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

MSC:

 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)