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VariationalCalculus[EulerLagrange] - construct the Euler-Lagrange equations

Calling Sequence

EulerLagrange(f, t, x(t))

Parameters

f

-

expression in t, x(t), and x'(t)

t

-

independent variable

x(t)

-

unknown function (or list of functions)

Description

• 

The EulerLagrange(f, t, x(t)) command computes the Euler-Lagrange equations of a functional J=abft,xt,x'tⅆt subject to xa=A and xb=B.

  

In general, the Euler-Lagrange equations are not independent.

  

The Euler-Lagrange equations are returned as expressions.

• 

If they can be calculated, the trivial first integrals are also returned.

  

The first integrals are set equal to generated global indexed variables Ki that denote arbitrary constants.

• 

For higher-order functionals, for example, f(t, y(t), y'(t), y''(t)), use variables to represent derivatives. For example, set x1(t) = y(t) and x2(t)=y'(t), and then determine the Euler-Lagrange equations of the functional f + L*( x1'(t) - x2(t) )^2. To find the equations for the higher-order problem, substitute x2(t) = x1'(t) into the result.

Examples

withVariationalCalculus

ConjugateEquation,Convex,EulerLagrange,Jacobi,Weierstrass

(1)

Geodesics in the plane

f:=ⅆⅆtxt2+ⅆⅆtyt212

f:=ⅆⅆtxt2+ⅆⅆtyt2

(2)

EulerLagrangef,t,xt,yt

12ⅆⅆtxt2ⅆⅆtxtⅆ2ⅆt2xt+2ⅆⅆtytⅆ2ⅆt2ytⅆⅆtxt2+ⅆⅆtyt23/2ⅆ2ⅆt2xtⅆⅆtxt2+ⅆⅆtyt2,12ⅆⅆtyt2ⅆⅆtxtⅆ2ⅆt2xt+2ⅆⅆtytⅆ2ⅆt2ytⅆⅆtxt2+ⅆⅆtyt23/2ⅆ2ⅆt2ytⅆⅆtxt2+ⅆⅆtyt2,ⅆⅆtxtⅆⅆtxt2+ⅆⅆtyt2=K1,ⅆⅆtytⅆⅆtxt2+ⅆⅆtyt2=K2,ⅆⅆtxt2+ⅆⅆtyt2ⅆⅆtxt2ⅆⅆtxt2+ⅆⅆtyt2ⅆⅆtyt2ⅆⅆtxt2+ⅆⅆtyt2=K3

(3)

Brachistochrone

g:=1+ⅆⅆtyt212yt12

g:=1+ⅆⅆtyt2yt

(4)

EulerLagrangeg,t,yt

121+ⅆⅆtyt2yt3/2+ⅆⅆtyt2ⅆ2ⅆt2yt1+ⅆⅆtyt23/2yt+12ⅆⅆtyt21+ⅆⅆtyt2yt3/2ⅆ2ⅆt2yt1+ⅆⅆtyt2yt,1+ⅆⅆtyt2ytⅆⅆtyt21+ⅆⅆtyt2yt=K1

(5)

See Also

dsolve, solve, VariationalCalculus


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