##
**Symmetries, topology and resonances in Hamiltonian mechanics. Transl. from the Russian by S. V. Bolotin, Y. Fedorov, D. Treshchev.**
*(English)*
Zbl 0843.58068

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 31. Berlin: Springer-Verlag. xi, 378 p. (1996).

This book summarizes the results on integrability of Hamiltonian systems in the two last decades, with special emphasis on those obtained by the author and his collaborators. All the scientific production of the author is written in Russian and only a few of their papers, mainly the recent ones, are translated into English. So, this publication makes the fundamental work of the author accessible to the non Russian audience.

The book is addressed to researchers and graduate students either in physics or mathematics. It has a good supply of interesting examples developed with full detail. Also, several open questions and non developed fields of research (of possible interest) are indicated through the text.

The first part consists of the two first chapters. Chapter one recalls the definitions of the basic concepts of Hamiltonian systems and shows the main examples. Chapter two reviews known results on the exact integration of Hamiltonian systems from the modern point of view, including the use of symmetry groups, perturbation theory and normal forms.

The second part, which is the core of the book, is the study of the nonintegrability behaviour of Hamiltonian systems. Chapter three deals with topological and geometrical obstructions to complete integrability. This is a new branch of research; it was neglected before due to the preference for the local study of dynamical systems. As an example of a topological obstruction we can find the following: a closed analytic surface with genus greater than 1 cannot be the configuration space of an integrable analytic system. Examples of geometric obstructions are obtained by means of conditions on the geodesics of the surface.

Chapter four studies the integrability problem of perturbed Hamiltonian systems. The main result is that the destruction of a large number of resonant tori of the unperturbed problem for small values of the perturbation parameter prevents integrability of the Hamiltonian system.

Chapter five contains methods of proving nonintegrability by analysing asymptotic surfaces of Hamiltonian systems which are close to completely integrable systems. The double asymptotic solutions of a completely integrable systems are divided into two classes: homoclinic and heteroclinic solutions. It was PoincarĂ© who first noticed that in the homoclinic case, the asymptotic surfaces may split when perturbed. This splitting is an obstruction to the integrability of the perturbed Hamiltonian system. The results are applied to the Kirchhoff equations governing the rotation of a rigid body in an ideal fluid.

Chapter six is devoted to another method of proving nonintegrability: nonintegrability in the vicinity of an equilibrium solution.

Chapter seven studies the phenomenon of branching of solutions of convex Hamiltonian systems with complex time as an obstruction to the existence of new single-valued first integrals.

Finally, Chapter eight presents some specific methods to search for Hamiltonian systems admitting first integrals which are polynomial in momenta. The problem of looking for polynomial integrals whose degree is not fixed in advance is only solved for some classes of Hamiltonian systems such as, systems with one and a half degree of freedom or systems with exponential interaction.

The book is addressed to researchers and graduate students either in physics or mathematics. It has a good supply of interesting examples developed with full detail. Also, several open questions and non developed fields of research (of possible interest) are indicated through the text.

The first part consists of the two first chapters. Chapter one recalls the definitions of the basic concepts of Hamiltonian systems and shows the main examples. Chapter two reviews known results on the exact integration of Hamiltonian systems from the modern point of view, including the use of symmetry groups, perturbation theory and normal forms.

The second part, which is the core of the book, is the study of the nonintegrability behaviour of Hamiltonian systems. Chapter three deals with topological and geometrical obstructions to complete integrability. This is a new branch of research; it was neglected before due to the preference for the local study of dynamical systems. As an example of a topological obstruction we can find the following: a closed analytic surface with genus greater than 1 cannot be the configuration space of an integrable analytic system. Examples of geometric obstructions are obtained by means of conditions on the geodesics of the surface.

Chapter four studies the integrability problem of perturbed Hamiltonian systems. The main result is that the destruction of a large number of resonant tori of the unperturbed problem for small values of the perturbation parameter prevents integrability of the Hamiltonian system.

Chapter five contains methods of proving nonintegrability by analysing asymptotic surfaces of Hamiltonian systems which are close to completely integrable systems. The double asymptotic solutions of a completely integrable systems are divided into two classes: homoclinic and heteroclinic solutions. It was PoincarĂ© who first noticed that in the homoclinic case, the asymptotic surfaces may split when perturbed. This splitting is an obstruction to the integrability of the perturbed Hamiltonian system. The results are applied to the Kirchhoff equations governing the rotation of a rigid body in an ideal fluid.

Chapter six is devoted to another method of proving nonintegrability: nonintegrability in the vicinity of an equilibrium solution.

Chapter seven studies the phenomenon of branching of solutions of convex Hamiltonian systems with complex time as an obstruction to the existence of new single-valued first integrals.

Finally, Chapter eight presents some specific methods to search for Hamiltonian systems admitting first integrals which are polynomial in momenta. The problem of looking for polynomial integrals whose degree is not fixed in advance is only solved for some classes of Hamiltonian systems such as, systems with one and a half degree of freedom or systems with exponential interaction.

Reviewer: Juan Monterde (Burjasot)

### 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 |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |

37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |

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

70H07 | Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics |

70H09 | Perturbation theories for problems in Hamiltonian and Lagrangian mechanics |

70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |