Jacobi equation

### 12.8.4 Jacobi equation

When determining whether a solution y0 to the Euler-Lagrange equations is an extremum, checking the convexity of the Lagrangian f does not always work. In such cases you may use 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. (4)

for an 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 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.

- jacobi_equation takes five or six arguments:
  - f(y,y′,x), an expression involving an independent variable, a dependent variable, and the dependent variable derivative.
  - depvar, the independent variable.
  - indvar, the dependent variable. This argument and the previous one can be combined to a single argument depvar(indvar), which case the call has five arguments.
  - expression y0 representing a function in C1[a,b] which is stationary for the functional F(y)=∫ab f(y,y′,x) dx.
  - h, a symbol for the unknown function in the Jacobi equation.
  - a, a real number which is the lower bound for x.
- jacobi_equation(f(y,y′,x),x,y,y0,a) returns the Jacobi equation and possibly the solution.

If the Jacobi equation can be solved by dsolve (see Section 12.4.1), a sequence containing the equation (4) and its solution is returned. Otherwise, if (4) cannot be solved immediately, only the Jacobi equation is returned.

#### Example

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

− |
d2
dt2
| h | ⎛ ⎝ | t | ⎞ ⎠ | −h | ⎛ ⎝ | t | ⎞ ⎠ | =0,c0 sint