Mathematical Foundations of Classical Variational Theory

Abstract

Near the end of the eighteenth century, Lagrange observed that a function u*(x) ∈ C ¹₀ [0,1] which minimizes the functional K: C ¹₀ [0,1] → ℝ given by

$$ {\text{K}}\left( {\text{u}} \right) = \int_0^1 {{\text{F}}\left( {{\text{x,u}}\left( {\text{x}} \right),{\text{u'}}\left( {\text{x}} \right)} \right){\text{dx}}} $$

where u′ = du/dx, also makes the bivariate functional δK(u,η) vanish, where

$$ \delta {\text{K}}\left( {{\text{u,}}\,\eta } \right) = \mathop {\lim }\limits_{\alpha \to 0} \frac{{\partial {\text{K}}\left( {{\text{u}}\,\,{\text{ + }}\,\,\alpha \eta } \right)}}{{\partial \alpha }} $$

$$ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \int_0^1 {\left( {\frac{{\left( {\partial {\text{F}}\left( {{\text{x}},\,\,{\text{u,}}\,\,{\text{u'}}} \right)} \right.}}{{\partial {\text{u}}}}\eta \,\, + \,\,\frac{{\partial {\text{F}}\left( {{\text{x,}}\,\,{\text{u,}}\,\,{\text{u'}}} \right)}}{{\partial {\text{u'}}}}\eta '} \right)} {\text{dx}} $$

and η is an arbitrary element in C ¹₀ [0,l]. Here C ¹₀ [0,l] is the linear space of functions continuously differentiable on the interval [O,1] and which vanish at 0 and 1, F: ℝ³ → ℝ has continuous partial derivatives of order ≥ 2 with respect to each argument (ℝ is the real line and ℝ³ = ℝ x ℝ x ℝ), and α ∈ ℝ.