Momentum Maps and Hamiltonian Reduction / Juan-Pablo Ortega, Tudor S. Ratiu

Bibliogr. p. [443]-476. Index

Publisher’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.