## Dynamical systems: Symplectic geometry and its applicationsThis book takes a snapshot of the mathematical foundations of classical and quantum mechanics from a contemporary mathematical viewpoint. It covers a number of important recent developments in dynamical systems and mathematical physics and places them in the framework of the more classical approaches; the presentation is enhanced by many illustrative examples concerning topics which have been of especial interest to workers in the field, and by sketches of the proofs of the major results. The comprehensive bibliographies are designed to permit the interested reader to retrace the major stages in the development of the field if he wishes. Not so much a detailed textbook for plodding students, this volume, like the others in the series, is intended to lead researchers in other fields and advanced students quickly to an understanding of the 'state of the art' in this area of mathematics. As such it will serve both as a basic reference work on important areas of mathematical physics as they stand today, and as a good starting point for further, more detailed study for people new to this field. |

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### Contents

3 Families of Quadratic Hamiltonians | 12 |

Lagrangian and Legendre Cobordisms | 113 |

2 Cobordisms | 121 |

Copyright | |

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### Common terms and phrases

action Anal arbitrary boundary canonical caustics characteristic classical coadjoint cobordism cohomology coincide commuting compact complex condition conﬁguration connected constant construction contact structure Corollary corresponding cotangent bundle critical points curve F deﬁned deﬁnition deformation denote differential dimension Dokl eigenvalues English translation equal equivalent Euclidean example ﬁbre ﬁnite gap ﬁnite-dimensional ﬁxed points ﬂow formula front Funct functions F Funkts geodesic geometric quantization germs given Grassmann manifold group G Hamiltonian system Hermitian hyperplane hypersurface invariant inverse scattering method isomorphic KdV equation Lagrangian ﬁbration Lagrangian submanifold Lemma Let us consider level surface Lie algebra Lie group linear Math matrix momentum mapping Nauk neighbourhood operator orbit parameter phase space Poisson bracket Poisson structure polarization poles polynomials prequantization Prilozh problem representation Riemann sect singularities solutions subgroup subspace symmetry symplectic group symplectic manifold symplectic space symplectic structure symplectomorphism Theorem theory torus transformation variables vector ﬁeld zero