Geometry of Integrable Systems

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.