 Jacobi - Maple Help

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)
 > $f≔-\frac{{\mathrm{diff}\left(y\left(t\right),t\right)}^{2}}{2}+\frac{{y\left(t\right)}^{2}}{2}$
 ${f}{≔}{-}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}}{{2}}{+}\frac{{{y}{}\left({t}\right)}^{{2}}}{{2}}$ (2)
 > $\mathrm{Jacobi}\left(f,t,y\left(t\right),\mathrm{sin}\left(t\right),h,0\right)$
 $\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{t}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{h}{}\left({t}\right){+}{h}{}\left({t}\right){,}{h}{}\left({t}\right){=}{\mathrm{_C1}}{}{\mathrm{sin}}{}\left({t}\right)$ (3)