24小时热门版块排行榜

返回列表

【奖励】本帖被评价56次，作者pkusiyuan增加金币 44.2 个

pkusiyuan

银虫 (正式写手)

应助: 0 (幼儿园)
金币: 15204.9
帖子: 773
在线: 130小时
虫号: 3451204

[资源] Oxford2006Analytical Mechanics

Contents
1 Geometric and kinematic foundations
of Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Curves in the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Length of a curve and natural parametrisation . . . . . . . . . . 3
1.3 Tangent vector, normal vector and curvature
of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Curves in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Vector fields and integral curves . . . . . . . . . . . . . . . . . . . . 15
1.6 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Differentiable Riemannian manifolds . . . . . . . . . . . . . . . . . 33
1.8 Actions of groups and tori . . . . . . . . . . . . . . . . . . . . . . . . 46
1.9 Constrained systems and Lagrangian coordinates . . . . . . . . . 49
1.10 Holonomic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.11 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.12 Accelerations of a holonomic system . . . . . . . . . . . . . . . . . 57
1.13 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.14 Additional remarks and bibliographical notes . . . . . . . . . . . 61
1.15 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 62
2 Dynamics: general laws and the dynamics
of a point particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.1 Revision and comments on the axioms of classical mechanics . 69
2.2 The Galilean relativity principle and interaction forces . . . . . 71
2.3 Work and conservative fields . . . . . . . . . . . . . . . . . . . . . . 75
2.4 The dynamics of a point constrained by smooth holonomic
constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.5 Constraints with friction . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.6 Point particle subject to unilateral constraints . . . . . . . . . . . 81
2.7 Additional remarks and bibliographical notes . . . . . . . . . . . 83
2.8 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 83
3 One-dimensional motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2 Analysis of motion due to a positional force . . . . . . . . . . . . 92
3.3 The simple pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4 Phase plane and equilibrium . . . . . . . . . . . . . . . . . . . . . . 98
3.5 Damped oscillations, forced oscillations. Resonance . . . . . . . . 103
3.6 Beats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.8 Additional remarks and bibliographical notes . . . . . . . . . . . 112
3.9 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 113
viii Contents
4 The dynamics of discrete systems. Lagrangian formalism . . . . . . 125
4.1 Cardinal equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.2 Holonomic systems with smooth constraints . . . . . . . . . . . . 127
4.3 Lagrange’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4 Determination of constraint reactions. Constraints
with friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.5 Conservative systems. Lagrangian function . . . . . . . . . . . . . 138
4.6 The equilibrium of holonomic systems
with smooth constraints . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.7 Generalised potentials. Lagrangian of
an electric charge in an electromagnetic field . . . . . . . . . . . . 142
4.8 Motion of a charge in a constant
electric or magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.9 Symmetries and conservation laws.
Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.10 Equilibrium, stability and small oscillations . . . . . . . . . . . . 150
4.11 Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.13 Additional remarks and bibliographical notes . . . . . . . . . . . 165
4.14 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 165
5 Motion in a central field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.1 Orbits in a central field . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.2 Kepler’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.3 Potentials admitting closed orbits . . . . . . . . . . . . . . . . . . . 187
5.4 Kepler’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.5 The Lagrange formula . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.6 The two-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.7 The n-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.9 Additional remarks and bibliographical notes . . . . . . . . . . . 207
5.10 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 208
6 Rigid bodies: geometry and kinematics . . . . . . . . . . . . . . . . . . 213
6.1 Geometric properties. The Euler angles . . . . . . . . . . . . . . . 213
6.2 The kinematics of rigid bodies. The
fundamental formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
6.3 Instantaneous axis of motion . . . . . . . . . . . . . . . . . . . . . . 219
6.4 Phase space of precessions . . . . . . . . . . . . . . . . . . . . . . . . 221
6.5 Relative kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6.6 Relative dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6.7 Ruled surfaces in a rigid motion . . . . . . . . . . . . . . . . . . . . 228
6.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
6.9 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 231
7 The mechanics of rigid bodies: dynamics . . . . . . . . . . . . . . . . . 235
7.1 Preliminaries: the geometry of masses . . . . . . . . . . . . . . . . 235
7.2 Ellipsoid and principal axes of inertia . . . . . . . . . . . . . . . . 236
Contents ix
7.3 Homography of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.4 Relevant quantities in the dynamics
of rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.5 Dynamics of free systems . . . . . . . . . . . . . . . . . . . . . . . . 244
7.6 The dynamics of constrained rigid bodies . . . . . . . . . . . . . . 245
