus class :)

For the geodesics on a sphere you need to consider the differential line element in spherical coordinates
$$ds^2 = dr^2 + r^2 d\theta^2 + r^2\sin^2(\theta) d\phi^2$$
and remember that for a geodesic the quantity you are trying to minimize is
$$\int ds$$.

On the surface of a sphere $r$ is constant, so that simplifies things a bit, and you can then factor out the $d\theta^2$ and follow the standard calculus of variations method using the Euler-Lagrange equation.

Last edited: Oct 11, 2006