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5.19.3  Jacobi equation : jacobi_equation

jacobi_equation takes five or six arguments :

The return value contains the Jacobi equation

d
dt
 
fy′ y(y0,y0′,th
+


fy y(y0,y0′,t)−
d
dt
 fy y(y0,y0′,t)


 h=0.     (3)

If the Jacobi equation has a solution such that h(a)=0, h(c)=0 for some c∈(a,b] and h not identically zero on [a,c], then y0 does not minimize the functional F. It is said that c is conjugate to a. The function y0 minimizes F if fy′ y(y0,y0′,t)>0 for all t∈[a,b] and there are no points conjugate to a in (a,b].

If the Jacobi equation can be solved by dsolve, a sequence containing the equation (3) and its solution is returned. Otherwise, if (3) cannot be solved immediately, only the Jacobi equation is returned.

For example, input :

jacobi_equation(-1/2*y’(t)^2+y(t)^2/2,t,y,sin(t),h,0)

Output :

(-diff(h(t),t,2)-h(t))=0, c_0*sin(t)

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