The problem of integrable discretization : Hamiltonian approach / Yuri B. Suris

...The book consists of three parts. Part I “General Theory” has two chapters. The first chapter reviews the fundamental concepts and techniques in the Hamiltonian theory on Poisson manifolds, which include Poisson bracket, Hamiltonian flow, bi-Hamiltonian system, Euler-Lagrangian equations on Lie groups, and Lagrangian reductions and Euler-Poincaré equations. The second one provides a compact exposition of the r-matrix theory containing linear and quadratic r-matrix structures and related R-operators. It then presents a general recipe for constructing an integrable discretization, through making Bäcklund transformations close to the identity transformation within the r-matrix theory.

The rest of the book, consisting of Part II “Lattice Systems” (16 chapters) and Part III “Systems of Classical Mechanics” (9 chapters), contains a detailed account of various classes of integrable finite-dimensional dynamical systems. Particular examples are Toda lattice, Volterra lattice, relativistic Toda lattice, relativistic Volterra lattice, Garnier system, Neumann system, and Hénon-Heiles system. Each chapter is devoted to one class of systems of the same genealogy. It starts with an introduction, which displays the typical systems and their discretizations presented in the chapter and contains a short overview on the contents of its text. This allows quick browsing of the primary results of the whole chapter. The main text of each chapter includes elaborating on the Hamiltonian and r-matrix interpretation of the systems under consideration, and their discretizations by the general recipe. ... (zbMath)