The conjugate_equation computes conjugate points.
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| − |
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| , (4) |
To find any conjugate points, set the returned expression to zero and solve.
Example.
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:
Input:
Output:
| y | ⎛ ⎝ | x | ⎞ ⎠ | =− |
| −y | ⎛ ⎝ | x | ⎞ ⎠ |
The general solution is:
Input:
Output:
c0 cosx+c1 sinx − |
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The stationary function depends on two parameters c0 and c1
which are fixed by the boundary conditions:
Input:
Output:
⎡ ⎢ ⎢ ⎣ | ⎡ ⎢ ⎢ ⎣ | 0, |
| π | ⎤ ⎥ ⎥ ⎦ | ⎤ ⎥ ⎥ ⎦ |
Input:
Output:
sinx |
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:
| π sinx− |
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