conjugate_equation takes four arguments :
The function y0(x) is assumed to be stationary for the problem of minimizing some functional F(y)=∫abf(x,y,y′) dx. The return value is the expression
|
| − |
|
| , (4) |
at α=A and β=B, which is zero if and only if t is conjugate to a. To find any conjugate points, set the returned expression to zero and solve.
For example, we find a minimum for the functional
F(y)= | ∫ |
| ⎛ ⎝ | y′(x)2−x y(x)−y(x)2 | ⎞ ⎠ | dx |
on D={y∈ C1[0,π/2]:y(0)=y(π/2)=0}. The corresponding Euler-Lagrange equation is :
^
2-x*y(x)-y(x)^
2,y(x))
Output :
The general solution is :
Output :
The stationary function depends on two parameters c0 and c1 which are fixed by the boundary conditions :
Output :
Input :
Output :
The above expression obviously has no zeros in (0,π/2], hence there are no points conjugate to 0. Since fy′ y′=2>0, where f(y,y′,x) is the integrand in F(y) (the strong Legendre condition), y0 minimizes F on D. To obtain y0 explicitly, input :
Output :