13.8.4  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:

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 f_{y′ 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)=∫_a^b 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 13.4.1), 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