Advertisement

From Trajectories to Ensembles in Classical Mechanics

  • Ángel S. Sanz
  • Salvador Miret-Artés
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 850)

Abstract

The time-evolution of interacting “small” systems is usually described by means of quantum mechanics. However, very often, it happens that a first approach to these dynamics by means of classical mechanics provides a better understanding of the processes and phenomena involved. Here, an overview on classical mechanics is presented, ranging from regular motions to chaos which induces intrinsic stochasticity. This allows to establish a direct link to classical statistical mechanics within the Liouvillian framework when dealing with particle ensembles, where motion is still deterministic but unpredictable. On these grounds, some criteria to discern whether a system has to be considered as classical or quantum mechanical will be analyzed. Finally, different aspects of the continuum limit of classical mechanics of many bodies will also be presented in order to better contextualize later on in this Volume the quantum and optical hydrodynamical interpretations.

Keywords

Phase Space Periodic Orbit Hamiltonian System Classical Mechanic Chaotic Dynamic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Copernicus, N.: De Revolutionibus Orbium Coelestium (1543). Translated into English by Wallis, C.G.: On the Revolutions of the Heavenly Spheres (St John’s College Bookstore, Annapolis, MD, 1939, republished by Prometheus Books, New York, 1995)Google Scholar
  2. 2.
    Kepler, J.: Astronomia Nova (1609). Translated into English by Donahue, W.H.: New Astronomy (Cambridge University Press, Cambridge, 1992)Google Scholar
  3. 3.
    Galilei, G.: Dialogo sopra i due massimi sistemi del mondo tolemaico e copernicano (1632). Translated into English by Drake, S.: Dialogue Concerning the Two Chief World Systems, Ptolemaic and Copernican (University of California Press, Los Angeles, CA, 1981)Google Scholar
  4. 4.
    Descartes, R.: Principia Philosophiae (1644). Translated into English by Miller, V.R., Miller, R.P.: Principles of Philosophy (Kluwer Academic, Amsterdam, 1991)Google Scholar
  5. 5.
    Newton, I.: Philosophiae Naturalis Principia Mathematica. Royal Society, London (1687). Published in English by Cohen, I.B., Whitman, A., Newton, I.: The Principia: Mathematical Principles of Natural Philosophy. A New Translation (University of California Press, Los Angeles, CA, 1999)Google Scholar
  6. 6.
    Bell, J.S.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964). Reprinted in Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987)Google Scholar
  7. 7.
    Aspect, A., Grangier, P., Roger, G.: Experimental tests of realistic theories via Bell’s theorem. Phys.Rev. Lett. 47, 460–463 (1981)ADSCrossRefGoogle Scholar
  8. 8.
    Margenau, H., Murphy, G.M.: The Mathematics of Physics and Chemistry, 2nd edn. D.van Nostrand Company, New York (1956)Google Scholar
  9. 9.
    Landau, L.D., Lifschitz, E.M.: Mechanics. Pergamon Press, Oxford (1960)Google Scholar
  10. 10.
    Goldstein, H.: Classical Mechanics. Addison-Wesley Publising Company, Reading (1980)zbMATHGoogle Scholar
  11. 11.
    Landau, L.D., Lifschitz, E.M.: The Classical Theory of Fields, 4th edn. Butterworth–Heinemann, London (1975)Google Scholar
  12. 12.
    Parnovsky, A.S.: Some generalisations of Brachistochrone problem. Acta Phys. Pol. A 93, S55–S64 (1998)Google Scholar
  13. 13.
    Erlichson, H.: Johann Bernoulli’s Brachistochrone solution using Fermat’s principle of least time. Eur. J. Phys. 20, 299–304 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Tannery, P., Henry, C. (eds.): \(\OE \hbox{uvres}\) de Fermat, pp. 354–457. Gauthier–Villars, Paris (1894)Google Scholar
  15. 15.
    Hero of Alexandria, Catoptrica (circa 60 AC)Google Scholar
  16. 16.
    Ibn al-Haytham “Alhazen”, Book of Optics (1021)Google Scholar
