Introduction to the Brachistochrone Problem — Finding the Time to Slide Along a Path

It seems like the most common introduction to the Calculus of Variations is to talk about the Brachistochrone problem. It goes like this.

Imagine you have a bead that can slide along a frictionless wire under the influence of gravity only. You want the bead to slide (starting from rest) from point 1 to point 2 along some path. What path would produce the shortest time?

So, let’s do this.

Bead on a Straight Wire

I’m going to start with the simplest path possible — a straight line. It looks like this.

Oh, why did I pick weird starting and ending points? You will see why soon (hopefully). But here is how to find this sliding time. I’ll start by breaking the path into small pieces — where each piece has a length of ds. I need to find the time it takes the bead to move along this short piece. That time depends on the velocity. So, using the definition of the velocity, I can find the time.

Now that I have the time for each piece, the total time will just be the sum of short times. Of course I will take the limit as…