Optimal Halfpipe - dynamic programing - Education

An optimal halfpipe is one that has the fastest line between 2 horizontal points only driven by gravity.

Here is a "half" of an optimal halfpipe - if you have the fastest way to the middle, the other half have to look the same!

I added a straight comparison curve and it is much slower!

It it a "calculus of variations"-problem, see Leonhard Euler und Joseph-Louis Lagrange.

But I calculate it with "Finite Elements" and a "Dynamic Programming" Idea, see Richard Ernest Bellman. I neglect rotation and rolling resistance as well as aerodynamic drag of the ball.

The size of model doesn't matter, it is always the same curve! I made also a little startbox to start the two iron ball simultaneous.