LDR 03467 a2200349 4500 010 _a9781475738131 _bbr. 090 _a16471 101 _aeng 102 _aUS 100 _a20180711 frey50 200 _bM _aMomentum Maps and Hamiltonian Reduction _fJuan-Pablo Ortega, Tudor S. Ratiu 210 _cBirkhauser _aBoston _dcop. 2004 215 _a1 vol. (XXXIV-497 p.) _cill., portr _d24 225 _9169097 _aProgress in mathematics _v222 _x0743-1643 320 _aBibliogr. p. [443]-476. Index 330 _aPublisherâ€™s description: The use of symmetries and conservation laws in the qualitative description of dynamics has a long history going back to the founders of classical mechanics. In some instances, the symmetries in a dynamical system can be used to simplify its kinematical description via an important procedure that has evolved over the years and is known generically as reduction. The focus of this work is a comprehensive and self-contained presentation of the intimate connection between symmetries, conservation laws, and reduction, treating the singular case in detail. The exposition reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds. This is followed by a discussion of momentum maps and the geometry of conservation laws that are used in the development of symplectic reduction. Table of contents: Introduction. Manifolds and smooth structures. Lie group actions. Pseudogroups and groupoids. The standard momentum map. Generalizations of the momentum map. Regular symplectic reduction theory. The Symplectic Slice Theorem. Singular reduction and the stratification theorem. Optimal reduction. Poisson reduction. Dual Pairs. Bibliography. Index. The book can serve as a resource for graduate courses and seminars in Hamiltonian mechanics and symmetry, symplectic and Poisson geometry, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers. 410 _9169095 _aBirkhĂ¤user _tProgress in mathematics _x0743-1643 676 _a2010 686 _20 _9164943 _a53D20 _bDifferential geometry -- Symplectic geometry, contact geometry _xMomentum maps; symplectic reduction 686 _20 _9163918 _a37J05 _bDynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems _xGeneral theory, relations with symplectic geometry and topology 686 _20 _9163920 _a37J15 _bDynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems _xSymmetries, invariants, invariant manifolds, momentum maps, reduction 686 _20 _9166006 _a70G45 _bMechanics of particles and systems -- General models, approaches, and methods _xDifferential-geometric methods 686 _20 _9166027 _a70H33 _bMechanics of particles and systems -- Hamiltonian and Lagrangian mechanics _xSymmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction 700 _4070 _9179926 _aOrtega _bJuan-Pablo 701 _4070 _9169041 _aRatiu _bTudor Stefan _f1950- 856 _uhttp://link.springer.com/book/10.1007/978-1-4757-3811-7 _zSpringerlink 856 _uhttp://www.ams.org/mathscinet-getitem?mr=2021152 _zMSN 856 _uhttps://zbmath.org/?q=an:1241.53069 _zzbMath 905 _aaw _b2018 906 _aaw _b2018-08-30 001 16471 995 _f12504-01 _xachat Ebsco _918663 _cCMI _20 _k53 ORT _o0 _eSalle R _z72 _bCMI