INTRODUCTION TO MECHANICS AND SYMMETRY
CONTENTS INTRODUCTION AND OVERVIEW LAGRANGIAN AND HAMILTONIAN FORMALISMS THE RIGID BODY LIE–POISSON BRACKETS, POISSON MANIFOLDS, MOMENTUM MAPS THE HEAVY TOP INCOMPRESSIBLE FLUIDS THE MAXWELL–VLASOV SYSTEM NONLINEAR STABILITY BIFURCATION THE POINCAR´E–MELNIKOV METHOD RESONANCES, GEOMETRIC PHASES, AND CONTROL HAMILTONIAN SYSTEMS ON LINEAR SYMPLECTIC SPACES INTRODUCTION SYMPLECTIC FORMS ON VECTOR SPACES CANONICAL TRANSFORMATIONS, OR SYMPLECTIC MAPS THE GENERAL HAMILTON EQUATIONS WHEN ARE EQUATIONS HAMILTONIAN? HAMILTONIAN FLOWS POISSON BRACKETS A PARTICLE IN A ROTATING HOOP AN INTRODUCTION TO IN?NITE-DIMENSIONAL SYSTEMS LAGRANGE’S AND HAMILTON’S EQUATIONS FOR FIELD THEORY EXAMPLES: HAMILTON’S EQUATIONS EXAMPLES: POISSON BRACKETS AND CONSERVED QUANTITIES MANIFOLDS, VECTOR FIELDS, AND DI?ERENTIAL FORMS MANIFOLDS DI?ERENTIAL FORMS THE LIE DERIVATIVE STOKES’ THEOREM HAMILTONIAN SYSTEMS ON SYMPLECTIC MANIFOLDS SYMPLECTIC MANIFOLDS SYMPLECTIC TRANSFORMATIONS COMPLEX STRUCTURES AND K¨AHLER MANIFOLDS HAMILTONIAN SYSTEMS POISSON BRACKETS ON SYMPLECTIC MANIFOLDS COTANGENT BUNDLES THE LINEAR CASE THE NONLINEAR CASE COTANGENT LIFTS LIFTS OF ACTIONS GENERATING FUNCTIONS FIBER TRANSLATIONS AND MAGNETIC TERMS A PARTICLE IN A MAGNETIC FIELD LAGRANGIAN MECHANICS HAMILTON’S PRINCIPLE OF CRITICAL ACTION THE LEGENDRE TRANSFORM EULER–LAGRANGE EQUATIONS HYPERREGULAR LAGRANGIANS AND HAMILTONIANS GEODESICS THE KALUZA–KLEIN APPROACH TO CHARGED PARTICLES MOTION IN A POTENTIAL FIELD THE LAGRANGE–D’ALEMBERT PRINCIPLE THE HAMILTON–JACOBI EQUATION VARIATIONAL PRINCIPLES, CONSTRAINTS, & ROTATING SYSTEMS A RETURN TO VARIATIONAL PRINCIPLES THE GEOMETRY OF VARIATIONAL PRINCIPLES CONSTRAINED SYSTEMS CONSTRAINED MOTION IN A POTENTIAL FIELD DIRAC CONSTRAINTS CENTRIFUGAL AND CORIOLIS FORCES THE GEOMETRIC PHASE FOR A PARTICLE IN A HOOP MOVING SYSTEMS ROUTH REDUCTION AN INTRODUCTION TO LIE GROUPS BASIC DE?NITIONS AND PROPERTIES SOME CLASSICAL LIE GROUPS ACTIONS OF LIE GROUPS 10 POISSON MANIFOLDS THE DE?NITION OF POISSON MANIFOLDS HAMILTONIAN VECTOR FIELDS AND CASIMIR FUNCTIONS PROPERTIES OF HAMILTONIAN FLOWS THE POISSON TENSOR QUOTIENTS OF POISSON MANIFOLDS THE SCHOUTEN BRACKET GENERALITIES ON LIE–POISSON STRUCTURES 11 MOMENTUM MAPS CANONICAL ACTIONS AND THEIR IN?NITESIMAL GENERATORS MOMENTUM MAPS AN ALGEBRAIC DE?NITION OF THE MOMENTUM MAP CONSERVATION OF MOMENTUM MAPS EQUIVARIANCE OF MOMENTUM MAPS 12 COMPUTATION AND PROPERTIES OF MOMENTUM MAPS MOMENTUM MAPS ON COTANGENT BUNDLES EXAMPLES OF MOMENTUM MAPS EQUIVARIANCE AND IN?NITESIMAL EQUIVARIANCE EQUIVARIANT MOMENTUM MAPS ARE POISSON POISSON AUTOMORPHISMS MOMENTUM MAPS AND CASIMIR FUNCTIONS 13 LIE–POISSON AND EULER–POINCAR´E REDUCTION THE LIE–POISSON REDUCTION THEOREM PROOF OF THE LIE–POISSON REDUCTION THEOREM FOR GL(N) LIE–POISSON REDUCTION USING MOMENTUM FUNCTIONS REDUCTION AND RECONSTRUCTION OF DYNAMICS THE EULER–POINCAR´E EQUATIONS THE LAGRANGE–POINCAR´E EQUATIONS 14 COADJOINT ORBITS EXAMPLES OF COADJOINT ORBITS TANGENT VECTORS TO COADJOINT ORBITS THE SYMPLECTIC STRUCTURE ON COADJOINT ORBITS THE ORBIT BRACKET VIA RESTRICTION OF THE LIE–POISSON BRACKET THE SPECIAL LINEAR GROUP OF THE PLANE THE EUCLIDEAN GROUP OF THE PLANE THE EUCLIDEAN GROUP OF THREE-SPACE 15 THE FREE RIGID BODY MATERIAL, SPATIAL, AND BODY COORDINATES THE LAGRANGIAN OF THE FREE RIGID BODY THE LAGRANGIAN AND HAMILTONIAN IN BODY REPRESENTATION KINEMATICS ON LIE GROUPS POINSOT’S THEOREM EULER ANGLES THE HAMILTONIAN OF THE FREE RIGID BODY THE ANALYTICAL SOLUTION OF THE FREE RIGID-BODY PROBLEM RIGID-BODY STABILITY 1510 HEAVY TOP STABILITY 1511 THE RIGID BODY AND THE PENDULUM