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. 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. This Ferran Sunyer i Balaguer Prize-winning monograph is the first self-contained and thorough presentation of the theory of Hamiltonian reduction in the presence of singularities. It can serve as a resource for graduate courses and seminars in symplectic and Poisson geometry, mechanics, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers.

Winner of the Ferran Sunyer i Balaguer Prize in 2000

Reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds

Currently in classroom use in Europe

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

Inhalt
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