Analytical Mechanics: An Introduction

Contents......Page 8
1.1 Curves in the plane......Page 16
1.2 Length of a curve and natural parametrisation......Page 18
1.3 Tangent vector, normal vector and curvature of plane curves......Page 22
1.4 Curves in R[sup(3)]......Page 27
1.5 Vector fields and integral curves......Page 30
1.6 Surfaces......Page 31
1.7 Differentiable Riemannian manifolds......Page 48
1.8 Actions of groups and tori......Page 61
1.9 Constrained systems and Lagrangian coordinates......Page 64
1.10 Holonomic systems......Page 67
1.11 Phase space......Page 69
1.12 Accelerations of a holonomic system......Page 72
1.13 Problems......Page 73
1.14 Additional remarks and bibliographical notes......Page 76
1.15 Additional solved problems......Page 77
2.1 Revision and comments on the axioms of classical mechanics......Page 84
2.2 The Galilean relativity principle and interaction forces......Page 86
2.3 Work and conservative fields......Page 90
2.4 The dynamics of a point constrained by smooth holonomic constraints......Page 92
2.5 Constraints with friction......Page 95
2.6 Point particle subject to unilateral constraints......Page 96
2.8 Additional solved problems......Page 98
3.1 Introduction......Page 106
3.2 Analysis of motion due to a positional force......Page 107
3.3 The simple pendulum......Page 111
3.4 Phase plane and equilibrium......Page 113
3.5 Damped oscillations, forced oscillations. Resonance......Page 118
3.6 Beats......Page 122
3.7 Problems......Page 123
3.8 Additional remarks and bibliographical notes......Page 127
3.9 Additional solved problems......Page 128
4.1 Cardinal equations......Page 140
4.2 Holonomic systems with smooth constraints......Page 142
4.3 Lagrange’s equations......Page 143
4.4 Determination of constraint reactions. Constraints with friction......Page 151
4.5 Conservative systems. Lagrangian function......Page 153
4.6 The equilibrium of holonomic systems with smooth constraints......Page 156
4.7 Generalised potentials. Lagrangian of an electric charge in an electromagnetic field......Page 157
4.8 Motion of a charge in a constant electric or magnetic field......Page 159
4.9 Symmetries and conservation laws. Noether’s theorem......Page 162
4.10 Equilibrium, stability and small oscillations......Page 165
4.11 Lyapunov functions......Page 174
4.12 Problems......Page 177
4.14 Additional solved problems......Page 180
5.1 Orbits in a central field......Page 194
5.2 Kepler’s problem......Page 200
5.3 Potentials admitting closed orbits......Page 202
5.4 Kepler’s equation......Page 208
5.5 The Lagrange formula......Page 212
5.6 The two-body problem......Page 215
5.7 The n-body problem......Page 216
5.8 Problems......Page 220
5.9 Additional remarks and bibliographical notes......Page 222
5.10 Additional solved problems......Page 223
6.1 Geometric properties. The Euler angles......Page 228
6.2 The kinematics of rigid bodies. The fundamental formula......Page 231
6.3 Instantaneous axis of motion......Page 234
6.4 Phase space of precessions......Page 236
6.5 Relative kinematics......Page 238
6.6 Relative dynamics......Page 241
6.7 Ruled surfaces in a rigid motion......Page 243
6.8 Problems......Page 245
6.9 Additional solved problems......Page 246
7.1 Preliminaries: the geometry of masses......Page 250
7.2 Ellipsoid and principal axes of inertia......Page 251
7.3 Homography of inertia......Page 254
7.4 Relevant quantities in the dynamics of rigid bodies......Page 257
7.5 Dynamics of free systems......Page 259
7.6 The dynamics of constrained rigid bodies......Page 260
7.7 The Euler equations for precessions......Page 265
7.8 Precessions by inertia......Page 266
7.9 Permanent rotations......Page 269
7.10 Integration of Euler equations......Page 271
7.11 Gyroscopic precessions......Page 274
7.12 Precessions of a heavy gyroscope (spinning top)......Page 276
7.13 Rotations......Page 278
7.14 Problems......Page 280
7.15 Additional solved problems......Page 281
8.1 Legendre transformations......Page 294
8.2 The Hamiltonian......Page 297
8.3 Hamilton’s equations......Page 299
8.4 Liouville’s theorem......Page 300
8.5 Poincaré recursion theorem......Page 302
8.6 Problems......Page 303
8.8 Additional solved problems......Page 306
9.1 Introduction to the variational problems of mechanics......Page 316
9.2 The Euler equations for stationary functionals......Page 317
9.3 Hamilton’s variational principle: Lagrangian form......Page 327
9.4 Hamilton’s variational principle: Hamiltonian form......Page 329
9.5 Principle of the stationary action......Page 331
9.6 The Jacobi metric......Page 333
9.7 Problems......Page 338
9.9 Additional solved problems......Page 339
10.1 Symplectic structure of the Hamiltonian phase space......Page 346
10.2 Canonical and completely canonical transformations......Page 355
10.3 The Poincaré–Cartan integral invariant. The Lie condition......Page 367
10.4 Generating functions......Page 379
10.5 Poisson brackets......Page 386
10.6 Lie derivatives and commutators......Page 389
10.7 Symplectic rectification......Page 395
10.8 Infinitesimal and near-to-identity canonical transformations. Lie series......Page 399
10.9 Symmetries and first integrals......Page 408
10.10 Integral invariants......Page 410
10.11 Symplectic manifolds and Hamiltonian dynamical systems......Page 412
10.12 Problems......Page 414
