Formal calculus of variations on fibered manifolds. (English) Zbl 0696.49075

Folia Facultatis Scientiarum Naturalium Universitatis Brunensis. Mathematica, 2. Brně: Univerzita J. E. Purkyně v Brně. 109 p. Kčs 15.50 (1989).
Some fundamental concepts of the classical calculus of variations are studied, without special interest in the proper existence theorems for the extremals. Each chapter is self-contained. The first one gives some generalities on critical points. The standpoint of this chapter is systematically examined in the next one, where the general Lagrange problem is introduced in invariant terms and a general variant of the Hamiltonian theory is also discussed. The classical variational calculus of multiple integrals of arbitrary order appears as a particular case of the multiple integral dependent on variable functions and their partial derivatives where the mentioned functions are subordinate to a general system of partial differential equations. In the third chapter the Euler- Lagrange system, Hamiltonian system, the Cartan-Poincaré form and its boundary counterpart, transversality conditions and the Hamilton-Jacobi equation are considered. After giving more on the concept of standardness, the last chapter is devoted to the variational bicomplex and contains the fundamental notions of the jet theory including an elementary introduction to Spencer’s homologies of partial differential equations, applications to reduction principles of Cartan-Poincaré forms and Euler-Lagrange equations, infinitesimal symmetries of variational problems and some other topics. The leading motive of the present book is the analysis of the concept of the Hamiltonian approach to the classical calculus of variations.
Reviewer: N.Papaghiuc


49Q20 Variational problems in a geometric measure-theoretic setting
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control