Euler-Lagrange differential equation (elementary)

Let q(t) be a twice differentiable function from to and let L be a twice differentiable function from 3 to . Let q˙ denote ddtq.

Define the functional I as follows:


Suppose we regard the function L and the limits of integratiuon a and b as fixed and allow q to vary. Then we could ask for which functions q (if any) this integral attains an extremal (minimum or maximum) value. (Note: especially in Physics literature, the function L is known as the Lagrangian.)

Suppose that a differentiable function  q0:[a,b]  is an extremumMathworldPlanetmath of I. Then, for every differentiable function  f:[-1,+1]×[a,b]  such that  f(0,x)=q0(x), the function  g:[-1,+1],  defined as


will have an extremum at λ=0. If this function is differentiableMathworldPlanetmath, then  dg/dλ=0  when  λ=0.

By studying the condition dg/dλ=0 (see the addendum to this entry for details), one sees that, if a function q is to be an extremum of the integral I, then q must satisfy the following equation:

qL-ddt(q˙L)=0. (1)

This equation is known as the Euler–Lagrange differential equationMathworldPlanetmath or the Euler-Lagrange condition. A few comments on notation might be in . The notations qL and q˙L denote the partial derivativesMathworldPlanetmath of the function L with respect to its second and third arguments, respectively. The notation ddt means that one is to first make the argument a function of t by replacing the second argument with q(t) and the third argument with q˙(t) and secondly, differentiate the resulting function with respect to t. Using the chain ruleMathworldPlanetmath, the Euler-Lagrange equation can be written as follows:

qL-2tq˙L-q˙2qq˙L-q¨2q˙2L=0 (2)

This equation plays an important role in the calculus of variationsMathworldPlanetmath. In using this equation, it must be remembered that it is only a necessary condition and, hence, given a solution of this equation, one cannot to the conclusion that this solution is a local extremum of the functional F. More work is needed to determine whether the solution of the Euler-Lagrange equation is an extremum of the integral I or not.

In the special case tL=0, the Euler-Lagrange equation can be replaced by the Beltrami identityMathworldPlanetmath.

Title Euler-Lagrange differential equation (elementary)
Canonical name EulerLagrangeDifferentialEquationelementary
Date of creation 2013-03-22 12:21:49
Last modified on 2013-03-22 12:21:49
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 34
Author rspuzio (6075)
Entry type Definition
Classification msc 47A60
Classification msc 70H03
Classification msc 49K05
Synonym Euler-Lagrange condition
Related topic CalculusOfVariations
Related topic BeltramiIdentity
Related topic VersionOfTheFundamentalLemmaOfCalculusOfVariations
Defines Euler-Lagrange differential equation
Defines Lagrangian