An aspect of Lagrangian systems. (Q2771071)

This is an introductory course on Lagrangian systems consisting of five sections. The first three sections include classic material, and are addressed to readers facing Lagrangian systems for the first time. The contents included in sections 4 and 5 are more interesting for readers familiarized with this topic, since they include material not so standard in introductory works describing Lagrangian systems (or variational calculus). In the first section, the reader can find an introduction to Lagrangian systems from the very beginning. Lagrangian functions, actions and related Euler-Lagrange equations are described. Section 2 follows the same process for Hamiltonian systems. The Legendre map is introduced, and the equivalence between Euler-Lagrange equations and Hamiltonian equations is stated. The section finishes with a description of Maupertuis principle. Section 3, entitled ``Global minimizings'', is devoted to both the Weierstrass theorem (for local existence of a minimizing curve) and Tonelli theorem (on the global existence of minimizing curves). The Tonelli theorem is proven. In section 4, ``Minimizing measures'', given a compact manifold and a Lagrangian function, convex and positive-definite on fibres and superlinear, the author defines a global minimizing curve as a curve minimizing the action functional within the class of absolutely continuous curves in the same homology class. For the analysis of these solutions, the author gives a series of results concerning minimizing measures. Finally, in section 5, critical values of a Lagrangian are described and related to minimizing measures.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00039].