## Integrability and Nonintegrability in Geometry and MechanicsApproach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . • 1111 Oulik'. n. . Chi" •. • ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. |

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### Contents

II | 1 |

III | 6 |

IV | 10 |

V | 12 |

VI | 17 |

VII | 20 |

VIII | 30 |

IX | 32 |

XXXIX | 171 |

XL | 173 |

XLII | 176 |

XLIII | 178 |

XLIV | 181 |

XLV | 182 |

XLVI | 184 |

XLVII | 186 |

X | 34 |

XI | 39 |

XII | 44 |

XIII | 46 |

XIV | 55 |

XV | 68 |

XVI | 70 |

XVII | 73 |

XVIII | 78 |

XIX | 91 |

XXI | 92 |

XXII | 97 |

XXIII | 98 |

XXIV | 99 |

XXV | 103 |

XXVI | 109 |

XXVII | 111 |

XXVIII | 116 |

XXIX | 123 |

XXX | 126 |

XXXI | 129 |

XXXII | 143 |

XXXIII | 146 |

XXXIV | 149 |

XXXV | 157 |

XXXVI | 160 |

XXXVII | 165 |

XXXVIII | 167 |

### Common terms and phrases

algebra G analytic arbitrary assertion basis Bott integral boundary Brailov Cartan subalgebra coadjoint representation coincide commutative algebra compact completely integrable Consider const constant-energy surface constructed coordinates corresponding covector critical points cylinders defined definition denote diffeomorphic differential dimension element equations of motion Euler equations exists fibre finite follows Fomenko formula full tori function f geodesic flow glued grad Hamiltonian field Hamiltonian system homeomorphic ind G integrable systems integral f integral trajectories invariant K3 surface LEMMA Let G Lie group linear commutative algebra Liouville Liouville tori manifold Q matrix maximal linear commutative momentum mapping noncommutative nondegenerate nonintegrability nonorientable nonsingular obtained operator orbits plane Poisson bracket polynomials proved quadratic representation Riemannian metric rigid body saddle circle semisimple Lie algebra separatrix separatrix diagram sgrad H singular smooth function space sphere submanifold subspace surface Q surgery symplectic manifold symplectic structure tangent three-dimensional manifolds topological transformation two-dimensional vector field zero