Brachistochrone for Specific Ratios

[Joe Wolf]

It is commonly known that the solution to the brachistochrone problem is a cycloid.

However, in order for a solution curve to be a cycloid, the ratio between points A on the y-axis and B on the x-axis has to be r/pi*r, since that is the ratio between the "height" of the cycloid and half of its "length". However, what is the solution curve to the brachistochrone problem if points A and B share a different ratio to each other - say, 1/2 ?

Two possible solutions that I have considered:

1. The curve is an affine function of the cycloid; the curve is stretched by a factor k along one of the directions.
2. The curve is a segment of a larger cycloid.

How would I approach this problem?

 

[ibkev]

Have you looked into the "Calculus of Variations"? Where you can use single variable calculus to find points on a curve corresponding to the extrema (ie. min/max), the Calculus of Variations can be used to find entire functions that are extrema for some characteristic. Taylor's Classical Mechanics text for example has a chapter that introduces this and I believe the Brachistochrone is one of the solutions presented (Morin's Mechanics text also touches on this.)

I suspect having gone through that you would be able to learn how to introduce new constraints that produce different curves.

 

[Joe Wolf]

I don't quite understand how I would go about adding such a constraint as I mentioned above