ELEMENTARY DYNAMICS OF PARTICLES AND SOLIDS
1 Mechanical systems with one degree of freedom 1 11 Newton’s laws and qualitative characterization of motion on a line 1 12 Time period of conservative oscillations between turning points 7 13* Inverse problem: determination of potential from time period 8 14* Time delay in unbounded ‘scattering’ trajectories 10 15 Simple pendulum: basic properties and small oscillations 13 16 Problems 17 2 Kepler’s gravitational two-body problem 21 21 Inverse problem: universal law of gravity from Kepler’s laws 22 22 Direct problem: center of mass & relative vectors and conservation laws 26 23 Planetary orbits 29 24 Time period of elliptical orbits 33 25* LRL vector and relations among conserved quantities 34 26* Collision of two gravitating point masses 38 27* Rutherford scattering cross section 42 28* The three-body problem: Euler and Lagrange solutions 49 29 Problems 51 3 Newtonian to Lagrangian and Hamiltonian mechanics 55 31 Time, space, light, simultaneity, causality, homogeneity and isotropy 55 32 Degrees of freedom and instantaneous configurations 56 33 Newton’s laws and Galileo’s relativity and equivalence principles 57 34 Phase space, dynamical variables, conserved quantities, collisions 65 35 Principle of extremal action and Euler-Lagrange equations 68 36 Nonuniqueness of Lagrangian 74 37 Conjugate momenta, their geometric meaning and cyclic coordinates 74 38 Coordinate invariance of the form of Lagrange’s equations 76 39 Hamiltonian and its conservation 77 310 Symmetries to conserved quantities: Noether’s theorem 79 311* Noether’s theorem when Lagrangian changes by a time derivative 83 312 Inertial frames of reference and Galilean invariance 84 313* Polar vectors, axial vectors, true scalars and pseudoscalars 89 314 Hamilton’s equations 91 315 Legendre transform: Hamiltonian from Lagrangian 93 316* Lagrange multipliers and constrained extremization 96 317* Singular Lagrangians and constraints 98 318* Action as a function along a trajectory 100 319 Variational principles for Hamilton’s equations 102 320* Coordinate invariance of Lagrange and Hamilton equations 104 321 Canonical Poisson brackets 105 CONTENTS 322 Properties of the Poisson bracket 107 323* Canonical formulation of charged particle in electromagnetic field 111 324* Poisson algebra of conserved quantities in the Kepler problem 113 325* Functional independence of conserved quantities 116 326* Noncanonical Poisson brackets, Poisson and symplectic manifolds 121 327* Free particle trajectories as geodesics on configuration space 127 328* Euler-Maupertuis principle and the Jacobi-Maupertuis metric 131 329 Problems 134 4 Introduction to special relativistic mechanics 149 41 Difficulties with Newtonian mechanics 149 42 Postulates of special relativity 150 43 Synchronization of clocks and simultaneity 151 44 Lorentz transformations 153 45 Time dilation, length contraction, proper length and time 156 46 Space-like, time-like and light-like intervals and causality 158 47 Relativistic addition of velocities 159 48 Relativistic momentum from two particle collision 160 49 Relativistic energy and energy-momentum dispersion relation 162 410* Minkowski space-time and relativistic dynamics 163 411 Problems 169 5 Dynamics viewed as a vector field on state space 171 51 Vector fields from Newtonian and Hamiltonian dynamics 171 52* Vector fields in one dimension 175 53 Existence and uniqueness of solutions 180 54* Vector fields on the phase plane 184 55 Problems 192 6 Small oscillations for one degree of freedom 195 61 Linear harmonic oscillator in 1d and neutral stability 195 62 Linear vector fields on the phase plane 197 63 Phase portrait from spectrum of coefficient matrix 199 64 Damped harmonic oscillator: view from the phase plane 201 65 Critically damped oscillator: deficient coefficient matrix 205 66* Trace-determinant classification of linear fixed points 207 67* Robustness of the linear theory 209 68 Driven or forced oscillations 210 69* Driven damped oscillations 217 610 Parametric oscillations and resonant amplification 218 611 Problems 219 7 Nonlinear oscillations: pendulum and anharmonic oscillator 225 71 Simple pendulum: view from phase space 225 72* Introduction to Jacobi elliptic functions 229 73* Time-dependence of pendulum