Lectures on Mechanics, Vol. 174
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...
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Based on the 1991 LMS Invited Lectures given by Professor Marsden, this book discusses and applies symmetry methods to such areas as bifurcations and chaos in mechanical systems.
Preface1Introduction11.1The Classical Water Molecule and the Ozone Molecule11.2Hamiltonian Formulation31.3Geometry, Symmetry, and Reduction91.4Stability121.5Geometric Phases161.6The Rotation Group and the Poincare Sphere232A Crash Course in Geometric Mechanics272.1Symplectic and Poisson Manifolds272.2The Flow of a Hamiltonian Vector Field292.3Cotangent Bundles292.4Lagrangian Mechanics312.5Lie-Poisson Structures322.6The Rigid Body332.7Momentum Maps342.8Reduction362.9Singularities and Symmetry392.10A Particle in a Magnetic Field403Cotangent Bundle Reduction433.1Mechanical G-systems433.2The Classical Water Molecule463.3The Mechanical Connection503.4The Geometry and Dynamics of Cotangent Bundle Reduction543.5Examples593.6Lagrangian Reduction663.7Coupling to a Lie group724Relative Equilibria774.1Relative Equilibria on Symplectic Manifolds774.2Cotangent Relative Equilibria794.3Examples824.4The Rigid Body875The Energy-Momentum Method935.1The General Technique935.2Example: The Rigid Body975.3Block Diagonalization1015.4The Normal Form for the Symplectic Structure1075.5Stability of Relative Equilibria for the Double Spherical Pendulum1106Geometric Phases1156.1A Simple Example1156.2Reconstruction1176.3Cotangent Bundle Phases--a Special Case1196.4Cotangent Bundles--General Case1206.5Rigid Body Phases1226.6Moving Systems1256.7The Bead on the Rotating Hoop1277Stabilization and Control1317.1The Rigid Body with Internal Rotors1317.2The Hamiltonian Structure with Feedback Controls1327.3Feedback Stabilization of a Rigid Body with a Single Rotor1347.4Phase Shifts1377.5The Kaluza-Klein Description of Charged Particles1417.6Optimal Control and Yang-Mills Particles1448Discrete reduction1478.1Fixed Point Sets and Discrete Reduction1498.2Cotangent Bundles1558.3Examples1578.4Sub-Block Diagonalization with Discrete Symmetry1628.5Discrete Reduction of Dual Pairs1689Mechanical Integrators1719.1Definitions and Examples1719.2Limitations on Mechanical Integrators1759.3Symplectic Integrators and Generating Functions1779.4Symmetric Symplectic Algorithms Conserve J1789.5Energy-Momentum Algorithms1809.6The Lie-Poisson Hamilton-Jacobi Equation1829.7Example: The Free Rigid Body1869.8Variational Considerations18710Hamiltonian Bifurcation18910.1Some Introductory Examples18910.2The Role of Symmetry19610.3The One to One Resonance and Dual Pairs20210.4Bifurcations in the Double Spherical Pendulum20410.5Continuous Symmetry Groups and Solution Space Singularities20510.6The Poincare-Melnikov Method20710.7The Role of Dissipation217References225Index250
