Brachistochrone curve - Gravity and other influences

[ehrenfest]

Homework Statement

http://content.answers.com/main/content/wp/en/d/d4/Brachistochrone.png
Go to the link and look at this brachistochrone curve. The brachistochrone problem is supposed to find the fastest path between to points for an object under only the influence of gravity, correct?

Then how does the problem even have a solution when the coordinates are at the same height. How can the the particle swoop back up under only the influence of gravity?

Homework Equations

The Attempt at a Solution

 

[fikus]

There is nothing wrong if they are at the same height. There are no damping forces like friction or air resistance. It's simply the energy law, it comes to the same height and then swoop back, like pendulum.

 

[ehrenfest]

So, it is always assumed that the particle is attached to wire or has some sort of constraint? Is the constraint always the same?

 

[fikus]

brachistochrone is a constraint. You want to calculate what shape should constraint have that time of traveling will be minimal. It doesn't matter what type of constraint is it, it's just a mathematical problem. It is also important that there are no damping forces, which is of course just ideal case.

 

[ehrenfest]

I see, and the constraint is almost always in the form of a normal force, such as in the classic example where they are at the same height, right?

 

[fikus]

yes, for the fastest path the constraint (brachistochrone) is in the plane parallel to force. But also you have for example the skier on the slope, the shortest path is still brachistochrone, only that now it lies in the plane of the slope.