# Jacobi equation: jacobi_equation

### 5.22.4 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′,t) h′

⎞
⎠

+

⎛
⎜
⎜
⎝

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.

- jacobi_equation takes five or six arguments:
  - f(y,y′,x), an expression involving an independent variable, a dependent variable, and the dependent variable prime.
  - 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 5.57.1), 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