Hamiltonian Systems and Their Integrability
This book serves as an introduction to mathematical aspects of integrable Hamiltonian systems. In the first chapter, the author introduces symplectic manifolds, Poisson brackets on a symplectic manifold, Hamiltonians and integrable Hamiltonian systems (together with basic examples), as well as the classical Darboux theorem and its proof. The second chapter contains a description of symplectic action and the local Arnold-Liouville theorem on action-angle variables, as well as a global version of the theorem. In the third chapter, a relationship of integrability and the differential Galois group is treated. The author proves a strictly stronger version of the Zeglin lemma and the Morales-Ramis theorem, giving a necessary condition for integrability in terms of the Galois group. In the fourth chapter, the author explains a relationship of the Arnold-Liouville theorem and algebraic curves via the Lax-pair approach. The last two chapters are appendices containing basics of differential Galois theory and algebraic geometry (algebraic curves and the Riemann-Roch theorem), which are used in the two previous chapters. The book is a nice introduction to integrability of Hamiltonian systems connecting it with two mentioned branches of mathematics, which were developed independently but which are deeply related to the studied topic. The book contains many nice illustrative examples (including the simple and spherical pendulum, the Euler-Poinsot rigid body, the Lagrange and the Kowalevski top, harmonic and anharmonic oscillators and the Hénon-Heiles system). It also contains many exercises, the solutions of which help the reader to understand the theory. The book offers a nice, short and concise introduction to this attractive topic for mathematicians as well as physicists. For the latter, it also represents a natural opportunity to learn more not only about integrable systems but also about the basics of differential Galois theory and algebraic curves.