THEORETICAL MECHANICS: AN INTRODUCTORY TREATISE ON THE PRINCIPLES OF DYNAMICS
Part I Newtonian Mechanics in Moving Coordinate Systems 1 Newton’s Equations in a Rotating Coordinate System 3 11 Introduction of the Operator D 6 12 Formulation of Newton’s Equation in the Rotating Coordinate System 7 13 Newton’s Equations in Systems with Arbitrary Relative Motion 7 2 Free Fall on the Rotating Earth 9 21 PerturbationCalculation 11 22 Method of Successive Approximation 12 23 Exact Solution 14 3 Foucault’s Pendulum 23 31 Solutionof theDifferentialEquations 26 32 Discussion of theSolution 28 Part II Mechanics of Particle Systems 4 Degrees of Freedom 41 41 Degrees of Freedom of a Rigid Body 41 5 Center of Gravity 43 6 Mechanical Fundamental Quantities of Systems of Mass Points 65 61 Linear Momentum of the Many-Body System 65 62 Angular Momentum of the Many-Body System 65 63 EnergyLawof theMany-BodySystem 68 64 Transformation toCenter-of-MassCoordinates 70 65 Transformation of theKineticEnergy 72 Part III Vibrating Systems 7 Vibrations of Coupled Mass Points 81 71 TheVibratingChain 88 8 The Vibrating String 101 81 Solutionof theWaveEquation 103 82 NormalVibrations 105 9 Fourier Series 121 10 The Vibrating Membrane 133 101 Derivationof theDifferentialEquation 133 102 Solutionof theDifferentialEquation 135 103 Inclusion of the Boundary Conditions 136 104 Eigenfrequencies 137 105 Degeneracy 137 106 Nodal Lines 138 107 General Solution 138 108 Superposition of Node Line Figures 140 109 TheCircularMembrane 141 1010 SolutionofBessel’sDifferentialEquation 144 Part IV Mechanics of Rigid Bodies 11 Rotation About a Fixed Axis 161 111 Momentof Inertia 162 112 The Physical Pendulum 166 12 Rotation About a Point 185 121 Tensorof Inertia 185 122 KineticEnergyof aRotatingRigidBody 187 123 ThePrincipalAxesof Inertia 188 124 Existence and Orthogonality of the Principal Axes 189 125 Transformationof theTensor of Inertia 193 126 Tensorof Inertia intheSystemofPrincipalAxes 195 127 Ellipsoid of Inertia 196 13 Theory of the Top 209 131 TheFreeTop 209 132 Geometrical Theory of the Top 210 133 Analytical Theory of the Free Top 213 134 TheHeavySymmetricTop:ElementaryConsiderations 224 135 FurtherApplicationsof theTop 228 136 TheEulerAngles 238 137 Motionof theHeavySymmetricTop 241 Part V Lagrange Equations 14 Generalized Coordinates 259 141 Quantities of Mechanics in Generalized Coordinates 264 15 D’Alembert Principle and Derivation of the Lagrange Equations 267 151 Virtual Displacements 267 16 Lagrange Equation for Nonholonomic Constraints 301 17 Special Problems 311 171 Velocity-Dependent Potentials 311 172 Nonconservative Forces and Dissipation Function (Friction Function) 315 173 Nonholonomic Systems and Lagrange Multipliers 317 Part VI Hamiltonian Theory 18 Hamilton’s Equations 327 181 The Hamilton Principle 337 182 General Discussion of Variational Principles 340 183 Phase Space and Liouville’s Theorem 350 184 The Principle of Stochastic Cooling 355 19 Canonical Transformations 365 20 Hamilton–Jacobi Theory 383 201 Visual Interpretation of the Action Function S 397 202 Transition to Quantum Mechanics 407 21 Extended Hamilton–Lagrange Formalism 415 211 Extended Set of Euler–Lagrange Equations 415 212 Extended Set of Canonical Equations 419 213 Extended Canonical Transformations 428 22 Extended Hamilton–Jacobi Equation 455 Part VII Nonlinear Dynamics 23 Dynamical Systems 463 231 Dissipative Systems: Contraction of the Phase-Space Volume 465 232 Attractors 467 233 Equilibrium Solutions 469 234 LimitCycles 475 24 Stability of Time-Dependent Paths 485 241 PeriodicSolutions 486 242 Discretization and Poincaré Cuts 487 25 Bifurcations 495 251 StaticBifurcations 495 252 Bifurcations of Time-Dependent Solutions 499 26 Lyapunov Exponents and Chaos 503 261 One-Dimensional Systems 503 262 Multidimensional Systems 505 263 Stretching and Folding in Phase Space 508 264 FractalGeometry 509 27 Systems with Chaotic Dynamics 517 271 Dynamics of Discrete Systems 517 272 One-Dimensional Mappings 518 Part VIII On the History of Mechanics 28 Emergence of Occidental Physics in the Seventeenth Century 555 Notes 561 Recommendations for Further Reading on Theoretical Mechanics 573