FORCE AND ENERGY
1 A brief history of classical mechanics 1 11 Introduction 1 12 Greek antiquity 1 13 Middle Ages 2 14 Age of Enlightenment 2 15 Variationalmethods in physics 5 16 The 20?? century revolution in physics 7 2 Review of Newtonian mechanics 9 21 Introduction 9 22 Newton’s Laws ofmotion 9 23 Inertial frames of reference 10 24 First-order integrals in Newtonianmechanics 11 241 LinearMomentum 11 242 Angular momentum 11 243 Kinetic energy 12 25 Conservation laws in classicalmechanics 12 26 Motion of finite-sized andmany-body systems 12 27 Center ofmass of amany-body system 13 28 Total linear momentumof a many-body system 14 281 Center-of-mass decomposition 14 282 Equations ofmotion 14 29 Angularmomentumof amany-body system 16 291 Center-of-mass decomposition 16 292 Equations ofmotion 16 210 Work and kinetic energy for amany-body system 18 2101 Center-of-mass kinetic energy 18 2102 Conservative forces and potential energy 18 2103 Totalmechanical energy 19 2104 Totalmechanical energy for conservative systems 20 211 Virial Theorem 22 212 Applications of Newton’s equations ofmotion 24 2121 Constant force problems 24 2122 Linear Restoring Force 25 2123 Position-dependent conservative forces 25 2124 Constrainedmotion 27 2125 Velocity Dependent Forces 28 2126 Systems with VariableMass 29 2127 Rigid-body rotation about a body-fixed rotation axis 31 2128 Time dependent forces 34 213 Solution ofmany-body equations ofmotion 37 2131 Analytic solution 37 2132 Successive approximation 37 2133 Perturbationmethod 37 214 Newton’s Law of Gravitation 38 2141 Gravitational and inertial mass 38 2142 Gravitational potential energy ? 39 2143 Gravitational potential ? 40 2144 Potential theory 41 2145 Curl of the gravitational field 41 2146 Gauss’s Law for Gravitation 43 2147 Condensed forms of Newton’s Law of Gravitation 44 215 Summary 46 3 Linear oscillators 49 31 Introduction 49 32 Linear restoring forces 49 33 Linearity and superposition 50 34 Geometrical representations of dynamical motion 51 341 Configuration space (??? ?? ? ?) 51 342 State space, (??? ?????) 52 343 Phase space, (??? ????) 52 344 Plane pendulum 53 35 Linearly-damped free linear oscillator 54 351 General solution 54 352 Energy dissipation 57 36 Sinusoidally-drive, linearly-damped, linear oscillator 58 361 Transient response of a driven oscillator 58 362 Steady state response of a driven oscillator 59 363 Complete solution of the driven oscillator 60 364 Resonance 61 365 Energy absorption 61 37 Wave equation 64 38 Travelling and standing wave solutions of the wave equation 65 39 Waveformanalysis 66 391 Harmonic decomposition 66 392 The free linearly-damped linear oscillator 66 393 Damped linear oscillator subject to an arbitrary periodic force 67 310 Signal processing 68 311 Wave propagation 69 3111 Phase, group, and signal velocities of wave packets 70 3112 Fourier transformof wave packets 75 3113 Wave-packet Uncertainty Principle 76 312 Summary 78 4 Nonlinear systems and chaos 81 41 Introduction 81 42 Weak nonlinearity 82 43 Bifurcation, and point attractors 84 44 Limit cycles 85 441 Poincaré-Bendixson theorem 85 442 van der Pol damped harmonic oscillator: 86 45 Harmonically-driven, linearly-damped, plane pendulum 89 451 Close to linearity 89 452 Weak nonlinearity 91 453 Onset of complication 92 454 Period doubling and bifurcation 92 455 Rolling motion 92 456 Onset of chaos 93 46 Differentiation between ordered and chaoticmotion 94 461 Lyapunov exponent 94 462 Bifurcation diagram 95 463 Poincaré Section 96 47 Wave propagation for non-linear systems 97 471 Phase, group, and signal velocities 97 472 Soliton wave propagation 99 48 Summary 100 5 Calculus of variations 103 51 Introduction 103 52 Euler’s differential equation 104 53 Applications of Euler’s equation 106 54 Selection of the independent variable 109 55 Functions with several independent variables ??(?) 