10.13 Additional remarks and bibliographical notes......Page 419
10.14 Additional solved problems......Page 420
11.1 The Hamilton–Jacobi equation......Page 428
11.2 Separation of variables for the Hamilton–Jacobi equation......Page 436
11.3 Integrable systems with one degree of freedom: action-angle variables......Page 446
11.4 Integrability by quadratures. Liouville’s theorem......Page 454
11.5 Invariant l-dimensional tori. The theorem of Arnol’d......Page 461
11.6 Integrable systems with several degrees of freedom: action-angle variables......Page 468
11.7 Quasi-periodic motions and functions......Page 473
11.8 Action-angle variables for the Kepler problem. Canonical elements, Delaunay and Poincaré variables......Page 481
11.9 Wave interpretation of mechanics......Page 486
11.10 Problems......Page 492
11.11 Additional remarks and bibliographical notes......Page 495
11.12 Additional solved problems......Page 496
12.1 Introduction to canonical perturbation theory......Page 502
12.2 Time periodic perturbations of one-dimensional uniform motions......Page 514
12.3 The equation D[sub(ω)]u = v. Conclusion of the previous analysis......Page 517
12.4 Discussion of the fundamental equation of canonical perturbation theory. Theorem of Poincaré on the non-existence of first integrals of the motion......Page 522
12.5 Birkhoff series: perturbations of harmonic oscillators......Page 531
12.6 The Kolmogorov–Arnol’d–Moser theorem......Page 537
12.7 Adiabatic invariants......Page 544
12.8 Problems......Page 547
12.9 Additional remarks and bibliographical notes......Page 549
12.10 Additional solved problems......Page 550
13.1 The concept of measure......Page 560
13.2 Measurable functions. Integrability......Page 563
13.3 Measurable dynamical systems......Page 565
13.4 Ergodicity and frequency of visits......Page 569
13.5 Mixing......Page 578
13.6 Entropy......Page 580
13.7 Computation of the entropy. Bernoulli schemes. Isomorphism of dynamical systems......Page 586
13.8 Dispersive billiards......Page 590
13.9 Characteristic exponents of Lyapunov. The theorem of Oseledec......Page 593
13.10 Characteristic exponents and entropy......Page 596
13.11 Chaotic behaviour of the orbits of planets in the Solar System......Page 597
13.12 Problems......Page 599
13.13 Additional solved problems......Page 601
13.14 Additional remarks and bibliographical notes......Page 605
14.1 Distribution functions......Page 606
14.2 The Boltzmann equation......Page 607
14.3 The hard spheres model......Page 611
14.4 The Maxwell–Boltzmann distribution......Page 614
14.5 Absolute pressure and absolute temperature in an ideal monatomic gas......Page 616
14.6 Mean free path......Page 619
14.7 The ‘H theorem’ of Boltzmann. Entropy......Page 620
14.8 Problems......Page 624
14.9 Additional solved problems......Page 625
14.10 Additional remarks and bibliographical notes......Page 626
15.1 The concept of a statistical set......Page 628
15.2 The ergodic hypothesis: averages and measurements of observable quantities......Page 631
15.3 Fluctuations around the average......Page 635
15.4 The ergodic problem and the existence of first integrals......Page 636
15.5 Closed isolated systems (prescribed energy). Microcanonical set......Page 639
15.6 Maxwell–Boltzmann distribution and fluctuations in the microcanonical set......Page 642
15.7 Gibbs’ paradox......Page 646
15.8 Equipartition of the energy (prescribed total energy)......Page 649
15.9 Closed systems with prescribed temperature. Canonical set......Page 651
15.10 Equipartition of the energy (prescribed temperature)......Page 655
15.11 Helmholtz free energy and orthodicity of the canonical set......Page 660
15.12 Canonical set and energy fluctuations......Page 661
15.13 Open systems with fixed temperature. Grand canonical set......Page 662
15.14 Thermodynamical limit. Fluctuations in the grand canonical set......Page 666
15.15 Phase transitions......Page 669
15.16 Problems......Page 671
15.17 Additional remarks and bibliographical notes......Page 674
15.18 Additional solved problems......Page 677
16.1 Brief summary of the fundamental laws of continuum mechanics......Page 686
16.2 The passage from the discrete to the continuous model. The Lagrangian function......Page 691
16.3 Lagrangian formulation of continuum mechanics......Page 693
16.4 Applications of the Lagrangian formalism to continuum mechanics......Page 695
16.5 Hamiltonian formalism......Page 699
16.6 The equilibrium of continua as a variational problem. Suspended cables......Page 700
16.7 Problems......Page 705
16.8 Additional solved problems......Page 706
A1.1 General results......Page 710
A1.2 Systems of equations with constant coeffcients......Page 712
A1.3 Dynamical systems on manifolds......Page 716
Appendix 2: Elliptic integrals and elliptic functions......Page 720
Appendix 3: Second fundamental form of a surface......Page 724
A4.1 Algebraic forms......Page 730
A4.2 Differential forms......Page 734
A4.3 Stokes’ theorem......Page 739
A4.4 Tensors......Page 741
Appendix 5: Physical realisation of constraints......Page 744
Appendix 6: Kepler’s problem, linear oscillators and geodesic......Page 748
Appendix 7: Fourier series expansions......Page 756
Appendix 8: Moments of the Gaussian distribution and the Euler Γ function......Page 760
Bibliography......Page 764
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