Introduction To Mechanics And Symmetry

This is page xv Printer: Opaque this Contents Preface ix About the Authors xiii 1 Introduction and Overview 1.1 Lagrangian and Hamiltonian Formalisms . . . . . . . 1.2 The Rigid Body . . . . . . . . . . . . . . . . . . . . . 1.3 Lie–Poisson Brackets, Poisson Manifolds, Momentum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Heavy Top . . . . . . . . . . . . . . . . . . . . . 1.5 Incompressible Fluids . . . . . . . . . . . . . . . . . . 1.6 The Maxwell–Vlasov System . . . . . . . . . . . . . . 1.7 Nonlinear Stability . . . . . . . . . . . . . . . . . . . 1.8 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . 1.9 The Poincar´e–Melnikov Method . . . . . . . . . . . . 1.10 Resonances, Geometric Phases, and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 16 18 22 29 43 47 50 2 Hamiltonian Systems on Linear Symplectic Spaces 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 Symplectic Forms on Vector Spaces . . . . . . . . . 2.3 Canonical Transformations, or Symplectic Maps . . 2.4 The General Hamilton Equations . . . . . . . . . . 2.5 When Are Equations Hamiltonian? . . . . . . . . . 2.6 Hamiltonian Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 61 66 69 74 77 80 . . . . . . . . . . . . 1 1 6 xvi Contents 2.7 2.8 2.9 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . . A Particle in a Rotating Hoop . . . . . . . . . . . . . . . . The Poincar´e–Melnikov Method . . . . . . . . . . . . . . . 82 87 94 3 An Introduction to Infinite-Dimensional Systems 105 3.1 Lagrange’s and Hamilton’s Equations for Field Theory . . 105 3.2 Examples: Hamilton’s Equations . . . . . . . . . . . . . . 107 3.3 Exa