Journal ArticleDOI
Reduction of symplectic manifolds with symmetry
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In this paper, a unified framework for the construction of symplectic manifolds from systems with symmetries is presented, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.About:
This article is published in Reports on Mathematical Physics.The article was published on 1974-02-01. It has received 1467 citations till now. The article focuses on the topics: Moment map & Symplectomorphism.read more
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The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories
TL;DR: In this article, the Lagrangian analogue of Lie-Poisson Hamiltonian systems is defined on semidirect product Lie algebras, and an abstract Kelvin-Noether theorem for these equations is derived.
MonographDOI
A Tour of Subriemannian Geometries, Their Geodesics and Applications
TL;DR: Geodesics in subriemannian manifolds: Dido meets Heisenberg Chow's theorem: Getting from A to B A remarkable horizontal curve Curvature and nilpotentization Singular curves and geodesics A zoo of distributions Cartan's approach The tangent cone and Carnot groups Discrete groups tending to Carnot geometries Open problems Mechanics and geometry of bundles: Metrics on bundles Classical particles in Yang-Mills fields Quantum phases Falling, swimming, and orbiting Appendices: Geometric mechanics Bundles and the Hopf fibration The S
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Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
TL;DR: In this article, Kobayashi et al. introduced a new family of quiver varieties, which they call quiver variety, and studied their geometric structures, such as a natural *-action, symplectic geometry, topology, and so on.
Journal ArticleDOI
The Symplectic Nature of Fundamental Groups of Surfaces
TL;DR: Soit #7B-G la categorie des groupes de Lie G avec une forme bilineaire symetrique non singuliere B sur l'algebre de Lie.
Journal ArticleDOI
Hamiltonian description of the ideal fluid
TL;DR: The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described in this paper, in the context of the passive advection of a scalar or tracer field by a fluid.
References
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Book
Foundations of mechanics
TL;DR: In this article, Ratiu and Cushman introduce differential theory calculus on manifolds and derive an overview of qualitative and topological properties of differentiable properties of topological dynamics.
Journal ArticleDOI
Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) are defined, i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Journal ArticleDOI
Groups of diffeomorphisms and the motion of an incompressible fluid
David G. Ebin,Jerrold E. Marsden +1 more
TL;DR: In this article, the authors studied the manifold structure of certain groups of diffeomorphisms, and used this structure to obtain sharp existence and uniqueness theorems for the classical equations for an incompressible viscous and non-viscous fluid on a compact C^∞ riemannian, oriented n-manifold, possibly with boundary.