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Bogatov E. M., Mukhin R. R. The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, iss. 5, pp. 69-87. DOI: 10.18500/0869-6632-2017-25-5-69-87

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The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others

Bogatov Egor Mihajlovich, 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)
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)

The main moments of the historical development of one of the basic methods of nonlinear systems investigating (the averaging method) are traced. This method is understood as a transition from the so-called exact equation dx/dt = εX(t, x) (ε is small parameter), to the averaging equation dξ/dt = εX0(ξ) + ε2P2(ξ) + ... + εmPm(ξ) by corresponding variable substitution.Bogolyubov–Krylov’s approach to the problem of justifying the averaging method, based on the invariant measure theorem, is analyzed. The paper presents the evolution of views on a physical pendulum with a vibrating suspension, beginning with the description of its simple motions (A. Stephenson, G. Jeffreys, N.N. Bogolyubov, P.L. Kapitza, V.N. Chelomey, etc.) and ending with complex movements. In the latter case, various characteristic features of the complex behavior of nonlinear systems is appeared – bifurcations, chaotic regimes, etc., (J. Blackburn, M. Bartuccelli, and others). A number of analogs of a pendulum with a vibrating suspension point outside of classical mechanics are described (A.V. Gaponov, M.A. Miller – localization of a particle in an electric field; S.M. Osovets – stabilization of hot plasma; V. Paul, N. Ramsey, H. Dehmelt – confinement of particles in an alternating electromagnetic field). An important part of the work is historical information about N.M. Krylov, N.N. Bogolyubov, P.L. Kapitza, which makes possible to more clearly show the motivation of the studies, their conditionality.   

