Introduction to Mechanics and Symmetry Q

By Marsden, Jerrold E.

Description
Physics Literature

Excerpt
Excerpt: Symmetry and mechanics have been close partners since the time of the founding masters: Newton, Euler, Lagrange, Laplace, Poisson, Jacobi, Hamilton, Kelvin, Routh, Riemann, Noether, Poincare, Einstein, Schraodinger, Cartan, Dirac, and to this day, symmetry has continued to play a strong role, especially with the modern work of Kolmogorov, Arnold, Moser, Kirillov, Kostant, Smale, Souriau, Guillemin, Sternberg, and many others. This book is about these developments, with an emphasis on concrete applications that we hope will make it accessible to a wide variety of readers, especially senior undergraduate and graduate students in science and engineering.

Table of Contents
Preface ix About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . xii I The Book xiv 1 Introduction and Overview 1 1.1 Lagrangian and Hamiltonian Formalisms . . . . . . . . . . . 1 1.2 The Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Lie{Poisson Brackets, Poisson Manifolds, Momentum Maps 9 1.4 The Heavy Top . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Incompressible Fluids . . . . . . . . . . . . . . . . . . . . . 18 1.6 The Maxwell{Vlasov System . . . . . . . . . . . . . . . . . 22 1.7 Nonlinear Stability . . . . . . . . . . . . . . . . . . . . . . . 29 1.8 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.9 The Poincar?e{Melnikov Method . . . . . . . . . . . . . . . . 46 1.10 Resonances, Geometric Phases, and Control . . . . . . . . . 49 2 Hamiltonian Systems on Linear Symplectic Spaces 61 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Symplectic Forms on Vector Spaces . . . . . . . . . . . . . . 65 2.3 Canonical Transformations or Symplectic Maps . . . . . . . 69 2.4 The General Hamilton Equations . . . . . . . . . . . . . . . 73 2.5 When Are Equations Hamiltonian? . . . . . . . . . . . . . . 76