111 56 Euler’s integral equation 113 57 Constrained variational systems 114 571 Holonomic constraints 114 572 Geometric (algebraic) equations of constraint 114 573 Kinematic (differential) equations of constraint 114 574 Isoperimetric (integral) equations of constraint 115 575 Properties of the constraint equations 115 576 Treatment of constraint forces in variational calculus 116 58 Generalized coordinates in variational calculus 117 59 Lagrangemultipliers for holonomic constraints 118 591 Algebraic equations of constraint 118 592 Integral equations of constraint 120 510 Geodesic 122 511 Variational approach to classicalmechanics 123 512 Summary 124 6 Lagrangian dynamics 125 61 Introduction 125 62 Newtonian plausibility argument for Lagrangian mechanics 126 63 Lagrange equations fromd’Alembert’s Principle 128 631 d’Alembert’s Principle of VirtualWork 128 632 Transformation to generalized coordinates 129 633 Lagrangian 130 64 Lagrange equations fromHamilton’s Action Principle 131 65 Constrained systems 132 651 Choice of generalized coordinates 132 652 Minimal set of generalized coordinates 132 653 Lagrangemultipliers approach 132 654 Generalized forces approach 134 66 Applying the Euler-Lagrange equations to classicalmechanics 134 67 Applications to unconstrained systems 136 68 Applications to systems involving holonomic constraints 138 69 Applications involving non-holonomic constraints 151 610 Velocity-dependent Lorentz force 158 611 Time-dependent forces 159 612 Impulsive forces 160 613 The Lagrangian versus the Newtonian approach to classicalmechanics 161 614 Summary 162 7 Symmetries, Invariance and the Hamiltonian 165 71 Introduction 165 72 Generalizedmomentum 165 73 Invariant transformations and Noether’s Theorem 167 74 Rotational invariance and conservation of angularmomentum 169 75 Cyclic coordinates 170 76 Kinetic energy in generalized coordinates 171 77 Generalized energy and the Hamiltonian function 172 78 Generalized energy theorem 173 79 Generalized energy and total energy 173 710 Hamiltonian invariance 174 711 Hamiltonian for cyclic coordinates 179 712 Symmetries and invariance 179 713 Hamiltonian in classicalmechanics 179 714 Summary 180 8 Hamiltonian mechanics 183 81 Introduction 183 82 Legendre Transformation between Lagrangian and Hamiltonianmechanics 184 83 Hamilton’s equations ofmotion 185 831 Canonical equations ofmotion 186 84 Hamiltonian in different coordinate systems 187 841 Cylindrical coordinates ?? ?? ? 187 842 Spherical coordinates, ?? ?? ? 188 85 Applications of Hamiltonian Dynamics 189 86 Routhian reduction 194 861 R?????? - Routhian is a Hamiltonian for the cyclic variables 195 862 R????????? - Routhian is a Hamiltonian for the non-cyclic variables 196 87 Variable-mass systems 200 871 Rocket propulsion: 200 872 Moving chains: 201 88 Summary 203 9 Hamilton’s Action Principle 205 91 Introduction 205 92 Hamilton’s Principle of Stationary Action 921 Stationary-action principle in Lagrangianmechanics 206 922 Stationary-action principle in Hamiltonianmechanics 207 923 Abbreviated action 208 924 Hamilton’s Principle applied using initial boundary conditions 209 93 Lagrangian 212 931 Standard Lagrangian 212 932 Gauge invariance of the standard Lagrangian 212 933 Non-standard Lagrangians 214 934 Inverse variational calculus 214 94 Application of Hamilton’s Action Principle tomechanics 215 95 Summary 216 10 Nonconservative systems 219 101 Introduction 219 102 Origins of nonconservativemotion 219 103 Algebraicmechanics for nonconservative systems 220 104 Rayleigh’s dissipation function 220 1041 Generalized dissipative forces for linear velocity dependence 221 1042 Generalized dissipative forces for nonlinear velocity dependence 222 1043 Lagrange equations ofmotion 222 1044 Hamiltonianmechanics 222 105 Dissipative Lagrangians 225 106 Summary 227 11 Conservative two-body central forces 229 111 Introduction 229 112 Equivalent one-body representation for two-bodymotion 230 113 Angular momentum L 232 114 Equations of motion 233 115 Differential orbit equation: 234 116 Hamiltonian 235 117 General features of the orbit solutions 236 118 Inverse-square, two-body, central force 237 1181 Bound orbits 238 1182 Kepler’s laws for bound planetarymotion 