Euler-Lagrange Differential Equation

Author

Eric W. Weisstein
December 12, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/Euler-LagrangeDifferentialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Euler-LagrangeDifferentialEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Calculus`VariationalMethods`

Usage

?EulerEquations

Fermat's Principle

EulerEquations[(Sqrt[1+((y'[x])^2)]),y[x],x]//Simplify

Schwarzschild Metric

reqn=FullSimplify[EulerEquations[((Sqrt[(((1-2((r[ϕ])^(-1))))^(-1))r'[ϕ]^2+((r[ϕ])^2)])/(Sqrt[1-2((r[ϕ])^(-1))])),r[ϕ],ϕ],r[ϕ]>0]

MapAt[Numerator,reqn,{1}]

DSolve[%,r[ϕ],ϕ]

Change of Variables

ueqn=FullSimplify[((((r[ϕ])^2)+2 ((r'[ϕ])^2)-r[ϕ] (3+r''[ϕ]))/((Sqrt[1-(2/(r[ϕ]))]) (Sqrt[r[ϕ] (r[ϕ]+(((r'[ϕ])^2)/(-2+r[ϕ])))])))==0/.r->Function[ϕ,1/u[ϕ]],u[ϕ]>0]

MapAt[Numerator,ueqn,{1}]

DSolve[%,u[ϕ],ϕ]

FullSimplify[%]