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Brachistochrone

The approach to choose a path along which any system progresses (light propagates, planet moves around the Sun etc.) based on minimizing some numeric function defined for each path appears to be very valuable in Physics, and it helps to solve certain tasks faster and more efficiently than using only the classic Newton's Laws.

Before generalizing this idea, let's consider a specific problem suggested by Johann Bernoulli in 1696.
It's called the Brachistochrone problem (from Greek 'brachistos' + 'chronus' = 'short' + 'time') and is formulated as follows.

Consider two points A and B in the uniform gravitational field (like near the surface of the Earth) with force of gravity directed vertically down. These points are positioned on different heights and not on the same vertical.

A small object should slide from the top point A(a,A) to the lower point B(b,B) along some frictionless supporting track.

We use a standard Cartesian reference frame with Y-coordinates increasing upwards, and X-coordinates increasing from left to right on a picture above.
The vector of gravity force is directed down along Y-axis.
Therefore, Y-coordinate A is greater then B, and X-coordinate a is less than b.

The supporting track can go straight from A to B or take some curved form.
The straight brown line of descend on a picture above is shorter, but the curved blue or purple lines, while longer, allow for an object to gain speed faster and the resulting time of descend might still be shorter than for a straight line.

The problem is to determine the shape of a supporting track to minimize the time of sliding.

Mathematically speaking, we have to consider all smooth functions f(x) on a segment [a,b] that satisfy the conditions:
f(a) = A and f(b) = B
Then, out of all these functions, we have to find such that represents the curve of fastest descend from A(a,A) to B(b,B).

This simply formulated problem is far from having a simple solution.
Best mathematicians of 17th century worked on it and solved using different methodologies.

Let's solve it using the apparatus developed for finding a minimum of a functional - the Euler-Lagrange equation. This methodology was discussed in the previous lectures of this course.

We have to express the time T of moving from point A to point B as a functional of a trajectory represented by function f(x):
T = Φ[f(x)]
and find a function y=f0(x) that minimizes this functional.

Hopefully, our functional will look like
Φ[f(x)] = ∫[a,b] F[x,f(x),f '(x)]dx
where F[...] is some known smooth real function of three arguments - real variable x, real value of function f(x) and real value of derivative f '(x)
and we will be able to apply Euler-Lagrange equation to find y=f0(x) as its solution.