THE PROJECT PHYSICS COURSE: CONCEPTS OF MOTION

By EMMA WHITE

1 A brief history of classical mechanics 1 11 Introduction 1 12 Prehistoric astronomy 1 13 Greek antiquity 1 14 Middle Ages 2 15 Age of Enlightenment 3 16 19?? century 5 17 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 Workshop exercises 48 Problems 51 3 Linear oscillators 53 31 Introduction 53 32 Linear restoring forces 53 33 Linearity and superposition 54 34 Geometrical representations of dynamical motion 55 341 Configuration space (??? ?? ? ?) 55 342 State space, (??? ?????) 56 343 Phase space, (??? ????) 56 344 Plane pendulum 57 35 Linearly-damped free linear oscillator 58 351 General solution 58 352 Energy dissipation 61 36 Sinusoidally-drive, linearly-damped, linear oscillator 62 361 Transient response of a driven oscillator 62 362 Steady state response of a driven oscillator 63 363 Complete solution of the driven oscillator 64 364 Resonance 65 365 Energy absorption 65 37 Wave equation 68 38 Travelling and standing wave solutions of the wave equation 69 39 Waveformanalysis 70 391 Harmonic decomposition 70 392 The free linearly-damped linear oscillator 70 393 Damped linear oscillator subject to an arbitrary periodic force 71 310 Signal processing 72 311 Wave propagation 73 3111 Phase, group, and signal velocities of wave packets 74 3112 Fourier transformof wave packets 79 3113 Wave-packet Uncertainty Principle 80 312 Summary 82 Workshop exercises 85 Problems 88 4 Nonlinear systems and chaos 89 41 Introduction 89 42 Weak nonlinearity 90 43 Bifurcation, and point attractors 92 44 Limit cycles 93 441 Poincaré-Bendixson theorem 93 442 van der Pol damped harmonic oscillator: 94 45 Harmonically-driven, linearly-damped, plane pendulum 97 451 Close to linearity 97 452 Weak nonlinearity 99 453 Onset of complication 100 454 Period doubling and bifurcation 100 455 Rolling motion 100 456 Onset of chaos 101 46 Differentiation between ordered and chaoticmotion 102 461 Lyapunov exponent 102 462 Bifurcation diagram 103 463 Poincaré Section 104 47 Wave propagation for non-linear systems 105 471 Phase, group, and signal velocities 105 472 Soliton wave propagation 107 48 Summary 108 Workshop exercises 110 Problems 110 5 Calculus of variations 111 51 Introduction 111 52 Euler’s differential equation 112 53 Applications of Euler’s equation 114 54 Selection of the independent variable 117 55 Functions with several independent variables ??(?) 119 56 Euler’s integral equation 121 57 Constrained variational systems 122 571 Holonomic constraints 122 572 Geometric (algebraic) equations of constraint 122 573 Kinematic (differential) equations of constraint 122 574 Isoperimetric (integral) equations of constraint 123 575 Properties of the constraint equations 123 576 Treatment of constraint forces in variational calculus 124 58 Generalized coordinates in variational calculus 125 59 Lagrangemultipliers for holonomic constraints 126 591 Algebraic equations of constraint 126 592 Integral equations of constraint 128 510 Geodesic 130 511 Variational approach to classicalmechanics 131 512 Summary 132 Workshop exercises 133 Problems 134 6 Lagrangian dynamics 135 61 Introduction 135 62 Newtonian plausibility argument for Lagrangian mechanics 136 63 Lagrange equations fromd’Alembert’s Principle 138 631 d’Alembert’s Principle of virtual work 138 632 Transformation to generalized coordinates 139 633 Lagrangian 140 64 Lagrange equations fromHamilton’s Principle 141 65 Constrained systems 142 651 Choice of generalized coordinates 142 652 Minimal set of generalized coordinates 142 653 Lagrangemultipliers approach 142 654 Generalized forces approach 144 66 Applying the Euler-Lagrange equations to classicalmechanics 144 67 Applications to unconstrained systems 146 68 Applications to systems involving holonomic constraints 148 69 Applications involving non-holonomic constraints 161 610 Velocity-dependent Lorentz force 168 611 Time-dependent forces 169 612 Impulsive forces 170 613 The Lagrangian versus the Newtonian approach to classicalmechanics 172 614 Summary 173 Workshop exercises 176 Problems 178 7 Symmetries, Invariance and the Hamiltonian 179 71 Introduction 179 72 Generalizedmomentum 179 73 Invariant transformations and Noether’s Theorem 181 74 Rotational invariance and conservation of angularmomentum 183 75 Cyclic coordinates 184 76 Kinetic energy in generalized coordinates 185 77 Generalized energy and the Hamiltonian function 186 78 Generalized energy theorem 187 79 Generalized energy and total energy 187 710 Hamiltonian invariance 188 711 Hamiltonian for cyclic coordinates 193 712 Symmetries and invariance 193 713 Hamiltonian in classicalmechanics 193 714 Summary 194 Workshop exercises 196 Problems 197 8 Hamiltonian mechanics 199 81 Introduction 199 82 Legendre Transformation between Lagrangian and Hamiltonianmechanics 200 83 Hamilton’s equations ofmotion 201 831 Canonical equations ofmotion 202 84 Hamiltonian in different coordinate systems 203 841 Cylindrical coordinates ?? ?? ? 