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Mukhin R. R. From the history of Hamiltonian chaos: billiards. Izvestiya VUZ. Applied Nonlinear Dynamics, 2008, vol. 16, iss. 6, pp. 86-98. DOI: 10.18500/0869-6632-2008-16-6-86-98

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Russian
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Personalia
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537

From the history of Hamiltonian chaos: billiards

Autors: 
Mukhin Ravil Rafkatovich, Stary Oskol technological Institute. A. A. Ugarov (branch) of Federal state educational institution of higher professional education "national research technological University "MISIS" (STI nust Misa)
Abstract: 

Problems of history of the Hamiltonian chaos discovery are considered. The example of Hamiltonian systems are free-moving particles with elastic collisions called mathematical billiards. The contribution from Russian scientists to chaos discovery in conservative systems (billiards are particular case of such systems) is especially large. Demonstration of billiard’s chaotic behaviour is one of the milestones in chaos history.

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Reference: 
  1. Bolzmann L. Über das Wärmegleichgewicht zwischen mehratomigen Gasmolekülen. Sitzber. Akad. Wiss. Wien. 1871;63(2):397–418 (in German).
  2. Lo Bello A. On the origin and history of ergodic theory. Bolletino di Storia delle Scienze Mathematiche. 1983;3(1):37–75 (in Italian).
  3. Vdovichenko NV. Development of the Fundamental Principles of Statistical Physics in the First Half of the Twentieth Century. Moscow: Nauka; 1986. 159 p. (in Russian).
  4. Kuznetsova OV. The History of the Justification of Statistical Mechanics. Moscow: Nauka; 1988. 182 p. (in Russian).
  5. Plancherel M. Beweis der Unmöglichkeit ergodischer mechanischer Systeme. Ann. Phys. 1913;42(4):1061–1063 (in German).
  6. Rosental A. Beweis der Unmöglichkeit ergodischer Gassysteme. Ann. Phys. 2006;347(14):796–806 (in German). DOI: 10.1002/andp.19133471407.
  7. Erenfest P, Erenfest T. Begriffische Grundlagen statistischen Auffassung in der Mechanik. Enzyklopedie der mathematischen Wissinschaften. 1911;4:32 (in German).
  8. Nemytskiy VV, Stepanov VV. Qualitative Theory of Differential Equations. Moscow-Leningrad: Gostekhizdat; 1947. Same, 2nd ed.: 1949. 448 p. (in Russian).
  9. Zaslavsky GM. Hamiltonian Chaos and Fractional Dynamics. Oxford: Oxford Univ. Press; 2004. 436 p.
  10. Birkhoff GD. Proof of recurrence theorem for strongly transitive systems and proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA. 1931;17(12):650–655. DOI: 10.1073/pnas.17.12.650.
  11. von Neumann J. Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. USA. 1932;18(1):70–82. DOI: 10.1073/pnas.18.1.70.
  12. Hopf E. Ergodentheorie. Berlin: Springer-Verlag; 1937. 83 p. (in German). DOI: 10.1007/978-3-642-86630-2.
  13. Khinchin AY. Zu Birkhoffs Lösung des Ergodeproblems. Math. Ann. 1931;107:485–488 (in German).
  14. Kolmogorov AN. A simplified proof of the Birkhoff – Khinchin ergodic theorem. Russian Mathematical Surveys. 1938;(5):52–56 (in Russian).
  15. Anosov DV. On the contribution of N.N. Bogolyubov to the theory of dynamical systems. Russian Mathematical Surveys. 1994;49(5):1–18. DOI: 10.1070/RM1994v049n05ABEH002417.
  16. Samoilenko AM. N.N. Bogolyubov and non-linear mechanics. Russian Mathematical Surveys. 1994;49(5):109–154. DOI: 10.1070/RM1994v049n05ABEH002432.
  17. Bogolyubov NN. On Some Statistical Methods in Mathematical Physics. Kiev: Publishing House of the Academy of Sciences of the Ukrainian SSR; 1945. 137 p. (in Russian).
  18. Kryloff N, Bogoliouboff N. La theorie generale de la mesure dans son application a l’etude des systeme dynamiques de la mecanique non lineaire. Ann. Math. 1937;38(1):65–113 (in French). DOI: 10.2307/1968511.
