Differential form methods in the theory of variational systems and Lagrangian field theories. (English) Zbl 0664.49018

The paper provides a clear introduction into the formal calculus of variations and applications to theoretical physics based on the theory of differential forms. A variational problem on a manifold Z is determined by a scalar-valued n-form \(\vartheta\) and an ideal \({\mathcal C}\) in the algebra of differential forms on Z; among the n-dimensional submanifolds \(\alpha\) :X\(\to Z\), satisfying \(\alpha^*{\mathcal C}=0\), we find that one for which the action \(\int \alpha^*\vartheta\) is stationary.
Besides the classical extremals (defined by vanishing variation) other ones (called Cartan’s are introduced in a purely algebraic manner by the use of the Poincaré-Cartan form \(\vartheta +\gamma\) (\(\gamma\in {\mathcal C})\) satisfying \(d(\vartheta +\gamma)\in {\mathcal C}\). The method is applied to the string theories in Riemannian manifolds, Maxwell equations, Yang- Mills theory and the definition of energy first-order field theories.
Reviewer’s remark: The P.-C. forms as defined above do not exist in general and the weaker condition \(d(\vartheta +\gamma)\in {\mathcal C}\) only along \(\alpha\) (X) is necessary.
Reviewer: J.Chrastina


49Q99 Manifolds and measure-geometric topics
49Q05 Minimal surfaces and optimization
81V99 Applications of quantum theory to specific physical systems
58A15 Exterior differential systems (Cartan theory)
78A25 Electromagnetic theory (general)