Jacobi equation : jacobi_equation

### 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 y0∈ C1[a,b] which is stationary for the functional F(y)=∫abf(y,y′,x) dx,
- 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′,t) h′ | ⎞ ⎠ | + | ⎛ ⎜ ⎜ ⎝ | 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)