VariationalCalculus
EulerLagrange
construct the Euler-Lagrange equations
Calling Sequence
Parameters
Description
Examples
EulerLagrange(f, t, x(t))
f
-
expression in t, x(t), and x'(t)
t
independent variable
x(t)
unknown function (or list of functions)
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.
withVariationalCalculus
ConjugateEquation,Convex,EulerLagrange,Jacobi,Weierstrass
Geodesics in the plane
f≔diffxt,t2+diffyt,t212
f≔ⅆⅆtxt2+ⅆⅆtyt2
EulerLagrangef,t,xt,yt
ⅆⅆtxt2ⅆⅆtxtⅆ2ⅆt2xt+2ⅆⅆtytⅆ2ⅆt2yt2ⅆⅆtxt2+ⅆⅆtyt232−ⅆ2ⅆt2xtⅆⅆtxt2+ⅆⅆtyt2,ⅆⅆtyt2ⅆⅆtxtⅆ2ⅆt2xt+2ⅆⅆtytⅆ2ⅆt2yt2ⅆⅆtxt2+ⅆⅆtyt232−ⅆ2ⅆt2ytⅆⅆtxt2+ⅆⅆtyt2,ⅆⅆtxtⅆⅆtxt2+ⅆⅆtyt2=K1,ⅆⅆtytⅆⅆtxt2+ⅆⅆtyt2=K2,ⅆⅆtxt2+ⅆⅆtyt2−ⅆⅆtxt2ⅆⅆtxt2+ⅆⅆtyt2−ⅆⅆtyt2ⅆⅆtxt2+ⅆⅆtyt2=K3
Brachistochrone
g≔1+diffyt,t212yt12
g≔1+ⅆⅆtyt2yt
EulerLagrangeg,t,yt
−1+ⅆⅆtyt22yt32+ⅆⅆtyt2ⅆ2ⅆt2yt1+ⅆⅆtyt232yt+ⅆⅆtyt221+ⅆⅆtyt2yt32−ⅆ2ⅆt2yt1+ⅆⅆtyt2yt,1+ⅆⅆtyt2yt−ⅆⅆtyt21+ⅆⅆtyt2yt=K1
See Also
dsolve
solve
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