principal bundle of periods of symplectic form
Symmetries and moment
This book has its origin in a summer course given by the author between 1996 and 1997. It gives a clear and motivated introduction to symplectic mechanics, based on the notion of momentum mapping. The book is strongly inspired by the classical book by \textit{J.-M. Souriau} [Structure of dynamical systems. A symplectic view of physics. Transl. from the French by C. H. Cushman-de Vries. Progress in Mathematics (Boston, Mass.) 149 Boston: Birkhäuser. XXXIV (1997; Zbl 0884.70001)].NEWLINENEWLINENEWLINEThe book is divided into five chapters and six appendices. The first chapter describes the theory of perturbations introduced by Lagrange, as a motivation to study symplectic geometry. Indeed, perturbation theory has the germ of the notions of Lagrange (and Poisson) brackets and symplectic form. In the second chapter, the author gives a brief introduction to mechanics using the d'Alembert principle. A variational derivation is also given, and the Euler-Lagrange equations are obtained. The author uses a homogeneous formalism which consists in extending the Lagrangian one by adding a new variable. The Legendre transformation is studied in this context. The formalism is applied in chapter 3 to several examples: the computation of symplectic manifold of geodesics of the two-sphere \(S^2\) and on the Poincaré disc. The equations of motion of a rigid solid are also obtained.NEWLINENEWLINENEWLINEChapters 4 and 5 discuss the notions of symmetry and momentum map. In chapter 4 the author considers Lagrangians which are invariant by a Lie group, and proves the Noether theorem. In chapter 5 he studies in great detail the existence and properties of the momentum map. Several cohomologies are studied, and the principal bundle of periods of a symplectic form is constructed. As applications, the Bott-Thurston and Gelfand-Fuchs cocycles are recovered.NEWLINENEWLINENEWLINEThe book ends with six appendices. The first one discusses the fascinating history of Lagrange and Poisson brackets through original papers of both mathematicians. Appendix B explains why the notation \(H\) for Hamiltonian function could be given in honor of Huyghens and not of Hamilton; indeed, it was Huyghens who discovered the theorem on the conservation of energy. Appendix C gives a fast and clear introduction to symplectic geometry, including a proof of Darboux theorem. A brief introduction to diffeological spaces is sketched in appendix D. A homotopy operator relating de Rham complex of a manifold \(X\) and differential complex on the differentiable space of arcs of \(X\) is defined in appendix E. In the last appendix the author proves a version of Stokes theorem in the context of differentiable spaces.
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