I plan on deriving a solution to this problem by **gradient descent**.By, this method of solving, we will be able to find not just the shortest time path, rather, *we can get any curve such that, the time taken will be as close as possible to any time value we want.*

To elaborate, if we want to find a path in the vertical plane such that the time taken by a friction-less bead traveling on the required path is exactly 1.75 times the time taken by the same friction-less bead traveling on a straight line, we can do that. All this is made possible by using gradient descent, backbone and primary working principle of all types of Neural Networks.

This** method of a traversing a function in a given domain is very powerful,** and it allows us to arrive at a solution without using integration or the Euler-Lagrange equation (which itself is derived after some heavy mathematics), i.e. we can arrive at a numerical solution for any path, given the amount of time taken, without ever having to integrate.Another way this method is very advantageous is it can quite easily take care of finding optimal solutions upon functions which are not integrable or are quite hard to integrate.

## What Can this Method accomplish?

Let me just show you just how powerful this method is, ahead of time.

The path of shortest time, exact cycloid curve joining start and end points.

Paths with customizable Times of descent:

One more:

## Brachistochrone Problem Mathematical Solution:

Before I use the numerical gradient descent method to solve the Bracistochrone problem, I will first solve the problem analytically, i.e. using Integration and the Euler-Lagrange Equation.The theoretical solution thus obtained is the cycloidal curve connecting the 2 points as shown below.