Generalized Weierstrass condition in the classical calculus of variations (Q2002535)

The paper refers to a classical problem of calculus of variation, the minimization of a functional \(J(x(.)) = \int_{0}^{1}L(t,x,dx/dt)dt\), \(x:[0,1]->R^n\), over a set of admissible trajectories with \(x(0)=x_o\), \(x(1)=x_1\). The generalized Weierstrass condition is defined. One proves that strong local minimums belonging to a special class of regular functions on \((0,1)\) and satisfying Weierstrass and Legendre conditions as equalities, fulfill the generalized Weierstrass condition. A set of examples illustrate the theoretical results.