203 842 Spherical coordinates, ?? ?? ? 204 85 Applications of Hamiltonian Dynamics 205 86 Routhian reduction 210 861 R?????? - Routhian is a Hamiltonian for the cyclic variables 211 862 R????????? - Routhian is a Hamiltonian for the non-cyclic variables 212 87 Dissipative dynamical systems 216 871 Generalized drag force 216 872 Rayleigh’s dissipation function 217 88 Summary 221 Workshop exercises 223 Problems 224 9 Conservative two-body central forces 227 91 Introduction 227 92 Equivalent one-body representation for two-bodymotion 228 93 Angular momentum L 230 94 Equations of motion 231 95 Differential orbit equation: 232 96 Hamiltonian 233 97 General features of the orbit solutions 234 98 Inverse-square, two-body, central force 235 981 Bound orbits 236 982 Kepler’s laws for bound planetarymotion 237 983 Unbound orbits 238 984 Eccentricity vector 239 99 Isotropic, linear, two-body, central force 241 991 Polar coordinates 242 992 Cartesian coordinates 243 993 Symmetry tensor A0 244 910 Closed-orbit stability 245 911 The three-body problem 250 912 Two-body scattering 251 9121 Total two-body scattering cross section 251 9122 Differential two-body scattering cross section 252 9123 Impact parameter dependence on scattering angle 252 9124 Rutherford scattering 254 913 Two-body kinematics 256 914 Summary 262 Workshop exercises 264 Problems 265 10 Non-inertial reference frames 267 101 Introduction 267 102 Translational acceleration of a reference frame 267 103 Rotating reference frame 268 1031 Spatial time derivatives in a rotating, non-translating, reference frame 268 1032 General vector in a rotating, non-translating, reference frame 269 104 Reference frame undergoing rotation plus translation 270 105 Newton’s law ofmotion in a non-inertial frame 270 106 Lagrangianmechanics in a non-inertial frame 271 107 Centrifugal force 272 108 Coriolis force 273 109 Routhian reduction for rotating systems 277 1010Effective gravitational force near the surface of the Earth 280 1011Freemotion on the earth 282 1012Weather systems 284 10121Low-pressure systems: 284 10122High-pressure systems: 286 1013Foucault pendulum 286 1014Summary 288 Workshop exercises 289 Problems 290 11 Rigid-body rotation 291 111 Introduction 291 112 Rigid-body coordinates 292 113 Rigid-body rotation about a body-fixed point 292 114 Inertia tensor 294 115 Matrix and tensor formulations of rigid-body rotation 295 116 Principal axis system 295 117 Diagonalize the inertia tensor 296 118 Parallel-axis theorem 297 119 Perpendicular-axis theoremfor plane laminae 300 1110General properties of the inertia tensor 301 11101Inertial equivalence 301 11102Orthogonality of principal axes 302 1111Angular momentum L and angular velocity ? vectors 303 1112Kinetic energy of rotating rigid body 305 1113Euler angles 307 1114Angular velocity ? 309 1115Kinetic energy in terms of Euler angular velocities 310 1116Rotational invariants 311 1117Euler’s equations ofmotion for rigid-body rotation 312 1118Lagrange equations ofmotion for rigid-body rotation 313 1119Hamiltonian equations ofmotion for rigid-body rotation 315 1120Torque-free rotation of an inertially-symmetric rigid rotor 315 11201Euler’s equations ofmotion: 315 11202Lagrange equations ofmotion: 319 1121Torque-free rotation of an asymmetric rigid rotor 321 1122Stability of torque-free rotation of an asymmetric body 322 1123Symmetric rigid rotor subject to torque about a fixed point 325 1124The rolling wheel 329 1125Dynamic balancing of wheels 333 1126Rotation of deformable bodies 334 1127Summary 335 Workshop exercises 337 Problems 340 12 Coupled linear oscillators 341 121 Introduction 341 122 Two coupled linear oscillators 341 123 Normalmodes 343 124 Center of mass oscillations 344 125 Weak coupling 345 126 General analytic theory for coupled linear oscillators 347 1261 Kinetic energy tensor T 347 1262 Potential energy tensor V 348 1263 Equations ofmotion 349 1264 Superposition 350 1265 Eigenfunction orthonormality 350 1266 Normal coordinates 351 127 Two-body coupled oscillator systems 352 128 Three-body coupled linear oscillator systems 358 129 Molecular coupled oscillator systems 363 1210Discrete Lattice Chain 366 12101Longitudinalmotion 366 12102Transversemotion 366 12103 Normalmodes 367 12104Travelling waves 370 12105Dispersion 370 12106Complex wavenumber 371 1211Damped coupled linear oscillators 372 1212Collective synchronization of coupled oscillators 373 1213Summary 377 Workshop exercises 379 Problems 380 13 Hamilton’s principle of least action 381 131 Introduction 381 132 Principle of Least Action 382 1321 Hamilton’s Principle 382 1322 Least-action principle in Hamiltonianmechanics 383 1323 Abbreviated action 384 133 Standard Lagrangian 385 134 Gauge invariance of the Lagrangian 385 135 Non-standard Lagrangians 387 136 Inverse variational calculus 387 137