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VariationalCalculus

 Weierstrass
 compute the Weierstrass excess function

 Calling Sequence Weierstrass(f, t, x(t), p)

Parameters

 f - expression in t, x(t), and x'(t) t - independent variable x(t) - unknown function (or list of functions) p - name to use for the vector p[1..n]

Description

 • The Weierstrass(f, t, x, x(t), p) command computes the Weierstrass excess function

$E\left(t,x,x\text{'},p\right)=L\left(t,x,x\text{'}\right)-L\left(t,x,p\right)-\left(x\text{'}-p\right){L}_{p}\left(t,x,p\right)$

 • The output is unsimplified. There are many techniques that can be used to simplify the output: factor, collect, combine, or complete the square.
 • If $E\left(t,x,x',p\right)$ is positive when $x$ is the extremal, except when $x'=p$, the extremal provides a strong local minimum.
 The Cauchy-Schwarz inequality can be used to determine the sign of the expression.

Examples

 > $\mathrm{with}\left(\mathrm{VariationalCalculus}\right)$
 $\left[{\mathrm{ConjugateEquation}}{,}{\mathrm{Convex}}{,}{\mathrm{EulerLagrange}}{,}{\mathrm{Jacobi}}{,}{\mathrm{Weierstrass}}\right]$ (1)

A Lagrange Multiplier Problem (Queen Dido's problem): Find the maximum area enclosed by a curve of length $\mathrm{\lambda }$.

 > $f≔y+\mathrm{λ}{\left(1+{\left(\frac{ⅆ}{ⅆt}y\left(t\right)\right)}^{2}\right)}^{\frac{1}{2}}$
 ${f}{:=}{y}{+}{\mathrm{λ}}{}\sqrt{{1}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{2}}}$ (2)
 > $\mathrm{Weierstrass}\left(f,t,y\left(t\right),p\right)$
 ${\mathrm{λ}}{}\sqrt{{1}{+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right)\right)}^{{2}}}{-}{\mathrm{λ}}{}\sqrt{{{p}}_{{1}}^{{2}}{+}{1}}{-}\frac{\left(\frac{{ⅆ}}{{ⅆ}{t}}{}{y}{}\left({t}\right){-}{{p}}_{{1}}\right){}{\mathrm{λ}}{}{{p}}_{{1}}}{\sqrt{{{p}}_{{1}}^{{2}}{+}{1}}}$ (3)