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Lectures on Mechanics Page i Lectures on Mechanics Jerrold E. Marsden First Published 1992; This Version: October, 2009 Page i Contents Preface v 1 Introduction 1 1.1 The Classical Water Molecule and the Ozone Molecule .. 1 1.2 Lagrangian and Hamiltonian Formulation ......... 3 1.3 The Rigid Body ........................ 5 1.4 Geometry, Symmetry and Reduction ............ 12 1.5 Stability ............................ 15 1.6 Geometric Phases ....................... 19 1.7 The Rotation Group and the Poincar´eSphere ....... 26 2 A Crash Course in Geometric Mechanics 29 2.1 Symplectic and Poisson Manifolds .............. 29 2.2 The Flow of a Hamiltonian Vector Field .......... 31 2.3 Cotangent Bundles ...................... 32 2.4 Lagrangian Mechanics .................... 33 2.5 Lie{Poisson Structures and the Rigid Body ........ 34 2.6 The Euler{Poincar´eEquations ................ 37 2.7 Momentum Maps ....................... 39 2.8 Symplectic and Poisson Reduction ............. 42 2.9 Singularities and Symmetry ................. 45 2.10 A Particle in a Magnetic Field ................ 46 3 Tangent and Cotangent Bundle Reduction 49 ii Contents 3.1 Mechanical G-systems .................... 50 3.2 The Classical Water Molecule ................ 52 3.3 The Mechanical Connection ................. 56 3.4 The Geometry and Dynamics of Cotangent Bundle Reduc- tion .............................. 61 3.5 Examples ........................... 65 3.6 Lagrangian Reduction and the Routhian .......... 72 3.7 The Reduced Euler{Lagrange Equations .......... 77 3.8 Coupling to a Lie group ................... 79 4 Relative Equilibria 85 4.1 Relative Equilibria on Symplectic Manifolds ........ 85 4.2 Cotangent Relative Equilibria ................ 88 4.3 Examples ........................... 90 4.4 The Rigid Body ........................ 95 5 The Energy{Momentum Method 101 5.1 The General Technique .................... 101 5.2 Example: The Rigid Body .................. 105 5.3 Block Diagonalization .................... 109 5.4 The Normal Form for the Symplectic Structure ...... 114 5.5 Stability of Relative Equilibria for the Double Spherical Pendulum ........................... 117 6 Geometric Phases 121 6.1 A Simple Example ...................... 121 6.2 Reconstruction ........................ 123 6.3 Cotangent Bundle Phases|a Special Case ......... 125 6.4 Cotangent Bundles|General Case ............. 126 6.5 Rigid Body Phases ...................... 128 6.6 Moving Systems ........................ 130 6.7 The Bead on the Rotating Hoop .............. 132 7 Stabilization and Control 135 7.1 The Rigid Body with Internal Rotors ............ 135 7.2 The Hamiltonian Structure with Feedback Controls .... 136 7.3 Feedback Stabilization of a Rigid Body with a Single Rotor 138 7.4 Phase Shifts .......................... 141 7.5 The Kaluza{Klein Description of Charged Particles .... 145 7.6 Optimal Control and Yang{Mills Particles ......... 148 8 Discrete Reduction 151 8.1 Fixed Point Sets and Discrete Reduction .......... 153 8.2 Cotangent Bundles ...................... 159 8.3 Examples ........................... 161 Contents iii 8.4 Sub-Block Diagonalization with Discrete Symmetry .... 165 8.5 Discrete Reduction of Dual Pairs .............. 168 9 Mechanical Integrators 173 9.1 Definitions and Examples .................. 173 9.2 Limitations on Mechanical Integrators ........... 177 9.3 Symplectic Integrators and Generating Functions ..... 179 9.4 Symmetric Symplectic Algorithms Conserve J ....... 180 9.5 Energy{Momentum Algorithms ............... 182 9.6 The Lie{Poisson Hamilton{Jacobi Equation ........ 184 9.7 Example: The Free Rigid Body ............... 187 9.8 PDE Extensions ....................... 188 10 Hamiltonian Bifurcation 191 10.1 Some Introductory Examples ................ 191 10.2 The Role of Symmetry .................... 198 10.3 The One-to-One Resonance and Dual Pairs ........ 203 10.4 Bifurcations in the Double Spherical Pendulum ...... 205 10.5 Continuous Symmetry Groups and Solution Space Singu- larities ............................. 206 10.6 The Poincar´e{Melnikov Method ............... 207 10.7 The Role of Dissipation ................... 217 10.8 Double Bracket Dissipation ................. 