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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 |
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37楼2015-04-13 11:40:12
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49楼2015-10-06 07:07:06
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2015-03-02 13:42
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五星好评 顶一下，感谢分享！
Quan.3楼
2015-03-03 06:53
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五星好评 顶一下，感谢分享！
2015-03-03 08:31
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五星好评 顶一下，感谢分享！
askuyue5楼
2015-03-03 09:15
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五星好评 顶一下，感谢分享！
2015-03-03 10:16
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五星好评 顶一下，感谢分享！
2015-03-03 10:56
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五星好评 顶一下，感谢分享！
supervb8楼
2015-03-03 11:48
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五星好评 顶一下，感谢分享！
2015-03-03 13:39
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五星好评 顶一下，感谢分享！
kendrick10楼
2015-03-03 20:52
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五星好评 顶一下，感谢分享！
muxinjin11楼
2015-03-03 22:07
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五星好评 顶一下，感谢分享！
macier12楼
2015-03-03 23:01
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五星好评 顶一下，感谢分享！
jyhustb13楼
2015-03-04 03:24
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五星好评 顶一下，感谢分享！
parkzhu14楼
2015-03-04 03:37
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五星好评 顶一下，感谢分享！
wmnick15楼
2015-03-04 07:07
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五星好评 顶一下，感谢分享！
anmingkang16楼
2015-03-04 08:43
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五星好评 顶一下，感谢分享！
peterflyer17楼
2015-03-04 10:14
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五星好评 顶一下，感谢分享！
断肠人走天涯18楼
2015-03-04 11:00
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一般 顶一下，感谢分享！
skywalker00919楼
2015-03-04 13:32
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五星好评 顶一下，感谢分享！
nihaoweiming20楼
2015-03-04 19:25
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五星好评 顶一下，感谢分享！
zym100321楼
2015-03-05 08:31
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五星好评 顶一下，感谢分享！
cyberspecter22楼
2015-03-05 11:43
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五星好评 顶一下，感谢分享！
我为玛丽狂23楼
2015-03-05 12:37
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五星好评 顶一下，感谢分享！
RubikLee24楼
2015-03-05 12:50
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感谢分享 [ 发自小木虫客户端 ]
果雨柄25楼
2015-03-05 16:16
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三星好评 顶一下，感谢分享！
得意地笑26楼
2015-03-05 16:20
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五星好评 顶一下，感谢分享！
qqchun27楼
2015-03-05 17:36
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五星好评 顶一下，感谢分享！
1093128楼
2015-03-05 20:35
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五星好评 顶一下，感谢分享！
zhangxiyu19929楼
2015-03-31 20:49
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chuntao11830楼
2015-04-05 10:29
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五星好评 顶一下，感谢分享！
anandaho31楼
2015-04-06 19:19
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五星好评 顶一下，感谢分享！
dollapym32楼
2015-04-08 20:13
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fenggaol33楼
2015-04-08 21:04
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五星好评 顶一下，感谢分享！
hylpy34楼
2015-04-08 21:08
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五星好评 感谢分享
highlywood35楼
2015-04-08 22:36
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五星好评 顶一下，感谢分享！
mantou052036楼
2015-04-11 22:13
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五星好评 顶一下，感谢分享！
du2008112938楼
2015-04-20 21:46
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五星好评 顶一下，感谢分享！
dongyue20840楼
2015-04-21 08:13
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roubao41楼
2015-05-24 16:00
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五星好评 顶一下，感谢分享！
EternalHeart42楼
2015-05-29 09:57
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五星好评 顶一下，感谢分享！
qinhuantxj43楼
2015-06-21 20:10
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五星好评 顶一下，感谢分享！
sunnymaxs44楼
2015-07-27 16:48
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hitkj45楼
2015-09-01 18:46
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五星好评 顶一下，感谢分享！
guli052046楼
2015-09-23 15:05
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五星好评 顶一下，感谢分享！
promin47楼
2015-09-28 09:06
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五星好评 顶一下，感谢分享！
du2008112948楼
2015-10-06 07:06
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顶一下，感谢分享！
wuhai66650楼
2015-12-28 12:57
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五星好评 顶一下，感谢分享！
