##
**The problem of integrable discretization: Hamiltonian approach.**
*(English)*
Zbl 1033.37030

Progress in Mathematics (Boston, Mass.) 219. Basel: Birkhäuser (ISBN 3-7643-6995-7/hbk). xxi, 1070 p. (2003).

This book is devoted to the problem of integrable discretization in the field of integrable systems. More concretely, it deals with the problem: how to discretize a given finite-dimensional integrable system, keeping its integrability property as in the theories of symplectic numerical algorithms and integrable couplings. The problem can be precisely described as follows:

Given an integrable Hamiltonian system \(\dot x =f(x)=\{H,x\}\) on an \(N\)-dimensional Poisson manifold \(\mathcal P\) with a Poisson bracket \(\{\cdot,\cdot\}\), which possesses \(N\) functionally independent integrals of motion \(I_k(x)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}\), find a one-parameter family of diffeomorphisms from \(\mathcal P\) to \( \mathcal P\): \[ \tilde x =\Phi(x;h),\quad h \text{ being a small positive parameter}, \] which satisfies \(\Phi(x,h)=x+hf(x)+\text{O}(h^2)\) and \(\{\tilde x,\tilde y\}_h= \{x,y\}_h\) under some Poisson bracket \(\{\cdot,\cdot\}_h\) with \(\{\cdot,\cdot\}_h=\{\cdot,\cdot\}+\text{O}(h)\), and possesses \(N\) functionally independent invariant functions \(I_k(x,h)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}_h\) and with \(I_k(x,h)=I_k(x)+\text{O}(h)\), \(1\leq k\leq N\).

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.

All chapters end with bibliographical remarks. They help the reader to trace the origin of the results established in the text, and provide the reader with a rich collection of related sources for further reading. The book also lists plenty of primary references, which the author explicitly refers to in the text.

Given an integrable Hamiltonian system \(\dot x =f(x)=\{H,x\}\) on an \(N\)-dimensional Poisson manifold \(\mathcal P\) with a Poisson bracket \(\{\cdot,\cdot\}\), which possesses \(N\) functionally independent integrals of motion \(I_k(x)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}\), find a one-parameter family of diffeomorphisms from \(\mathcal P\) to \( \mathcal P\): \[ \tilde x =\Phi(x;h),\quad h \text{ being a small positive parameter}, \] which satisfies \(\Phi(x,h)=x+hf(x)+\text{O}(h^2)\) and \(\{\tilde x,\tilde y\}_h= \{x,y\}_h\) under some Poisson bracket \(\{\cdot,\cdot\}_h\) with \(\{\cdot,\cdot\}_h=\{\cdot,\cdot\}+\text{O}(h)\), and possesses \(N\) functionally independent invariant functions \(I_k(x,h)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}_h\) and with \(I_k(x,h)=I_k(x)+\text{O}(h)\), \(1\leq k\leq N\).

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.

All chapters end with bibliographical remarks. They help the reader to trace the origin of the results established in the text, and provide the reader with a rich collection of related sources for further reading. The book also lists plenty of primary references, which the author explicitly refers to in the text.

Reviewer: Ma Wen-Xiu (Tampa)

### MSC:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37M15 | Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems |

70H06 | Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics |

39A12 | Discrete version of topics in analysis |