7.7 The Euler equations for precessions . . . . . . . . . . . . . . . . . . 250
7.8 Precessions by inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.9 Permanent rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
7.10 Integration of Euler equations . . . . . . . . . . . . . . . . . . . . . 256
7.11 Gyroscopic precessions . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.12 Precessions of a heavy gyroscope
(spinning top) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.13 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.14 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.15 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 266
8 Analytical mechanics: Hamiltonian formalism . . . . . . . . . . . . . . 279
8.1 Legendre transformations . . . . . . . . . . . . . . . . . . . . . . . . 279
8.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
8.3 Hamilton’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.4 Liouville’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.5 Poincar′e recursion theorem . . . . . . . . . . . . . . . . . . . . . . . 287
8.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.7 Additional remarks and bibliographical notes . . . . . . . . . . . 291
8.8 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 291
9 Analytical mechanics: variational principles . . . . . . . . . . . . . . . . 301
9.1 Introduction to the variational problems
of mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
9.2 The Euler equations for stationary functionals . . . . . . . . . . . 302
9.3 Hamilton’s variational principle: Lagrangian form . . . . . . . . 312
9.4 Hamilton’s variational principle: Hamiltonian form . . . . . . . . 314
9.5 Principle of the stationary action . . . . . . . . . . . . . . . . . . . 316
9.6 The Jacobi metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
9.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
9.8 Additional remarks and bibliographical notes . . . . . . . . . . . 324
9.9 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . . 324
10 Analytical mechanics: canonical formalism . . . . . . . . . . . . . . . . 331
10.1 Symplectic structure of the Hamiltonian phase space . . . . . . 331
10.2 Canonical and completely canonical transformations . . . . . . . 340
10.3 The Poincar′e–Cartan integral invariant.
The Lie condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
10.4 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
10.5 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
10.6 Lie derivatives and commutators . . . . . . . . . . . . . . . . . . . . 374
10.7 Symplectic rectification . . . . . . . . . . . . . . . . . . . . . . . . . . 380
x Contents
10.8 Infinitesimal and near-to-identity canonical
transformations. Lie series . . . . . . . . . . . . . . . . . . . . . . . 384
10.9 Symmetries and first integrals . . . . . . . . . . . . . . . . . . . . . 393
10.10 Integral invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
10.11 Symplectic manifolds and Hamiltonian
dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
10.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
10.13 Additional remarks and bibliographical notes . . . . . . . . . . . 404
10.14 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 405
11 Analytic mechanics: Hamilton–Jacobi theory
and integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
11.1 The Hamilton–Jacobi equation . . . . . . . . . . . . . . . . . . . . 413
11.2 Separation of variables for the
Hamilton–Jacobi equation . . . . . . . . . . . . . . . . . . . . . . . 421
11.3 Integrable systems with one degree of freedom:
action-angle variables . . . . . . . . . . . . . . . . . . . . . . . . . . 431
11.4 Integrability by quadratures. Liouville’s theorem . . . . . . . . 439
11.5 Invariant l-dimensional tori. The theorem of Arnol’d . . . . . . 446
11.6 Integrable systems with several degrees of freedom:
action-angle variables . . . . . . . . . . . . . . . . . . . . . . . . . . 453
11.7 Quasi-periodic motions and functions . . . . . . . . . . . . . . . . 458
11.8 Action-angle variables for the Kepler problem.
Canonical elements, Delaunay and Poincar′e variables . . . . . 466
11.9 Wave interpretation of mechanics . . . . . . . . . . . . . . . . . . 471
11.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
11.11 Additional remarks and bibliographical notes . . . . . . . . . . . 480
11.12 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 481
12 Analytical mechanics: canonical
perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
12.1 Introduction to canonical perturbation theory . . . . . . . . . . 487
12.2 Time periodic perturbations of one-dimensional uniform
motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
12.3 The equation Dωu = v. Conclusion of the
previous analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
12.4 Discussion of the fundamental equation
of canonical perturbation theory. Theorem of Poincar′e on the
non-existence of first integrals of the motion . . . . . . . . . . . 507
12.5 Birkhoff series: perturbations of harmonic oscillators . . . . . 516
12.6 The Kolmogorov–Arnol’d–Moser theorem . . . . . . . . . . . . . 522
12.7 Adiabatic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
12.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Contents xi
12.9 Additional remarks and bibliographical notes . . . . . . . . . . . 534
12.10 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 535
13 Analytical mechanics: an introduction to
ergodic theory and to chaotic motion . . . . . . . . . . . . . . . . . . . 545
13.1 The concept of measure . . . . . . . . . . . . . . . . . . . . . . . . . 545
13.2 Measurable functions. Integrability . . . . . . . . . . . . . . . . . 548
13.3 Measurable dynamical systems . . . . . . . . . . . . . . . . . . . . 550
13.4 Ergodicity and frequency of visits . . . . . . . . . . . . . . . . . . 554
13.5 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
13.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565
13.7 Computation of the entropy. Bernoulli schemes.
Isomorphism of dynamical systems . . . . . . . . . . . . . . . . . 571
13.8 Dispersive billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
13.9 Characteristic exponents of Lyapunov.
The theorem of Oseledec . . . . . . . . . . . . . . . . . . . . . . . . 578
13.10 Characteristic exponents and entropy . . . . . . . . . . . . . . . . 581
13.11 Chaotic behaviour of the orbits of planets
in the Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
13.12 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
13.13 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 586
13.14 Additional remarks and bibliographical notes . . . . . . . . . . . 590
14 Statistical mechanics: kinetic theory . . . . . . . . . . . . . . . . . . . . 591
14.1 Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . 591
14.2 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . 592
14.3 The hard spheres model . . . . . . . . . . . . . . . . . . . . . . . . 596
14.4 The Maxwell–Boltzmann distribution . . . . . . . . . . . . . . . . 599
14.5 Absolute pressure and absolute temperature
in an ideal monatomic gas . . . . . . . . . . . . . . . . . . . . . . . 601
14.6 Mean free path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604
14.7 The ‘H theorem’ of Boltzmann. Entropy . . . . . . . . . . . . . . 605
14.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
14.9 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 610
14.10 Additional remarks and bibliographical notes . . . . . . . . . . . 611
15 Statistical mechanics: Gibbs sets . . . . . . . . . . . . . . . . . . . . . . . 613
15.1 The concept of a statistical set . . . . . . . . . . . . . . . . . . . . 613
15.2 The ergodic hypothesis: averages and
measurements of observable quantities . . . . . . . . . . . . . . . 616
15.3 Fluctuations around the average . . . . . . . . . . . . . . . . . . . 620
15.4 The ergodic problem and the existence of first integrals . . . . 621
15.5 Closed isolated systems (prescribed energy).
Microcanonical set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
xii Contents
15.6 Maxwell–Boltzmann distribution and fluctuations
in the microcanonical set . . . . . . . . . . . . . . . . . . . . . . . . 627
15.7 Gibbs’ paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
15.8 Equipartition of the energy (prescribed total energy) . . . . . . 634
15.9 Closed systems with prescribed temperature.
Canonical set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
15.10 Equipartition of the energy (prescribed temperature) . . . . . 640
15.11 Helmholtz free energy and orthodicity
of the canonical set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
15.12 Canonical set and energy fluctuations . . . . . . . . . . . . . . . . 646
15.13 Open systems with fixed temperature.
Grand canonical set . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
15.14 Thermodynamical limit. Fluctuations
in the grand canonical set . . . . . . . . . . . . . . . . . . . . . . . 651
15.15 Phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
15.16 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
15.17 Additional remarks and bibliographical notes . . . . . . . . . . . 659
15.18 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 662
16 Lagrangian formalism in continuum mechanics . . . . . . . . . . . . . 671
16.1 Brief summary of the fundamental laws of
continuum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 671
16.2 The passage from the discrete to the continuous model. The
Lagrangian function . . . . . . . . . . . . . . . . . . . . . . . . . . . 676
16.3 Lagrangian formulation of continuum mechanics . . . . . . . . . 678
16.4 Applications of the Lagrangian formalism to continuum
mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 680
16.5 Hamiltonian formalism . . . . . . . . . . . . . . . . . . . . . . . . . 684
16.6 The equilibrium of continua as a variational problem.
Suspended cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
16.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
16.8 Additional solved problems . . . . . . . . . . . . . . . . . . . . . . 691
Appendices
Appendix 1: Some basic results on ordinary
differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
A1.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
A1.2 Systems of equations with constant coefficients . . . . . . . . 697
A1.3 Dynamical systems on manifolds . . . . . . . . . . . . . . . . . . 701
Appendix 2: Elliptic integrals and elliptic functions . . . . . . . . . . . 705
Appendix 3: Second fundamental form of a surface . . . . . . . . . . . 709
Appendix 4: Algebraic forms, differential forms, tensors . . . . . . . . 715
A4.1 Algebraic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
A4.2 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719
A4.3 Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
A4.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726
Contents xiii
Appendix 5: Physical realisation of constraints . . . . . . . . . . . . . . 729
Appendix 6: Kepler’s problem, linear oscillators
and geodesic flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
Appendix 7: Fourier series expansions . . . . . . . . . . . . . . . . . . . . 741
Appendix 8: Moments of the Gaussian distribution
and the Euler Γ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759

回复此楼

» 本帖附件资源列表

欢迎监督和反馈：小木虫仅提供交流平台，不对该内容负责。
本内容由用户自主发布，如果其内容涉及到知识产权问题，其责任在于用户本人，如对版权有异议，请联系邮箱：xiaomuchong@tal.com
附件 1 : Mathematics_-_Analytical_Mechanics_-_An_Introduction_(Oxford_Graduate_Texts,_Antonio_Fasano,_Stefano_Marmi,_Oxford_University_Press,_2006).pdf

2015-03-02 11:44:24, 3.39 M