Mechanics And Symmetry. Reduction Theory

Mechanics and Symmetry Reduction Theory Jerrold E. Marsden and Tudor S. Ratiu February 3, 1998 ii Preface Preface goes here. Pasadena, CA Spring, 1998 Jerry Marsden and Tudor Ratiu iii iv Preface Contents Preface iii 1 Introduction and Overview 1.1 Lagrangian and Hamiltonian Mechanics. 1.2 The Euler–Poincar´e Equations. . . . . . 1.3 The Lie–Poisson Equations. . . . . . . . 1.4 The Heavy Top. . . . . . . . . . . . . . . 1.5 Incompressible Fluids. . . . . . . . . . . 1.6 The Basic Euler–Poincar´e Equations. . . 1.7 Lie–Poisson Reduction. . . . . . . . . . . 1.8 Symplectic and Poisson Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 8 10 11 13 14 20 2 Symplectic Reduction 2.1 Presymplectic Reduction . . . . . . . . . . . . . 2.2 Symplectic Reduction by a Group Action . . . 2.3 Coadjoint Orbits as Symplectic Reduced Spaces 2.4 Reducing Hamiltonian Systems . . . . . . . . . 2.5 Orbit Reduction . . . . . . . . . . . . . . . . . 2.6 Foliation Orbit Reduction . . . . . . . . . . . . 2.7 The Shifting Theorem . . . . . . . . . . . . . . 2.8 Dynamics via Orbit Reduction . . . . . . . . . 2.9 Reduction by Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .