 
 
 
13.8.3  Solving the Brachistochrone Problem
To solve the brachistochrone problem (see Section 13.8.1),
you can first find the Euler-Lagrange equations for the Lagrangian
You can simplify this somewhat by assuming that you are using units
where 2g=1.
| assume(y>=0):; euler_lagrange(sqrt((1+y'^2)/y),x,y) | 
|  | | |  | ⎡ ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎣
 | − | | 1 |  |  |  | |  | |  |  | | ⎛ ⎜
 ⎜
 ⎜
 ⎜
 ⎜
 ⎝
 | ⎛ ⎜
 ⎜
 ⎝
 |  | y | ⎛ ⎝
 | x | ⎞ ⎠
 | ⎞ ⎟
 ⎟
 ⎠
 |  | +1 | ⎞ ⎟
 ⎟
 ⎟
 ⎟
 ⎟
 ⎠
 | y | ⎛ ⎝
 | x | ⎞ ⎠
 | 
 | 
 | 
 | 
 | =K2, |  | y | ⎛ ⎝
 | x | ⎞ ⎠
 | = | | | − | ⎛ ⎜
 ⎜
 ⎝
 |  | y | ⎛ ⎝
 | x | ⎞ ⎠
 | ⎞ ⎟
 ⎟
 ⎠
 |  | −1 | 
 |  |  |  |  | 
 | ⎤ ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎦
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
It is easier to solve the first equation for y which can be rewritten as
for appropriate C, which can be solved by separation of variables,
getting you the parametric equations
which parameterize a cycloid. This implicitly defines a function
y=y(x) as the only stationary function for L. The
problem is to prove that it minimizes T, which would be easy if the
integrand L was convex. However, it’s not the case here:
| assume(y>=0):;
 convex(sqrt((1+y'^2)/y),y(x)) | 
|  | | |  | ⎡ ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎣
 | − | ⎛ ⎜
 ⎜
 ⎝
 |  | y | ⎛ ⎝
 | x | ⎞ ⎠
 | ⎞ ⎟
 ⎟
 ⎠
 |  | +3≥ 0 | ⎤ ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎦
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
This is equivalent to |y′(t)|≤√3, which is certainly not
satisfied by the cycloid y near the point x=0.
Using the substitution y(x)=z(x)2/2, you get y′(x)=z′(x) z(x)
and
| L(x,y(x),y′(x))=P(x,z(x),z′(x))= | √ |  | . | 
The function P is convex:
| assume(z>=0):; convex(sqrt(2*(z^(-2)+z'^2)),z(x)) | 
Hence the function z(t)=√2 y(t),
stationary for P (verified directly), minimizes the
objective functional
| U(z)= | ∫ |  | P(x,z(x),z′(x)) dx. | 
From here and U(z)=T(y) it easily follows that y
minimizes T and is therefore the brachistochrone.
 
 