  19. Gibbs JW. Elementary Principles In Statistical Mechanics. New York: Charles Scribner's Sons; 1902. 207 p.
  20. Zaslavsky GM. Stochasticity of Dynamic Systems. Moscow: Nauka; 1984. 272 p. (in Russian).
  21. Krylov NS. Works on the Substantiation of Statistical Physics. Moscow, Leningrad: Publishing House of the Academy of Sciences of the USSR; 1950. 210 p. (in Russian).
  22. Chirikov BV. Research on the theory of nonlinear resonance and stochasticity. Preprint Budker Institute of Nuclear Physics of USSR Academy of Sciences. No. 267. Novosibirsk: Budker Institute of Nuclear Physics of USSR Academy of Sciences; 1969 (in Russian).
  23. Kuznetsova OV. N.S. Krylov's research on the substantiation of statistical mechanics. In: Research on the History of Physics and Mechanics. Moscow: Nauka; 1987. P. 80 (in Russian).
  24. Sinai YG. Development of Krylov’s ideas. In: N.S.Krylov. Works on Foundation of the Statistical Physics. Princeton: Princeton Univ. Press; 1980. P. 239–281.
  25. Krylov NS. Works on the Foundations of Statistical Physics. Princeton, NJ: Princeton Univ. Press; 1977. 314 p.
  26. Hadamard J. Les surfaces a courbures opposees et leurs lignes geodesiques. J. Math. pures et appl. 1898;4:27–74 (in French).
  27. Hedlund GA. The dynamic of geodesic flows. Bull. Amer. Math. Soc. 1939;45(4):241–260.
  28. Hopf E. Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sachs. Akad. Wiss. Leipzig. 1939;91:261–304 (in German).
  29. Boltzmann L. Lectures on the Theory of Gases. University of California Press; 1964. 483 p. DOI: 10.1525/9780520327474.
  30. Coriolis G. Mathematical Theory of Spin, Friction, and Collision in the Game of Billiards. David Nadler; 2005. 126 p.
  31. Sinai YG. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math. Surveys. 1970;25(2):137–189. DOI: 10.1070/RM1970v025n02ABEH003794.
  32. Sinai YG, Chernov NI. Ergodic properties of certain systems of two-dimensional discs and three-dimensional balls. Russian Math. Surveys. 1987;42(3):181–207. DOI: 10.1070/RM1987v042n03ABEH001421.
  33. Sinai J. On the concept of entropy for a dynamic system. Proc. Acad. Sci. USSR. 1959;124(4):768–771 (in Russian).
  34. Novikov SP et. al. Yakov Grigor'evich Sinai (on his sixtieth birthday). Russian Math. Surveys. 1996;51(4):765–778. DOI: 10.1070/RM1996v051n04ABEH002992.
  35. Shiryaeva AN, editor. Kolmogorov. Truth is Good. Moscow: Fizmatlit; 2003. 384 p. (in Russian).
  36. Sinai YG. To substantiate the ergodic hypothesis for one dynamical system of statistical mechanics. Proc. Acad. Sci. USSR. 1963;153(6):1261–1264 (in Russian).
  37. Arnol'd VI. Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys. 1963;18(6):85–191. DOI: 10.1070/RM1963v018n06ABEH001143.
  38. Sinai YG. Classical dynamical systems with countably multiple Lebesgue spectrum. II. Izvestiya: Mathematics. 1966;30(1):15–68 (in Russian).
  39. Kolmogorov AN. A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces. Proc. Acad. Sci. USSR. 1958;119(5):861–864 (in Russian).
  40. Mukhin RR. Kolmogorov's development of the entropy direction of ergodic theory. Historical and Mathematical Research. II Series. 2003;8(43):18–26 (in Russian).
  41. Sinai YG. Ergodic theory. In: A.N. Kolmogorov. Information Theory and Theory of Algorithms. Moscow: Nauka; 1987. P. 275–278 (in Russian).
  42. Bunimovich LA, Sinai YG. On a fundamental theorem in the theory of dispersing billiards. Mathematics of the USSR-Sbornik. 1973;19(3):407–423. DOI: 10.1070/SM1973v019n03ABEH001786.
  43. Sinai YG. Written message dated 26.03.2007 (in Russian).
  44. Diner S. Les voies du chaos deterministe dans l’ecole russe. In: Dahan Dalmedico A, Chabert JL, Chemla K. Chaos et Determinism. Edition du Seuil; 1992. P. 331–368 (in French).
Received: 
02.04.2008
Accepted: 
02.04.2008
Published: 
27.02.2009
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