Brachistochrone Problem/Calculus of Variations

  • Thread starter Megatron
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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
 
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Answers and Replies

  • #2
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If you make the substitution
[tex]y = \sin^2(\theta)/c^2[/tex]
then you also need to substitute for [itex]dy[/itex] the value
[tex]dy = 2/c^2 \sin(\theta) \cos(\theta) d\theta[/tex].
Grouping [itex]\theta[/itex] 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
[tex]ds^2 = dr^2 + r^2 d\theta^2 + r^2\sin^2(\theta) d\phi^2[/tex]
and remember that for a geodesic the quantity you are trying to minimize is
[tex]\int ds[/tex].

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