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6.22.4  Jacobi equation: jacobi_equation

To determine whether a solution y0 to the Euler-Lagrange equations is an extrema, checking the convexity of the Lagrangian f doesn’t always work. Another approach is to look at the Jacobi equation, which is

d
dt
 
fy′ y(y0,y0′,th
+


fy y(y0,y0′,t)−
d
dt
 fy y(y0,y0′,t)


 h=0.     (3)

for unknown function h. If the Jacobi equation has a solution such that h(a)=0, h(c)=0 for some c∈(a,b] (the interval given in the variational problem) and h not identically zero on [a,c], then c is called a conjugate to a. If a conjugate exists, then y0 does not minimize the functional F. But the function y0 minimizes F if fy′ y(y0,y0′,x)>0 for all x∈[a,b] and there are no points conjugate to a in (a,b].

The jacobi_equation command computes the Jacobi equation.

If the Jacobi equation can be solved by dsolve (see Section 6.57.1), a sequence containing the equation (3) and its solution is returned. Otherwise, if (3) cannot be solved immediately, only the Jacobi equation is returned.


Example.
Input:

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

Output:

d2
dt2
h
t
h
t
=0,c0 sint

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