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
− |
| ⎛ ⎝ | fy′ y′(y0,y0′,t) h′ | ⎞ ⎠ | + | ⎛ ⎜ ⎜ ⎝ | fy y(y0,y0′,t)− |
| 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.
If the Jacobi equation can be solved by dsolve (see Section 6.57.1), 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:
Output:
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| h | ⎛ ⎝ | t | ⎞ ⎠ | −h | ⎛ ⎝ | t | ⎞ ⎠ | =0,c0 sint |