derivation of Euler-Lagrange differential equation (advanced)

Suppose that x0D. Choose r such that the closed ball of radius r about x0 is contained in D. Let q be any function whose support lies in this closed ball.

By the definition of F,


By the condition imposed on q, the derivative of the second integral is zero. Since the integrand of the first integral and its first derivativesMathworldPlanetmath are continuousMathworldPlanetmath and the closed ball is compact, the integrand and its first derivatives are uniformly continuous, so it is permissible to interchange differentiation and integration. Hence,

Title derivation of Euler-Lagrange differential equationMathworldPlanetmathPlanetmath (advanced)
Canonical name DerivationOfEulerLagrangeDifferentialEquationadvanced
Date of creation 2013-03-22 14:45:13
Last modified on 2013-03-22 14:45:13
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 8
Author rspuzio (6075)
Entry type Derivation
Classification msc 47A60