derivation of Euler-Lagrange differential equation (advanced)

Suppose that $x_0\in D$. Choose $r$ such that the closed ball of radius $r$ about $x_0$ is contained in $D$. Let $q$ be any function whose support lies in this closed ball.

By the definition of $F$,

$$\frac{\partial }{\partial \lambda }F(q_0+\lambda q)=\frac{\partial }{\partial \lambda }{\int }_{D}L(x,q_0+\lambda q,d{q}_0+\lambda dq)d^{m}x$$

$$=\frac{\partial }{\partial \lambda }\left({\int }_{|x-x_0|\le r}L(x,q_0+\lambda q,d{q}_0+\lambda dq)d^{m}x+{\int }_{\genfrac{}{}{0pt}{}{x\in D}{|x-x_0|>r}}L(x,q_0+\lambda q,d{q}_0+\lambda dq)d^{m}x\right)$$

By the condition imposed on $q$, the derivative of the second integral is zero. Since the integrand of the first integral and its first derivatives are continuous 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,

$$\frac{\partial }{\partial \lambda }F(q_0+\lambda q)={\int }_{|x-x_0|\le r}\frac{\partial L(x,q_0+\lambda q,d{q}_0+\lambda dq)}{\partial \lambda }{d}^{m}x$$

Title	derivation of Euler-Lagrange differential equation (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