By Juan-Pablo Ortega
The use of symmetries and conservation legislation within the qualitative description of dynamics has a protracted heritage going again to the founders of classical mechanics. In a few cases, the symmetries in a dynamical procedure can be utilized to simplify its kinematical description through a tremendous process that has developed through the years and is understood generically as relief. the focal point of this paintings is a accomplished and self-contained presentation of the intimate connection among symmetries, conservation legislation, and relief, treating the singular case in detail.
The exposition stories the mandatory must haves, starting with an advent to Lie symmetries on Poisson and symplectic manifolds. this is often through a dialogue of momentum maps and the geometry of conservation legislation which are utilized in the advance of symplectic aid. The Symplectic Slice Theorem, a massive device that gave upward thrust to the 1st description of symplectic singular decreased areas, can be handled intimately, in addition to the Reconstruction Equations which were the most important in functions to the learn of symmetric mechanical platforms. The final a part of the publication includes extra complicated issues, akin to symplectic stratifications, optimum and Poisson relief, singular aid by means of levels, bifoliations and twin pairs. numerous attainable study instructions are mentioned within the advent and in the course of the textual content. an intensive bibliography and a close index around out the paintings.
This Ferran Sunyer i Balaguer Prize-winning monograph is the 1st self-contained and thorough presentation of the speculation of Hamiltonian relief within the presence of singularities. it will possibly function a source for graduate classes and seminars in symplectic and Poisson geometry, mechanics, Lie idea, mathematical physics, and as a finished reference source for researchers.
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Extra resources for Momentum Maps and Hamiltonian Reduction
A subset of a topological space is called residual if it is a countable intersection of open den se sets. The Baire category theorem states that a residual sub set of a locally compact topological space (hence of a finite-dimensional manifold) is dense. A topological space is called Lindelöl if every covering of the topological space by open sets admits a countable subcovering. Every c10sed subspace of a Lindelöf space is Lindelöf. Let I : X -+ Y be a continuous map with X Lindelöf. Then the subspace I(X) of Y is also Lindelöf.
Every open submanifold of a paracompact manifold 14 Chapter 1. Manifolds and Smooth Structures is paracompact. Second countable locally compact Hausdorff spaces are paracompact. For (Hausdorff finite-dimensional) manifolds, paracompactness and metrizability are equivalent. A connected (Hausdorff finite-dimensional) manifold admits a Riemannian metric if and only if it is second countable. 22 Proposition Let M be a smooth paracompact manifold. The presheaf of smooth functions on M is a sheaf Proof.
Such charts of Mare said to have the submanifold property. The intersection of N with the charts on M having the submanifold property endows N with a smooth manifold structure whose underlying topology coincides with the induced topology on N. It is important to note here that a submanifold is necessarily a locally closed subset of M. Recall that a subset N of a topological space M is said to be locally closed if each point n E N has an open neighborhood V in M such that V n N is closed in V, where V is endowed with the relative topology inherited from M.