The Classical Problems of the Calculus of Variations: Necessary Conditions and Sufficient Conditions; Convexity and Lower Semicontinuity

Abstract

We are concerned here with classical Lagrange problems of the calculus of variations (see Section 1.1). Precisely, we are concerned with minima and maxima of the functional

$$ I\left[ x \right] = \int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {{{f}_{0}}(t,x(t),x'(t))dt,x(t) = ({{x}^{1}}, \ldots ,{{x}^{n}})} , $$

(2.1.1)

with constraints and boundary conditions

$$\left( {t,x\left( t \right)} \right) \in A,{\text{ t}} \in \left[ {{t_1},{t_2}} \right],{\text{ }}\left( {{t_1},x\left( {{t_1}} \right),{t_2},x\left( {{t_2}} \right)} \right) \in B$$

(2.1.2)