derivation of Euler-Lagrange differential equation (elementary)

Let [e,c] be a finite subinterval of (a,b). Let the function h: be chosen so that a) h is twice differentiableMathworldPlanetmathPlanetmath, b) h(t)=0 when t[e,c], c) h(t)>0 when t(e,c), and d) ech(t)𝑑t=1.

Choose f(λ,t)=q(t)+λh(t). It is easy to see that this function satisfies the requirements for f laid out in the main entry. Then, we can write


Let us split the integration into three parts — the integral from a to e, the integral from e to c, and the integral from c to b. By the way h was chosen, the integrand reduces to L(t,q(t)(t),q˙(t)) when t(a,e) or t(c,b). Hence the pieces of the integral over the intervals (a,e) and (c,b) do not depend on λ and we have


Since [e,c] is closed and bounded, it is compact. By our assumption, the derivative of the integrand is continuousMathworldPlanetmath. Since continuous functions on compact sets are uniformly continuous, the derivative of the integrand is uniformly continuous. This imples that it is permissible to interchange differentiationMathworldPlanetmath and integration:


Using the chain ruleMathworldPlanetmath (several variables) and setting λ=0, we have


Integrating by parts and using the fact that h was chosen so as to vanish at the endpoints e qnd c, we find that


(The last equals sign defines EL as the quantity in the brackets in the first integral.)

I claim that requiring dg/dλ=0 for all finite intervals [e,c](a,b) implies that the EL(t) must equal zero for all t[a,b]. By our assumptions, EL is a continuous function. Hence, for every t0(a,b) and every ϵ>0, there must exist and [e,c](a,b) such that t0[e,c] and t1[e,c] implies that |EL(t0)-EL(t1)|<ϵ. Therefore,


Since this must be true for all ϵ>0, it follows that EL(t0)=0 for all t0(a,b). In other words, q satisfies the Euler-Lagrange equation.

Title derivation of Euler-Lagrange differential equationMathworldPlanetmathPlanetmath (elementary)
Canonical name DerivationOfEulerLagrangeDifferentialEquationelementary
Date of creation 2013-03-22 14:45:35
Last modified on 2013-03-22 14:45:35
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 16
Author rspuzio (6075)
Entry type Derivation
Classification msc 47A60