  17. 17.
    de Maupertuis P.L.M.: Accord de différentes lois de la nature qui avoient jusqu’ici paru incompatibles. Histoire de l’Académie Royale des Sciences et des Belles Lettres, pp. 417–426 (1744)Google Scholar
  18. 18.
    de Maupertuis P.L.M.: Les lois de mouvement et du repos, déduites d’un principle de métaphysique. Histoire de 1’Académie Royale des Sciences et des Belles Lettres, pp. 267–294 (1746)Google Scholar
  19. 19.
    Hamilton, W.R.: On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function. Phil. Trans. R. Soc. Lond. 124, 247–308 (1834)CrossRefGoogle Scholar
  20. 20.
    Hamilton, W.R.: Second essay on a general method in dynamics. Phil. Trans. R. Soc. Lond. 125, 95–144 (1835)CrossRefGoogle Scholar
  21. 21.
    de Laplace, P.S.: Theorie Analytique des Probabilités. In: \(\OE \hbox{uvres}\) Complètes de Laplace, vol. VII. Gauthier–Villars, Paris (1820)Google Scholar
  22. 22.
    Thornton, S.T., Marion, J.B.: Classical Dynamics of Particles and Systems, 5th edn. Thomson, Belmont (2004)Google Scholar
  23. 23.
    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989)Google Scholar
  24. 24.
    Born, M., Wolf, E.: Principles of Optics. Pergamon Press, Oxford (1980)Google Scholar
  25. 25.
    Elmore, W.C., Heald, M.A.: Physics of Waves. Dover Publications, New York (1985)Google Scholar
  26. 26.
    Moiseyev, N.: Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling. Phys. Rep. 302, 211–293 (1998)ADSCrossRefGoogle Scholar
  27. 27.
    Kosloff, R., Kosloff, D.: Absorbing boundaries for wave propagation problems. J. Comp. Phys. 63, 363–376 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Xavier, A.L., de Aguiar, M.A.M.: Complex trajectories in the quartic oscillator and its semiclassical coherent-state propagator. Ann. Phys. (N.Y.) 252, 458–478 (1996)ADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Xavier, A.L., de Aguiar, M.A.M.: Phase-space approach to the tunnel effect: a new semiclassical transversal time. Phys. Rev. Lett. 79, 3323–3326 (1997)ADSCrossRefGoogle Scholar
  30. 30.
    Leacock, R.A., Padgett, M.J.: Hamilton–Jacobi theory and the quantum action variable. Phys. Rev. Lett. 50, 3–6 (1983)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Leacock, R.A., Padgett, M.J.: Hamilton–Jacobi/action–angle quantum mechanics. Phys. Rev. D 28, 2491–2502 (1983)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \({\mathcal{PT}}\) symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)Google Scholar
  33. 33.
    Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401(1–4) (2002)Google Scholar
  34. 34.
    Bender, C.M., Boettcher, S., Meisinger, P.N.: quantum mechanics. J. Math. Phys. 40, 2201–2229 (1999)MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Degiovanni, L., Rastelli, G.: Complex variables for separation of the Hamilton–Jacobi equation on real pseudo-Riemannian manifolds. J. Math. Phys. 48, 073519(1–23) (2007)Google Scholar
  36. 36.
    Bender, C.M., Chen, J.-H., Darg, D.W., Milton, K.A.: Classical trajectories for complex Hamiltonians. J. Phys. A 39, 4219–4238 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    Bender C.M., Darg D.W.: Spontaneous breaking of classical \({\mathcal{PT}}\) symmetry. J. Math. Phys. 48, 042703(1–14) (2007)Google Scholar
  38. 38.
    Kaushal, R.S., Korsch, H.J.: Some remarks on complex Hamiltonian systems. Phys. Lett. A 276, 47–51 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  39. 39.
    Kaushal, R.S., Singh, S.: Construction of complex invariants for classical dynamical systems. Ann. Phys. 288, 253–276 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  40. 40.
    Singh, S., Kaushal, R.S.: Complex dynamical invariants for one-dimensional classical systems. Phys. Scr. 67, 181–185 (2003)MathSciNetADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Kaushal, R.S.: Classical and quantum mechanics of complex Hamiltonian systems: an extended complex phase space approach. Pramana J. Phys. 73, 287–297 (2009)CrossRefGoogle Scholar
