### 5.19.3  Jacobi equation : jacobi_equation

jacobi_equation takes five or six arguments :

• expression f(y,y′,x),
• independent variable x,
• dependent variable y (this argument and the previous one can be combined to a single argument y(x), in which case the call has five arguments),
• expression y0C1[a,b] which is stationary for the functional F(y)=∫abf(y,y′,xdx,
• symbol h for the unknown function in Jacobi equation,
• point a∈ℝ, which is the lower bound for x.

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)