This textbook explores differential geometrical aspects of the theory of completely integrable Hamiltonian systems. It provides a comprehensive introduction to the mathematical foundations and illustrates it with a thorough analysis of classical examples.
This book is organized into two parts. Part I contains a detailed, elementary exposition of the topics needed to start a serious geometrical analysis of complete integrability. This includes a background in symplectic and Poisson geometry, the study of Hamiltonian systems with symmetry, a primer on the theory of completely integrable systems, and a presentation of bi-Hamiltonian geometry.
Part II is devoted to the analysis of three classical examples of integrable systems. This includes the description of the (free) n-dimensional rigid-body, the rational Calogero-Moser system, and the open Toda system. In each case, ths system is described, its integrability is discussed, and at least one of its (known) bi-Hamiltonian descriptions is presented.
This work can benefit advanced undergraduate and beginning graduate students with a strong interest in geometrical methods of mathematical physics. Prerequisites include an introductory course in differential geometry and some familiarity with Hamiltonian and Lagrangian mechanics.
Part I: Preliminary material.- Symplectic Geometry.- Elements of Poisson
Geometry.- Hamiltonian G-actions and the Marsden-Weinstein-Meyer reduction.-
Lagrangian fibrations and integrable systems.- Elements of Bi-Hamiltonian
Geometry.- Bibliographical Notes.- Part II: A Sample of Classical Integrable
Systems.- Rigid bodies.- The Toda System.- Calogero-Moser Systems.- Appendix
A: Elements of Symplectic Linear Algebra.- Appendix B: Elements of
Differential Geometry.- Appendix C: Lie Groups, Lie Algebras, and Fiber
Bundles.- Appendix D: Cotangent Lifts.- Bibliography.- Index.
Igor Mencattini is an Associate Professor at the Institute of Mathematics and Computer Science (ICMC) at the University of São Paulo, São Carlos campus. His area of expertise is mathematical physics, with an emphasis on the geometrical and algebraic aspects of classical mechanical systems. He earned his PhD (2005) from Boston University, and conducted post-doc studies in Germany and Italy.
Alessandro Arsie is a Professor of Mathematics in the Department of Mathematics and Statistics at the University of Toledo, USA. His current research interests focus on Geometric and Algebraic techniques in Mathematical Physics and Differential Geometry. He earned his PhD (2001) from the Scuola Internazionale Superiore di Studi Avanzati (SISSA) in Trieste, Italy and conducted post-doctoral studies in Italy and in the USA.
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