in terms of elliptic functions 232 74 Anharmonic oscillations: quartic double-well potential 234 75* Quartic oscillator: exact solution and Lindstedt-Poincar´e method 236 76 Problems 240 8 Rigid body mechanics 243 81 Lab and comoving frames 245 82 Configuration space and degrees of freedom 245 83 Infinitesimal displacement and angular velocity of rigid body 246 84 Kinetic energy and inertia tensor 249 85 Types of rigid bodies 251 86 Angular momentum of a rigid body 253 87 Equations of motion of a rigid body 254 88 Force-free motion of rigid bodies 257 89 Euler angles and rotations 261 810 Angular velocity and kinetic energy in terms of Euler angles 264 811 Euler equations for a rigid body in body-fixed frame 267 812 Ellipsoid of inertia and qualitative description of free motion 270 813* Solution of force-free Euler equations via elliptic functions 275 814* Poisson bracket formulation of Euler’s equations 277 815 Motion of a heavy symmetrical (Lagrange) top 280 816 Problems 283 9 Motion in noninertial frames of reference 289 91 Uniformly accelerating frames and the equivalence principle 290 92 Nonuniformly accelerated frames: Lagrangian approach 291 93 Uniform rotation: Hamiltonian formulation and magnetic analog 294 94 Precession of Foucault’s pendulum 295 95* Circular restricted three-body problem 296 96 Problems 298 10 Canonical transformations 301 101 From point transformations to canonical transformations 303 102 Preservation of Hamilton’s equations and Poisson brackets 305 103* Comparison of classical and quantum mechanical formalisms 308 104 Canonical transformations and area-preserving maps 311 105 Canonical transformations preserve Poisson tensor 314 106 Generating function for infinitesimal canonical transformations 316 107 Symmetries & Noether’s theorem in the Hamiltonian framework 319 108 Liouville’s theorem 321 109* Poincar´e recurrence 324 1010 Generating functions for finite canonical transformations 329 1011 Problems 335 11 Angle-action variables 341 111 Angle-action variables for the harmonic oscillator 342 112 Generator of CT to angle-action variables: Hamilton-Jacobi equation 344 113 Generating function for oscillator angle-action variables 346 114 Angle-action variables for systems with one degree of freedom 347 115* Angle-action variables for libration of the simple pendulum 348 116* Bohr-Sommerfeld quantization rule 349 117* Liouville integrability and KAM tori 351 118* Liouville-Arnold theorem 353 119* Conserved quantities from a Lax pair 357 1191* Harmonic oscillator Lax pair 357 1192* Isospectral evolution and conserved quantities 358 1193* Toda chain: Flaschka’s variables and a Lax pair 359 1194* Euler-Poinsot top Lax pair: spectral parameter 361 1110 Problems 362 12 Hamilton-Jacobi equation 365 121 Time-dependent Hamilton-Jacobi evolution equation 367 122* Connection of Hamilton-Jacobi to Schr¨odinger and eikonal equations 370 123 Separation of variables in the Hamilton-Jacobi equation 372 124 Hamilton’s principal function as action along a trajectory 376 125* Geometric interpretation of Hamilton-Jacobi equation 377 126 Problems 380 13 Normal modes of oscillation and linear stability 383 131 Elementary examples of coupled small oscillations 384 1311 Normal modes of two weakly coupled pendula 384 1312 Normal modes of a diatomic molecule 386 132 Double pendulum: formulation and small oscillations 387 1321 Energy, Lagrangian and equations of motion 388 1322 Normal modes of a double pendulum 390 133* Normal modes around a static equilibrium: general framework 396 134 Small perturbations around a periodic solution 401 1341 Formulation as a system of first order equations 403 1342* Time evolution matrix 404 1343* Monodromy matrix 408 1344* Stability of a periodic solution 408 1345* Kapitza pendulum with oscillating support: Mathieu equation 410 135 Problems 413 14 Bifurcations: qualitative changes in dynamics 417 141 Bifurcations of vector fields on the real line 417 1411* Saddle-node bifurcation 418 1412* Transcritical bifurcation 419 1413* Pitchfork bifurcations 420 14131* Supercritical pitchfork bifurcation 421 14132* Subcritical pitchfork bifurcation 422 142 Bifurcations in two dimensions 425 1421* Saddle-node, transcritical and pitchfork bifurcations 425 1422* Hopf bifurcations 427 143 Problems 430