Lectures on Mechanics (London Mathematical Society Lecture Note Series, Series Number 174)

The Use Of Geometric Methods In Classical Mechanics Has Proven To Be A Fruitful Exercise, With The Results Being Of Wide Application To Physics And Engineering. Here Professor Marsden Concentrates On These Geometric Aspects, And Especially On Symmetry Techniques. The Main Points He Covers Are: The Stability Of Relative Equilibria, Which Is Analyzed Using The Block Diagonalization Technique; Geometric Phases, Studied Using The Reduction And Reconstruction Technique; And Bifurcation Of Relative Equilibria And Chaos In Mechanical Systems. A Unifying Theme For These Points Is Provided By Reduction Theory, The Associated Mechanical Connection And Techniques From Dynamical Systems. These Methods Can Be Applied To Many Control And Stabilization Situations, And This Is Illustrated Using Rigid Bodies With Internal Rotors, And The Use Of Geometric Phases In Mechanical Systems. To Illustrate The Above Ideas And The Power Of Geometric Arguments, The Author Studies A Variety Of Specific Systems, Including The Double Spherical Pendulum And The Classical Rotating Water Molecule. This Book, Based On The 1991 Lms Invited Lectures, Will Be Valued By Pure And Applied Mathematicians, Physicists And Engineers Who Work In Geometry, Nonlinear Dynamics, Mechanics, And Robotics. Jerrold E. Marsden. Includes Index. Includes Bibliographical References (p. 225-249). Preface......Page 6 The Classical Water Molecule and the Ozone Molecule......Page 11 Lagrangian and Hamiltonian Formulation......Page 13 The Rigid Body......Page 15 Geometry, Symmetry and Reduction......Page 22 Stability......Page 25 Geometric Phases......Page 29 The Rotation Group and the Poincaré Sphere......Page 36 Symplectic and Poisson Manifolds......Page 39 The Flow of a Hamiltonian Vector Field......Page 41 Cotangent Bundles......Page 42 Lagrangian Mechanics......Page 43 Lie--Poisson Structures and the Rigid Body......Page 44 The Euler--Poincaré Equations......Page 47 Momentum Maps......Page 49 Symplectic and Poisson Reduction......Page 52 Singularities and Symmetry......Page 55 A Particle in a Magnetic Field......Page 56 Tangent and Cotangent Bundle Reduction......Page 59 Mechanical G-systems......Page 60 The Classical Water Molecule......Page 62 The Mechanical Connection......Page 66 The Geometry and Dynamics of Cotangent Bundle Reduction......Page 71 Examples......Page 75 Lagrangian Reduction and the Routhian......Page 82 The Reduced Euler--Lagrange Equations......Page 87 Coupling to a Lie group......Page 89 Relative Equilibria on Symplectic Manifolds......Page 95 Cotangent Relative Equilibria......Page 98 Examples......Page 100 The Rigid Body......Page 105 The General Technique......Page 111 Example: The Rigid Body......Page 115 Block Diagonalization......Page 119 The Normal Form for the Symplectic Structure......Page 124 Stability of Relative Equilibria for the Double Spherical Pendulum......Page 127 A Simple Example......Page 131 Reconstruction......Page 133 Cotangent Bundle Phases---a Special Case......Page 135 Cotangent Bundles---General Case......Page 136 Rigid Body Phases......Page 138 Moving Systems......Page 140 The Bead on the Rotating Hoop......Page 142 The Rigid Body with Internal Rotors......Page 145 The Hamiltonian Structure with Feedback Controls......Page 146 Feedback Stabilization of a Rigid Body with a Single Rotor......Page 148 Phase Shifts......Page 151 The Kaluza--Klein Description of Charged Particles......Page 155 Optimal Control and Yang--Mills Particles......Page 158 Discrete Reduction......Page 161 Fixed Point Sets and Discrete Reduction......Page 163 Cotangent Bundles......Page 169 Examples......Page 171 Sub-Block Diagonalization with Discrete Symmetry......Page 175 Discrete Reduction of Dual Pairs......Page 178 Definitions and Examples......Page 183 Limitations on Mechanical Integrators......Page 187 Symplectic Integrators and Generating Functions......Page 189 Symmetric Symplectic Algorithms Conserve J......Page 190 Energy--Momentum Algorithms......Page 192 The Lie--Poisson Hamilton--Jacobi Equation......Page 194 Example: The Free Rigid Body......Page 197 PDE Extensions......Page 198 Some Introductory Examples......Page 201 The Role of Symmetry......Page 208 The One-to-One Resonance and Dual Pairs......Page 213 Bifurcations in the Double Spherical Pendulum......Page 215 Continuous Symmetry Groups and Solution Space Singularities......Page 216 The Poincaré--Melnikov Method......Page 217 The Role of Dissipation......Page 227 Double Bracket Dissipation......Page 233 Bibliography......Page 237 Index......Page 256 The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule. The use of geometric methods in classical mechanics has proven to be a fruitful exercise, with the results being of wide application to physics and engineering. Here Professor Marsden concentrates on these geometric aspects, and especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studies using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems.