## Dynamical Systems IV: Symplectic Geometry and its ApplicationsFrom the reviews of the first edition: "... In general the articles ... are well written in a style that enables one to grasp the ideas. The actual style is a readable mix of the important results, outlines of proofs and complete proofs when it does not take too long together with readable explanations of what is going on. Also very useful are the large lists of references which are important not only for their mathematical content but also because the references given also contain articles in the Soviet literature which may not be familiar or possibly accessible to readers." New Zealand Math. Soc. Newsletter 1991"... Here ... a wealth of material is displayed for us, too much to even indicate in a review. ... Your reviewer was very impressed by the contents of both volumes (EMS 2 and 4), recommending them without any restriction. As far as he could judge, most presentations seem fairly complete..." Mededelingen van het Wiskundig genootshap 1992 |

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### Contents

Fig 1 Transversality | 13 |

1 | 15 |

OK | 17 |

G+ | 22 |

ر ا ا ا | 48 |

Fig 15 The ray space | 49 |

F | 50 |

81 | 53 |

a | 92 |

Fig 41 Transversal fronts caustics | 100 |

Fig 43 A source with a boundary | 102 |

Z Z | 105 |

x | 107 |

A | 112 |

A3 | 116 |

Geometric Quantization | 139 |

Fig 20 Elliptical coordinates on the plane | 55 |

Fig 23 Hamiltonian flows on surfaces | 69 |

a0 | 78 |

Fig 28 The geometric meaning of Lagrangian self | 82 |

y | 86 |

12mm + P m | 170 |

Integrable Systems I | 177 |

61+Go | 219 |

V | 284 |

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### Common terms and phrases

action analogue arbitrary base boundary bundle called caustics characteristic classical closed cobordism coefficients coincide commuting compact complex condition connected constant construction coordinates corresponding cotangent curve defined definition deformation depend described differential dimension element equal equations equivalent example exists fibration fibre field finite fixed flow formula front function geometric germs given gives Hamiltonian system hypersurface identity integral invariant isomorphic Lagrangian Legendre Let us consider Lie algebra linear manifold mapping Math matrix means method multiplication neighbourhood normal obtained operator orbit parameter periodic phase space Poisson bracket polarization poles polynomials potential problem projection proof properties quadratic quantization reduced representation respect Riemann satisfy sect singularities smooth solutions space structure submanifold surface symplectic symplectic manifold symplectic structure takes tangent Theorem theory transformation variables vector zero