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 biHamiltonian geometry. Part II is devoted to the analysis of three classical examples of integrable systems. This includes the description of the (free) ndimensional rigidbody, the rational CalogeroMoser system, and the open Toda system. In each case, ths system is described, its integrability is discussed, and at least one of its (known) biHamiltonian 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.
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 biHamiltonian geometry. Part II is devoted to the analysis of three classical examples of integrable systems. This includes the description of the (free) ndimensional rigidbody, the rational CalogeroMoser system, and the open Toda system. In each case, ths system is described, its integrability is discussed, and at least one of its (known) biHamiltonian 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.
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