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VariationalCalculus

 ConjugateEquation
 compute a determining equation for conjugate points from a two-parameter family of extremals

 Calling Sequence ConjugateEquation(extremals, parms, parm_values, t, t0)

Parameters

 extremals - list of extremals, or algebraic expression for extremals parms - list of parameter names parm_values - list of parameter boundary values t - independent variable t0 - one end point for the independent variable

Description

 • The ConjugateEquation(extremals, parms, parm_values, t, t0) command computes an algebraic equation. Its roots are the conjugate points.
 • The extremals option can be specified as a list, for example, [a*sin(t) + b*cos(t), c*sinh(t/c)+d*cosh(t/d)].
 • The extremals option can also be expressed algebraically, for example, A*cosh((t-a)/A).
 • If there are $n$ extremals, there must be $2n$ parameters.
 • The parm_values options specifies the list of values of the parameters parms at the extremal points matching the boundary conditions of the problem.
 • The t0 option is an end point for the independent variable, for example, the left one.
 • To find any conjugate points, set the returned equation to 0 (zero) and solve.

Examples

 > $\mathrm{with}\left(\mathrm{VariationalCalculus}\right)$
 $\left[{\mathrm{ConjugateEquation}}{,}{\mathrm{Convex}}{,}{\mathrm{EulerLagrange}}{,}{\mathrm{Jacobi}}{,}{\mathrm{Weierstrass}}\right]$ (1)
 > $\mathrm{ConjugateEquation}\left(A\mathrm{sin}\left(t\right)+B\mathrm{cos}\left(t\right),\left[A,B\right],\left[1,0\right],t,0\right)$
 ${\mathrm{sin}}{}\left({t}\right)$ (2)

The conjugate points to 0 are the zeros of this.