223 References 227 Index 245 Index 246 iv Contents Page v Preface Many of the greatest mathematicians|Euler, Gauss, Lagrange, Rie- mann, Poincar´e,Hilbert, Birkhoff, Atiyah, Arnold, Smale|mastered mechanics and many of the greatest advances in mathematics use ideas from mechanics in a fundamental way. Hopefully the revival of it as a basic subject taught to mathematicians will continue and flourish as will the combined geometric, analytic and numerical ap- proach in courses taught in the applied sciences. Anonymous I venture to hope that my lectures may interest engineers, physi- cists, and astronomers as well as mathematicians. If one may accuse mathematicians as a class of ignoring the mathematical problems of the modern physics and astronomy, one may, with no less justice perhaps, accuse physicists and astronomers of ignoring departments of the pure mathematics which have reached a high degree of de- velopment and are fitted to render valuable service to physics and astronomy. It is the great need of the present in mathematical sci- ence that the pure science and those departments of physical science in which it finds its most important applications should again be brought into the intimate association which proved so fruitful in the work of Lagrange and Gauss. Felix Klein, 1896 Mechanics is a broad subject that includes, not only the mechanics of particles and rigid bodies, but also continuum mechanics (fluid mechanics, elasticity, plasma physics, etc), field theory (Maxwell's equations, the Ein- stein equations etc) and even, as the name implies, quantum mechanics. Rather than compartmentalizing these into separate subjects (and even separate departments in Universities), there is power in seeing these topics as a unified whole. For example, the fact that the flow of a Hamiltonian system consists of canonical transformations becomes a single theorem com- mon to all these topics. Numerical techniques one learns in one branch of mechanics are useful in others. These lectures cover a selection of topics from recent developments in the mathematical approach to mechanics and its applications. In partic- vi Preface ular, we emphasize methods based on symmetry, especially the action of Lie groups, both continuous and discrete, and their associated Noether conserved quantities viewed in the geometric context of momentum maps. In this setting, relative equilibria, the analogue of fixed points for systems without symmetry are especially interesting. In general, relative equilibria are dynamic orbits that are also one-parameter group orbits. For the rota- tion group SO(3), these are uniformly rotating states or, in other words, dynamical motions in steady rotation. Some of the main points treated in the lectures are as follows: • The stability of relative equilibria analyzed using the method of sep- aration of internal and rotational modes, also referred to as the block diagonalization or normal form technique. • Geometric phases, including the phases of Berry and Hannay, are studied using the technique of reduction and reconstruction. • Mechanical integrators, such as numerical schemes that exactly pre- serve the symplectic structure, energy, or the momentum map. • Stabilization & control using methods for mechanical systems. • Bifurcation of relative equilibria in mechanical systems, dealing with the appearance of new relative equilibria and their symmetry break- ing as parameters are varied, and with the development of complex (chaotic) dynamical motions. A unifying theme for many of these aspects is provided by reduction the- ory and the associated mechanical connection for mechanical systems with symmetry. When one does reduction, one sets the corresponding conserved quantity (the momentum map) equal to a constant, and quotients by the subgroup of the symmetry group that leaves this set invariant. One arrives at the reduced symplectic manifold that itself is often a bundle that carries a connection. This connection is induced by a basic ingredient in the the- ory, the mechanical connection on configuration space. This point of view is sometimes called the gauge theory of mechanics. The geometry of reduction and the mechanical connection is an impor- tant ingredient in the decomposition into internal and rotational modes in the block diagonalization method, a powerful method for analyzing the stability and bifurcation of relative equilibria. The holonomy of the con- nection on the reduction bundle gives geometric phases. When stability of a relative equilibrium is lost, one can get bifurcation, solution symmetry breaking, instability and chaos. The notion of system symmetry break- ing (also called forced symmetry breaking) in which not only the solutions, but the equations themselves lose symmetry, is also important but here is treated only by means of some simple examples. Preface vii Two related topics that are discussed are control and mechanical inte- grators. One would like to be able to control the geometric phases with the aim of, for example, controlling the attitude of a rigid body with internal rotors. With mechanical integrators one is interested in designing numeri- cal integrators that exactly preserve the conserved momentum (say angular momentum) and either the energy or symplectic structure, for the purpose of accurate long time integration of mechanical systems. Such integrators
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