  42. 42.
    Le Levier, R.E., Saxon, D.S.: An optical model for nucleon-nuclei scattering. Phys. Rev. 87, 40–41 (1952)ADSzbMATHCrossRefGoogle Scholar
  43. 43.
    Feshbach, H., Porter, C.E., Weisskopf, V.F.: Model for nuclear reactions with neutrons. Phys. Rev. 96, 448–464 (1954)ADSzbMATHCrossRefGoogle Scholar
  44. 44.
    Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131, 2766–2788 (1963)MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. Springer, New York (1992)zbMATHGoogle Scholar
  46. 46.
    Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations, 3rd edn. Oxford University Press, Oxford (1999)Google Scholar
  47. 47.
    Tabor, M.: Chaos and Integrability in Nonlinear Dynamics. Wiley, New York (1989)Google Scholar
  48. 48.
    Gutzwiller, M.C.: Chaos in Classical and Quantum Mechanics. Springer, Berlin (1990)zbMATHGoogle Scholar
  49. 49.
    Ott, E.: Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  50. 50.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)zbMATHGoogle Scholar
  51. 51.
    Gaspard, P.: Chaos, Scattering and Statistical Mechanics. Cambridge University Press, Cambridge (1998)Google Scholar
  52. 52.
    Weinstein, A.: Normal modes for nonlinear hamiltonian systems. Invent. Math. 20, 47–57 (1973)MathSciNetADSzbMATHCrossRefGoogle Scholar
  53. 53.
    Christiansen, F.: Fixed points, and how to get them classical and quantum chaos. In: Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay G. (eds.) The Chaos Book. Niels Bohr Institute, Copenhagen (2001). www.nbi.dk/ChaosBook/
  54. 54.
    Farantos, S.C.: Methods for locating periodic orbits in highly unstable systems. THEOCHEM 341, 91–100 (1995)CrossRefGoogle Scholar
  55. 55.
    Prosmiti, R., Farantos, S.C.: Periodic orbits bifurcation diagrams and the spectroscopy of C\(_2\)H\(_2\) system. J. Chem. Phys. 103, 3299–3314 (1995)Google Scholar
  56. 56.
    Seydel, R.: Practical Bifurcation and Stability Analysis. Springer, New York (1994)zbMATHGoogle Scholar
  57. 57.
    Meyer, K.R.: Generic bifurcation of periodic points. Trans. Amer. Math. Soc. 149, 95–107 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    de Aguiar, M.A.M., Malta, C.P., Baranger, M., Davies, K.T.R.: Bifurcations of periodic trajectories in non-integrable Hamiltonian systems with two degrees of freedom: numerical and analytical results. Ann. Phys. (NY) 180, 167–205 (1987)ADSzbMATHCrossRefGoogle Scholar
  59. 59.
    Mao, J.-M., Delos, J.B.: Hamiltonian bifurcation theory of closed orbits in the diamagnetic Kepler problem. Phys.Rev. A 45, 1746–1761 (1992)ADSCrossRefGoogle Scholar
  60. 60.
    Hénon, M., Heyles, C.: The applicability of the third integral of motion: some numerical experiments. Astron. J. 69, 73–79 (1964)ADSCrossRefGoogle Scholar
  61. 61.
    Dorfman, J.E.: An Introduction to Chaos in Non-Equilibrium Statistical Mechanics. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  62. 62.
    Vega, J.L., Guantes, R., Miret–Artés, S.: Chaos and transport properties of adatoms on solid surfaces. J. Phys.: Condens. Matter 14, 6193–6232 (2002)ADSCrossRefGoogle Scholar
  63. 63.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion. a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  64. 64.
    Klafter, J., Shlesinger, M.F., Zumofen, G.: Beyond Brownian motion. Phys. Today 49(2), 33–39 (1996)CrossRefGoogle Scholar
  65. 65.
    Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)zbMATHGoogle Scholar
  66. 66.
    Berry, M.: Waves and Thom’s theorem. Adv. Phys. 25, 1–26 (1976)ADSCrossRefGoogle Scholar
  67. 67.
    Guantes, R., Sanz, A.S., Margalef-Roig, J., Miret-Artés, S.: Atom-surface diffraction: a trajectoy description. Surf. Sci. Rep. 53, 199–330 (2004)ADSCrossRefGoogle Scholar
  68. 68.
    Pathria, R.K.: Statistical Mechanics. Pergamon Press, Oxford (1972)Google Scholar
