Geometric mechanics
This is an excellent monography on geometric mechanics which gives a geometric treatment of classical Newtonian mechanics, that is, mechanics in Riemannian manifolds. In fact, this is a topic where the bibliography is limited (see, for instance, the references quoted in this book), and thus this work contributes to alleviate this lack. The most standard geometric description of classical dynamical systems uses symplectic manifolds to describe both the Lagrangian and Hamiltonian formalisms of these systems. Nevertheless, Newtonian systems can also be described in an alternative but equivalent geometric way, taking into account that their configuration spaces can be considered as Riemannian manifolds. Then, using the covariant derivative induced by the corresponding Levi-Civita connection, the dynamical equations can be stated in terms of geodesics of the metrics. The book begins with the review of basic concepts of differential geometry (chapters 1 and 2), in particular (pseudo-) Riemanian manifolds (chapter 3). Then, after a brief classical presentation and discussion of Newtonian mechanics (chapter 4), the foundations of Riemannian mechanics are established (chapter 5). Next, the particular situations of systems with nonholonomic constraints and of vakonomic mechanics are treated in this framework (chapters 6 and 7). Finally, introductory remarks to the special and general relativity theories are given in chapters 8 and 9, using pseudo-Riemannian manifolds. Furthermore, some appendices are devoted to give insight into different subjects, namely: Lagrangian and Hamiltonian formalisms (appendix A), Möbius transformations and Lorentz group (appendix B), Maxwell formulation of Einstein's equations (appendix C), and viscosity solutions and Aubry-Mather theory (appendix D) (the last three are written in collaboration with other authors, Jose Natario and Diego Gomes).
scientific article; zbMATH DE number 1856264