## Topological Methods in HydrodynamicsThe first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics from a unified point of view. It describes the necessary preliminary notions both in hydrodynamics and pure mathematics with numerous examples and figures. The book is accessible to graduates as well as pure and applied mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential geometry. |

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### Contents

lowledgments | 1 |

The group setting of ideal magnetohydrodynamics | 49 |

12 The NavierStokes equation from the group viewpoint | 63 |

4 Asymptotic linking number | 139 |

B Asymptotic crossing number of knots and links | 155 |

7 Generalized helicities and linking numbers | 166 |

8 Asymptotic holonomy and applications | 184 |

Differential Geometry of Diffeomorphism Groups | 195 |

Kinematic Fast Dynamo Problems 259 | 258 |

constructions | 267 |

5 Dynamo exponents in terms of topological entropy | 299 |

chain equations | 331 |

### Common terms and phrases

algebra g arbitrary asymptotic boundary closed coadjoint action coadjoint orbits coadjoint representation commutator complex components coordinates corresponding coset curl curvature curve defined derivative Diff diffeomorphism group differential forms dimension divergence-free divergence-free vector fields domain dual space dynamo eigenvalue Euclidean Euler equation exists exponential finite foliation formula geodesic group G Hamiltonian function helicity hence hydrodynamics ideal fluid identity inertia operator infinite-dimensional initial inition integral intersection invariant knot lark left-invariant Lie algebra Lie group linear linking number magnetic field modulo motion mple nition nsional obtained ollary orem pair particles path plane Poisson bracket problem quadratic form Riemannian manifold Riemannian metric right-invariant rigid body Section smooth solid torus solutions space g stationary stream function submanifold surface tangent space theory three-dimensional topological torus trajectories two-dimensional vanishes variation Vect(M velocity field volume form volume-preserving diffeomorphisms vorticity vorticity field zero