Motivation: the Brachistochrone

13.8.1 Motivation: the Brachistochrone

The Brachistochrone problem is perhaps the original problem in the calculus of variations. The problem is to find the curve from two points in a plane such that an object falling under its own weight will get from the first point to the second in the shortest time.

If the points are (0,y₀) and (x₁,0), with y₀>0 and x₁>0, this becomes the problem of minimizing the objective functional

T(y)=

∫

x₁

L(t,y(x),y′(x)) dx

where the function L is defined by

L(t,y(x),y′(x))=

√

1+y′(x)²

2 g y(x)

for y:[0,x₁]→ℝ such that y(0)=y₀ and y(x₁)=0 (the constant g is the gravitational acceleration).

More generally, one type of problem in the Calculus of variations is to minimize (or maximize) a functional

F(y)=

∫

f(x,y,y′) dx

over all functions y∈ C²[a,b] with boundary conditions y(a)=A and y(b)=B, where A,B∈ℝ. The function f is called the Lagrangian.