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VariationalCalculus

  

Weierstrass

  

compute the Weierstrass excess function

 

Calling Sequence

Parameters

Description

Examples

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

Et,x,x',p=Lt,x,x'Lt,x,px'pLpt,x,p

• 

The output is unsimplified. There are many techniques that can be used to simplify the output: factor, collect, combine, or complete the square.

• 

If Et,x,x',p 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

withVariationalCalculus

ConjugateEquation,Convex,EulerLagrange,Jacobi,Weierstrass

(1)

A Lagrange Multiplier Problem (Queen Dido's problem): Find the maximum area enclosed by a curve of length λ.

fy+λ1+ⅆⅆtyt212

f:=y+λ1+ⅆⅆtyt2

(2)

Weierstrassf,t,yt,p

λ1+ⅆⅆtyt2λp12+1ⅆⅆtytp1λp1p12+1

(3)

See Also

collect

combine

eval

factor

Student[Precalculus][CompleteSquare]

VariationalCalculus

VariationalCalculus[EulerLagrange]

 


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