find the curve for which the body will follow such that the time of travel is a minimim.(adsbygoogle = window.adsbygoogle || []).push({});

Hints Minimize [tex] t_{12} = \int_{x_{1}}^{x_{2}} dt = \int_{x_{1}}^{x_{2}} \frac{ds}{v} = \int_{x_{1}}^{x_{2}} \sqrt{\frac{1+y'^2}{2gy}} dx [/tex]

since F does not depend on x i can use hte beltrami identity (from the previous post)

[tex] H = \frac{-1}{\sqrt{2gy} \sqrt{1+y'^2}}[/tex]

and

[tex] \frac{dy}{dx} = \frac{1}{2gyH^2} -1 [/tex]

this is where i am stuck

SOlving this creates an ugly mess! How can i get the parametric equations from this?

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# The Brachistochrone problem

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