PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 45(59), pp. 143151 (1989) 

NECESSARY CONDITIONS IN A PROBLEM OF CALCULUS OF VARIATIONSVladimir Jankovi\'cMatematicki fakultet, Beograd, YugoslaviaAbstract: Problem of the calculus of variations with Bolza functionals is considered. Constraints are of both types: equalities and inequalities. The Lagrange multipler rule type theorem, which gives necessary conditions for weak optimality, is proved. When applied to the simplest problem of the calculus of variations , this theorem gives that every smooth minimizing function must satisfy the well known Euler equation and also the differential equation $$ (d/dt)\,(L_{\dot x}\dot xL)=L_t. $$ It should be emphasized that both differential equations are obtained under the only condition that integrand $L$ is continuously differentiable. Keywords: Bolza functional, weak optimality, Lagrange multipliers Classification (MSC2000): 49B10 Full text of the article:
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