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.