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VariationalCalculus[ConjugateEquation] - compute a determining equation for conjugate points from a two-parameter family of extremals
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Calling Sequence
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ConjugateEquation(extremals, parms, parm_values, t, t0)
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Parameters
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extremals
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list of extremals, or algebraic expression for extremals
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parms
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list of parameter names
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parm_values
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list of parameter boundary values
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t
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independent variable
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t0
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one end point for the independent variable
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Description
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The ConjugateEquation(extremals, parms, parm_values, t, t0) command computes an algebraic equation. Its roots are the conjugate points.
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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)].
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The extremals option can also be expressed algebraically, for example, A*cosh((t-a)/A).
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If there are  extremals, there must be  parameters.
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The parm_values options specifies the list of values of the parameters parms at the extremal points matching the boundary conditions of the problem.
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The t0 option is an end point for the independent variable, for example, the left one.
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To find any conjugate points, set the returned equation to 0 (zero) and solve.
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Examples
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>
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![[ConjugateEquation, Convex, EulerLagrange, Jacobi, Weierstrass]](/support/helpjp/helpview.aspx?si=9078/file03353/math128.png)
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![ConjugateEquation(A*sin(t)+B*cos(t), [A, B], [1, 0], t, 0)](/support/helpjp/helpview.aspx?si=9078/file03353/math132.png)
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The conjugate points to 0 are the zeros of this.
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