compute and solve Jacobi's equation for conjugate points - Maple Help

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VariationalCalculus[Jacobi] - compute and solve Jacobi's equation for conjugate points

 Calling Sequence Jacobi(f, t, x(t), X(t), h, a)

Parameters

 f - integrand to be tested t - independent variable x(t) - dependent function or list of functions X(t) - expression for the extremal (found by solving the Euler-Lagrange equations) h - name for the unknown function in Jacobi's equation a - initial point (left end of the interval)

Description

 • The Jacobi(f, t, x(t), X(t), h, a) command finds Jacobi's equation and tries to find solutions of Jacobi's equation, that is, conjugate points.
 • The routine returns an expression sequence consisting of Jacobi's equation and any solutions found by dsolve.
 If dsolve encounters a problem, an error message is returned.
 If dsolve fails to find a solution, only Jacobi's equation is returned.
 • If the solution of Jacobi's equation has a zero on the region of interest, the extremal is not optimal.

Examples

 > $\mathrm{with}\left(\mathrm{VariationalCalculus}\right)$
 $\left[{\mathrm{ConjugateEquation}}{,}{\mathrm{Convex}}{,}{\mathrm{EulerLagrange}}{,}{\mathrm{Jacobi}}{,}{\mathrm{Weierstrass}}\right]$ (1)
 > $f:=-\frac{{\left(\frac{ⅆ}{ⅆt}y\left(t\right)\right)}^{2}}{2}+\frac{{y\left(t\right)}^{2}}{2}$
 ${f}{:=}{-}\frac{{1}}{{2}}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{2}}{+}\frac{{1}}{{2}}{}{{y}{}\left({t}\right)}^{{2}}$ (2)
 > $\mathrm{Jacobi}\left(f,t,y\left(t\right),\mathrm{sin}\left(t\right),h,0\right)$
 $\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}{}{h}{}\left({t}\right){+}{h}{}\left({t}\right){,}{h}{}\left({t}\right){=}{\mathrm{_C1}}{}{\mathrm{sin}}{}\left({t}\right)$ (3)