×

Necessary conditions in a problem of calculus of variations. (English) Zbl 0681.49019

Summary: The problem of the calculus of variations with Bolza functionals is considered. Constraints are of both types: equalities and inequalities. The Lagrange multiplier 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 \[ (d/dt)(L_{\dot x}\dot x-L)=-L_ t. \] It should be emphasized that both differential equations are obtained under the only condition that the integrand L is continuously differentiable.

MSC:

49K15 Optimality conditions for problems involving ordinary differential equations
PDFBibTeX XMLCite
Full Text: EuDML