239 1183 Unbound orbits 240 1184 Eccentricity vector 241 119 Isotropic, linear, two-body, central force 243 1191 Polar coordinates 244 1192 Cartesian coordinates 245 1193 Symmetry tensor A0 246 1110Closed-orbit stability 247 1111The three-body problem 252 1112Two-body scattering 253 11121Total two-body scattering cross section 253 11122Differential two-body scattering cross section 254 11123Impact parameter dependence on scattering angle 254 11124Rutherford scattering 256 1113Two-body kinematics 258 1114Summary 264 12 Non-inertial reference frames 267 121 Introduction 267 122 Translational acceleration of a reference frame 267 123 Rotating reference frame 268 1231 Spatial time derivatives in a rotating, non-translating, reference frame 268 1232 General vector in a rotating, non-translating, reference frame 269 124 Reference frame undergoing rotation plus translation 270 125 Newton’s law ofmotion in a non-inertial frame 270 126 Lagrangianmechanics in a non-inertial frame 271 127 Centrifugal force 272 128 Coriolis force 273 129 Routhian reduction for rotating systems 277 1210Effective gravitational force near the surface of the Earth 280 1211Freemotion on the earth 282 1212Weather systems 284 12121Low-pressure systems: 284 12122High-pressure systems: 286 1213Foucault pendulum 286 1214Summary 288 13 Rigid-body rotation 289 131 Introduction 289 132 Rigid-body coordinates 290 133 Rigid-body rotation about a body-fixed point 290 134 Inertia tensor 292 135 Matrix and tensor formulations of rigid-body rotation 293 136 Principal axis system 293 137 Diagonalize the inertia tensor 294 138 Parallel-axis theorem 295 139 Perpendicular-axis theoremfor plane laminae 298 1310General properties of the inertia tensor 299 13101Inertial equivalence 299 13102Orthogonality of principal axes 300 1311Angular momentum L and angular velocity ? vectors 301 1312Kinetic energy of rotating rigid body 303 1313Euler angles 305 1314Angular velocity ? 307 1315Kinetic energy in terms of Euler angular velocities 308 1316Rotational invariants 309 1317Euler’s equations ofmotion for rigid-body rotation 310 1318Lagrange equations ofmotion for rigid-body rotation 311 1319Hamiltonian equations ofmotion for rigid-body rotation 313 1320Torque-free rotation of an inertially-symmetric rigid rotor 313 13201Euler’s equations ofmotion: 313 13202Lagrange equations ofmotion: 317 1321Torque-free rotation of an asymmetric rigid rotor 319 1322Stability of torque-free rotation of an asymmetric body 320 1323Symmetric rigid rotor subject to torque about a fixed point 323 1324The rolling wheel 327 1325Dynamic balancing of wheels 330 1326Rotation of deformable bodies 331 1327Summary 332 14 Coupled linear oscillators 335 141 Introduction 335 142 Two coupled linear oscillators 335 143 Normalmodes 337 144 Center of mass oscillations 338 145 Weak coupling 339 146 General analytic theory for coupled linear oscillators 341 1461 Kinetic energy tensor T 341 1462 Potential energy tensor V 342 1463 Equations ofmotion 343 1464 Superposition 344 1465 Eigenfunction orthonormality 344 1466 Normal coordinates 345 147 Two-body coupled oscillator systems 346 148 Three-body coupled linear oscillator systems 352 149 Molecular coupled oscillator systems 357 1410Discrete Lattice Chain 360 14101Longitudinalmotion 360 14102Transversemotion 360 14103 Normalmodes 361 14104Travelling waves 364 14105Dispersion 364 14106Complex wavenumber 365 1411Damped coupled linear oscillators 366 1412Collective synchronization of coupled oscillators 367 1413Summary 370 15 Advanced Hamiltonian mechanics 373 151 Introduction 373 152 Poisson bracket representation of Hamiltonianmechanics 375 1521 Poisson Brackets 375 1522 Fundamental Poisson brackets: 375 1523 Poisson bracket invariance to canonical transformations 376 1524 Correspondence of the commutator and the Poisson Bracket 377 1525 Observables in Hamiltonianmechanics 378 1526 Hamilton’s equations ofmotion 381 1527 Liouville’s Theorem 385 153 Canonical transformations in Hamiltonianmechanics 387 1531 Generating functions 388 153