The Brachistochrone Problem

### 5.22.1 The Brachistochrone Problem

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
0
| 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)= | ∫ |
b
a
| 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.