On the Lagrange variational problem. (English) Zbl 1526.49016
Summary: We investigate the stationarity of variational integrals evaluated on solutions of a system of differential equations. First, the fundamental concepts are relieved of accidental structures and of hypothetical assumptions. The differential constraints, stationarity and the Euler-Lagrange equations related to Poincaré-Cartan forms do not require any reference to coordinates or deep existence theorems for boundary value problems. Then, by using the jet formalism, the Lagrange multiplier rule is proved for all higher-order variational integrals and arbitrary compatible systems of differential equations. The self-contained exposition is based on the standard theory of differential forms and vector fields.
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 35Q31 | Euler equations |
| 58A17 | Pfaffian systems |
| 58E30 | Variational principles in infinite-dimensional spaces |
| 49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |
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| [16] | Veronika Chrastinová, Václav Tryhuk Institute of Mathematics and Descriptive Geometry Faculty of Civil Engineering Brno University of Technology 602 00 Brno, Czech Republic E-mail: chrastinova.v@fce.vutbr.cz tryhuk.v@email.cz |
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