  69. 69.
    McQuarrie, D.A.: Statistical Mechanics. Harper and Row, New York (1976)Google Scholar
  70. 70.
    Hansen, J.P., McDonald, I.R.: Theory of Simple Liquids. Academic Press, New York (1986)Google Scholar
  71. 71.
    Gibbs, J.W.: Elementary Principles in Statistical Mechanics. Scribner’s Sons, New York (1902)zbMATHGoogle Scholar
  72. 72.
    Jaffé, C., Brumer, P.: Classical Liouville mechanics and intramolecular relaxation dynamics. J. Phys. Chem. 88, 4829–4839 (1984)CrossRefGoogle Scholar
  73. 73.
    Jaffé, C., Brumer, P.: Classical-quantum correspondence in the distribution dynamics of integrable systems. J. Chem. Phys. 82, 2330–2340 (1985)MathSciNetADSCrossRefGoogle Scholar
  74. 74.
    Wilkie, J., Brumer, P.: Quantum-classical correspondence via Liouville dynamics. I. Integrable systems and the chaotic spectral decomposition. Phys. Rev. A 55, 27–42 (1997)MathSciNetADSCrossRefGoogle Scholar
  75. 75.
    Wilkie, J., Brumer, P.: Quantum-classical correspondence via Liouville dynamics. II. Correspondence for chaotic Hamiltonian systems. Phys. Rev. A 55, 43–61 (1997)MathSciNetADSCrossRefGoogle Scholar
  76. 76.
    Bogoliubov, N.N.: Kinetic equations. J. Phys.USSR 10, 265–274 (1946)Google Scholar
  77. 77.
    Born, M., Green, H.S.: A general kinetic theory of liquids I. The molecular distribution functions. Proc. R. Soc. A 188, 10–18 (1946)MathSciNetADSzbMATHCrossRefGoogle Scholar
  78. 78.
    Kirkwood, J.G.: The statistical mechanical theory of transport processes I. General theory. J. Chem. Phys. 14, 180–201 (1946)ADSCrossRefGoogle Scholar
  79. 79.
    Kirkwood, J.G.: The statistical mechanical theory of transport processes II. Transport in gases. J. Chem. Phys. 15, 72–76 (1947)ADSCrossRefGoogle Scholar
  80. 80.
    Yvon, J.: Theorie Statistique des Fluides et l’Equation d’Etat, Actes Scientifique et Industrie, vol. 203. Hermann, Paris (1935)Google Scholar
  81. 81.
    Kubo, R., Toda, M., Hashitsume, N.: Nonequilibrium Statistical Mechanics Statistical Physics II. Springer, Berlin (1985)Google Scholar
  82. 82.
    Boon, J.P., Yip, S.: Molecular Hydrodynamics. Dover, New York (1991)Google Scholar
  83. 83.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166–179 (1952)MathSciNetADSCrossRefGoogle Scholar
  84. 84.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables. II. Phys. Rev. 85, 180–193 (1952)MathSciNetADSCrossRefGoogle Scholar
  85. 85.
    Holland, P.R.: The Quantum Theory of Motion. An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1993)CrossRefGoogle Scholar
  86. 86.
    Madelung, E.: Quantentheorie in hydrodynamischer Form. Z. Phys. 40, 322–326 (1926)ADSGoogle Scholar
  87. 87.
    Schiller, R.: Quasi-classical theory of the nonspinning electron. Phys. Rev. 125, 1100–1108 (1962)MathSciNetADSzbMATHCrossRefGoogle Scholar
  88. 88.
    Schiller, R.: Quasi-classical transformation theory. Phys. Rev. 125, 1109–1115 (1962)MathSciNetADSCrossRefGoogle Scholar
  89. 89.
    Rosen, N.: The relation between classical and quantum mechanics. Am. J. Phys. 32, 597–600 (1964)ADSCrossRefGoogle Scholar
  90. 90.
    Rosen, N.: Quantum particles and classical particles. Found. Phys. 16, 687–700 (1986)MathSciNetADSCrossRefGoogle Scholar
  91. 91.
    Louisell, W.H.: Quantum Statistical Properties of Radiation. Wiley, New York (1990)zbMATHGoogle Scholar
  92. 92.
    Weiss, U.: Quantum Dissipative Systems. (World Scientific, Singapore (1999)zbMATHCrossRefGoogle Scholar
  93. 93.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  94. 94.
    Razavy, M.: Classical and Quantum Dissipative Systems. Imperial College Press, London (2005)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg  2012

Authors and Affiliations

  • Ángel S. Sanz
    • 1
  • Salvador Miret-Artés
    • 1
  1. 1.Instituto de Física FundamentalConsejo Superior de Investigaciones CientíficasMadridSpain

Personalised recommendations