Finding conjugate points

13.8.5 Finding conjugate points

The conjugate_equation command computes conjugate points.

conjugate_equation takes four arguments:
- y₀, an expression which depends on an independent variable and two parameters. The expression y₀ is assumed to represent a stationary function for the problem of minimizing some functional F(y)=∫_a^bf(x,y,y′) dx.
- [α,β], a list of parameters which y₀ depends on.
- [A,B], a list of the values of parameters α and β, respectively.
- x, the independent variable.
- a, a real number equal to the lower or to the upper bound for x.
conjugate_equation(y₀,[α,β],[A,B],x,a) returns the expression

∂ y₀(t)

∂α

∂ y₀(a)

∂β

−
∂ y₀(a)

∂α

∂ y₀(t)

∂β

, (5)

at α=A and β=B, which is zero if and only if t is conjugate to a.

To find any conjugate points, solve for zeros of the returned expression.

Example

Find a minimum for the functional

F(y)=

∫

⎛
⎝

y′(x)²−x y(x)−y(x)²

⎞
⎠

on D={y∈ C¹[0,π/2]:y(0)=y(π/2)=0}. The corresponding Euler-Lagrange equation is:

eq:=euler_lagrange(y'(x)^2-x*y(x)-y(x)^2,y(x))

d²

dx²

⎛
⎝

⎞
⎠

=−

−y

⎛
⎝

⎞
⎠

The general solution is:

y0:=dsolve(eq,x,y)

c₀ cosx+c₁ sinx−

The stationary function depends on two parameters c₀ and c₁ which are fixed by the boundary conditions:

c:=solve([subs(y0,x,0)=0,subs(y0,x,pi/2)=0],[c_0,c_1])

⎡
⎢
⎢
⎣

⎤
⎥
⎥
⎦

conjugate_equation(y0,[c_0,c_1],c[0],x,0)

sinx

The above expression obviously has no zeros in (0,π/2], hence there are no points conjugate to 0. Since f_{y′ 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:

subs(y0,[c_0,c_1],c[0])

π sinx−