Brachistochrone

Recently, I studied the "Brachistochrone" problem for an Optimization course project. If, like me, you dont know Greek, the term requires an introduction. It literally means "shortest time". The problem is thus...

The brachistochrone curve, or curve of shortest time, is the curve between two points that is covered in the least time by a body that travels under the action of constant gravity.

In other words, given 2 points in a vertical plane, what is the path from the higher point to the lower point such that a particle starting from rest and traveling only under the force of gravity will take the shortest time. We will ignore friction in this case.

The problem is interesting in the sense that the solution is non intuitive and has surprising conclusions.

To add to the interesting things about the problem, it has a history too. The problem was first posed by Johann Bernoulli in 1696 in the following manner...

I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.

4 mathematicians solved the problem almost simultaneously... Isaac Newton, Jakob Bernoulli (Johann's brother), Leibniz and L'Hopital. The methods used to solve this problem were later refined into 'the calculus of variations'.

Have a look at a solution curve here...
http://en.wikipedia.org/wiki/Brachistochrone

The curve was shown to be part of a cycloid (path traced by a point on the circumference of a rolling circle).

Interesting Fact 1 : As is seen in the example curve on the wikipedia page, the curve goes below the destination point and comes back up.

Interesting Fact 2 : An extension to objects with initial velocity suggests the following...

Suppose there are 2 jet planes (same acceleration and capability) flying at the same height with same speed, which are racing each other to reach a point x km away at the same horizontal level. The one which accelerates directly towards the point and takes a horizontal path will not win. The path of shortest time actually requires the jet to dip to a lower level in order to speed up using gravity and then come back up. The curve of shortest time is never a straight line, unless the destination point is directly above or below.

I had lot of fun while solving this problem numerically using optimization techniques and tried to compile all the fun into the project report. If you wish to know more about it, or interested in simply looking at some nice pictures of this curve being solved through optimization, have a look at ...

http://www.stanford.edu/~nagrajan/Optimization_Project_Report.pdf (3.5 mb)

(warning - most of the file might not make any sense, but the pictures can be fun)