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:
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| ⎛ ⎝ | fy′ y′(y0,y0′,t) h′ | ⎞ ⎠ | + | ⎛ ⎜ ⎜ ⎝ | fy y(y0,y0′,t)− |
| 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.
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.
jacobi_equation(-1/2*y'(t)^2+y(t)^2/2,t,y,sin(t),h,0) |
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