On the Lagrange variational problem
Volume 130 / 2023
Annales Polonici Mathematici 130 (2023), 149-180
MSC: Primary 49-01; Secondary 49K15, 35Q31, 58A17, 58E30.
DOI: 10.4064/ap220330-30-1
Published online: 20 March 2023
Abstract
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.
