## Topological Methods in HydrodynamicsTopological hydrodynamics is a young branch of mathematics studying topological features of flows with complicated trajectories, as well as their applications to fluid motions. It is situated at the crossroad of hyrdodynamical stability theory, Riemannian and symplectic geometry, magnetohydrodynamics, theory of Lie algebras and Lie groups, knot theory, and dynamical systems. Applications of this approach include topological classification of steady fluid flows, descriptions of the Korteweg-de Vries equation as a geodesic flow, and results on Riemannian geometry of diffeomorphism groups, explaining, in particular, why longterm dynamical weather forecasts are not reliable. Topological Methods in Hydrodynamics is the first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics for a unified point of view. The necessary preliminary notions both in hydrodynamics and pure mathematics are described with plenty of examples and figures. The book is accessible to graduate students as well as to both pure and applied mathematicians working in the fields of 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