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Mathematics | Topological Methods in Hydrodynamics

Topological Methods in Hydrodynamics

Series: Applied Mathematical Sciences, Vol. 125

Arnold, Vladimir I., Khesin, Boris A.

1998, XV, 376 p.

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... ad alcuno, dico, di quelli, che troppo laconicamente vorrebbero vedere, nei piu ` angusti spazii che possibil fusse, ristretti i ?loso?ci insegnamenti,s´ ?chesempresiusassequellarigidaeconcisamaniera, spogliata di qualsivoglia vaghezza ed ornamento, che e ´ propria dei puri geometri, li quali ne ´ pure una parola proferiscono che dalla assoluta necessita ´ non sia loro suggerita. Ma io, all’incontro, non ascrivo a difetto in un trattato, ancorche ´ indirizzato ad un solo scopo, interserire altre varie notizie, purche ´ non siano totalmente separate e senza veruna coerenza annesse al ? principale instituto. Galileo Galilei “Lettera al Principe Leopoldo di Toscana” (1623) Hydrodynamics is one of those fundamental areas in mathematics where progress at any moment may be regarded as a standard to measure the real success of ma- ematicalscience.Manyimportantachievementsinthis?eldarebasedonprofound theoriesratherthanonexperiments.Inturn,thosehydrodynamicaltheoriessti- lateddevelopmentsinthedomainsofpuremathematics,suchascomplexanalysis, topology,stabilitytheory,bifurcationtheory,andcompletelyintegrabledynamical systems. In spite of all this acknowledged success, hydrodynamics with its sp- tacular empirical laws remains a challenge for mathematicians. For instance, the phenomenon of turbulence has not yet acquired a rigorous mathematical theory. Furthermore, the existence problems for the smooth solutions of hydrodynamic equations of a three-dimensional ?uid are still open.

Content Level » Research

Keywords » Knot theory - Lie - Morphism - Topology - algebra - differential equation - dynamical systems - dynamische Systeme - equation - fluid dynamics - geometry - magnetohydrodynamics - mathematics

Related subjects » Classical Continuum Physics - Complexity - Computational Intelligence and Complexity - Mathematics

Table of contents 

Group and Hamiltonian Structures of Fluid Dynamics.- Topology of Steady Fluid Flows.- Topological Properties of Magnetic and Vorticity Fields.- Differential Geometry of Diffeomorphism Groups.- Kinematic Fast Dynamo Problems.- Dynamical Systems with Hydrodynamical Background.

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