  1. Urbanskii V.M. PhD Thesis. Kiev, 1983 (in Russian).
  2. Bogolyubov A.N., Urbanskii V.M. Nikolai Mitrophanovich Krylov. Kiev, Naukova dumka, 1987 (in Russian).
  3. Volosov V.M. Averaging method in the theory of nonlinear vibration. In: Mechanics in USSR for 50 years, Nauka, Moscow, 1968. P. 115–136. (in Russian).
  4. Bogolyubov N.N., Mitropolsky Y.A. and Lykova O.B. Asimptoticheskie metody v nelinejnoj mehanike. Istorija otechestvennoj matematiki. Vol. 4, part 2. Kiev, Naukova dumka, 1970, P. 264–290. (in Russian)
  5. Samoilenko A.M. N.N. Bogolyubov and non-linear mechanics. Uspekhi Mat. Nauk. 1994. 299. Vol. 49, N5. P. 103–146.
  6. Nesterenko E.M. PhD Thesis. Moscow, 1970 (in Russian).
  7. Poincare H. Les methodes nouvelles de la mecanique celeste. V. 1–3. Paris: Gauthier-Villars, 1892–1899.
  8. Poincare H. Memoire sur les courbes definies par lesequations differenti elles, I–IV // J. Math. Pures Appl., 3 serie, 1881, 7, 375–422; 1882, 8, 251–286; 4 serie, 1885, 1, 167–244; 1886, 2, 151–217.
  9. Lyapunov A.M. The general problem of the stability of motion. Moscow; Leningrad, 1950 (in Russian).
  10. Kryloff N. et Bogoluboff N. Quelques exsemples d’oseillations non lineares. Comptes rendus des l’Acad. Sci. de Paris. 1932. Vol. 194.
  11. Krylov N.M., Bogolyubov N.N. New methods of nonlinear mechanics. Moscow-Leningrad, ONTI, 1934 (in Russian).
  12. Krylov N.M., Bogolyubov N.N. Introduction to nonlinear mechanics. Kiev, 1937 (in Russian).
  13. Kryloff N., Bogoliuboff N. La theorie generale de la mesure dans son application a l’etude des systemes dynamiques de la mecanique non lineaire. Ann. Math. 1937. Vol. 38. P. 65–113.
  14. Bogolyubov N.N. About some statistical methods in mathematical physics. Kiev, 1945 (in Russian).
  15. Kryloff N.M., Bogoliuboff N.N. Introduction to non-linear mechanics. Prinseton, NY: Prinseton Univ. Press, 1943.
  16. Bogolyubov N.N. Theory of perturbations in nonlinear mechanics. Coll. sci. works builds. Mechanics Inst. of the Ukrainian Academy of Sciences. Kiev. 1950. Vol. 14. P. 9–34 (in Russian).
  17. Aleksandrov P.S. First International topological conference in Moscow. Uspekhi Mat. Nauk. 1936. No.1. P. 260-262. (in Russian)
  18. Vek Lavrent’eva. Novosibirsk: SB RAS, 2000 (in Russian).
  19. Stephenson A. On a class of forced oscillations // Quart. J. Pure and Appl. Math. 1906. Vol. 37, N148. P. 353–360.
  20. Stephenson A. On the stability of the steady state of forced oscillation // Phil. Mag. and J. Sci. Ser. 6. 1907. Vol. 14, N84. P. 707–712.
  21. Stephenson A. On induced stability // Phil. Mag. and J. Sci. Ser. 6. 1908. Vol. 15, N86. P. 233–236.
  22. Stephenson A. On a new type of dynamical stability // Memoirs and Proceedings of the Manchester Literary and Philosophical Society. 1908. Vol. 52, N8.
  23. Jeffreys H. Methods of mathematical physics. Cambridge (C.U.P.). 2nd Edition, 1950.
  24. Van der Pol B. Stabiliseering door kleine trillingen // Physica. Bd. 1925. 5. P. 157–162.
  25. Strutt M.J. Stabiliseering en labiliseering door trillingen // Physica. Bd. 1927. 7. P. 265–271.
  26. Hirsh P. Das Pendel mit Oszillierendem Aufhangepunkt // Z. angew. Math. Mech.Bd. 1930. 10. P. 41–52.
  27. Erdelyi A. Uber die Kleinen Schwingungen eines Pendels mit oszillierendem Aufhan-gepunkt // Z. angew. Math. Mech. Bd. 1934. 14.
  28. Lowenstern E.R. The stabilizing effect of imposed oscillations of high frequency on a dynamical system // London, Edinburgh and Dublin Phil. Mag. 1932. Vol. 13. P. 458–486.
  29. Van der Pol B., Strutt M.J.O. On the stability of Mathieu equation // The London, Edinburgh and Dublin Phil. Mag. 7th series. 1928. Vol. 5. P. 23–28.
  30. Kapitza P.L. Dynamic stability of the pendulum with vibrating suspension point. Sov. Phys. JETP. 1951. Vol. 21. P. 588–597 (in Russian); see also Collected Papers of P.L. Kapitza, edited by D. Ter Haar, Pergamon, London, 1965. Vol. 2. P. 714–726.
  31. Kapitza P.L. Pendulum with an oscillating pivot. Sov. Phys. Uspekhi. 1951. Vol. 44, Iss. 1. P. 7–20 (in Russian).
  32. Vospominaniya ob akademike N.N. Bogolyubove. Moscow, 2009 (in Russian).
  33. Landau L. D. and Lifschitz E. M. Mechanics, Nauka, Moscow, 1965 (in Russian); Pergamon, New York, 1976.
  34. V.G. Shironosov. Resonance in physics, chemistry and biology. Department of BioMedPhysics, UdSU, Izhevsk, 2001. 92 p. (in Russian).
  35. Blekhman I.I. Vibration mechanics. Moscow: Nauka, 1994 (in Russian).
  36. Chelomey V.N. On the possibility of elastic systems stability increase by means of vibrations. DAN SSSR. 1956. Vol. 110, No.3. P. 345–347. (in Russian).
  37. Chelomey V.N. Paradoxes in mechanics caused by vibrations. DAN SSSR. 1983. Vol. 270, No.1. P. 62–67. (in Russian).
  38. Bogatov E.M., Mukhin R.R. The relation between the non-linear analysis, bifur- cations and nonlinear dynamics (on the example of Voronezh school of nonlinear functional analysis). Izvestiya VUZ. Applied Nonlinear Dynamics. 2015. Vol. 23, No.6. P. 74–88. (in Russian).
  39. Gaponov A.V., Miller M.A. Potential wells for charged particles in high-frequency fields. Sov. Phys. JETP. 1958. Vol. 34, Iss. 2. P. 242–243. (in Russian).
  40. Gaponov A.V., Miller M.A. Use of Moving High-Frequency Potential Wells for the Acceleration of Charged Particles. Sov. Phys. JETP. 1958. Vol. 34, Iss. 3. P. 751–752. (in Russian).
  41. Osovets S.M. Dynamic methods of confinement and stabilization of hot plasma. Sov. Phys. Uspekhi. 1974. Vol. 112, Iss. 4. P. 638–683. (in Russian).
  42. Blackburn J.A., Smith H.Y.T., Gronbech-Jensen N. Stability and Hopf bifurcation in an inverted pendulum // Amer. J. Phys. 1992. Vol. 60. P. 903–908.
  43. Bartuccelli M.V., Gentile G., Georgin K.V. On the dynamics of a vertically driven damped planar pendulum // Proc. Roy. Soc. Lond. 2001. Vol. 457. P. 3007–3022.
  44. Bartuccelli M.V., Gentile G., Georgin K.V. KAM theory, Linstedt series and the stability of the upside-down pendulum // Discrete and continuous dynamical systems. 2003. Vol. 9, No 2. P. 413–426.
  45. Burd V.Sh., Zabreiko P.P., Kolesov Yu.S., and Krasnosel’skii M.A., Principle of averaging and bifurcation almost periodic solutions. DAN SSSR. 1969. Vol. 187, N6. P. 1219–1221. (in Russian).
  46. Osberghaus O., Paul V., Fischer E. Forschungsberichte des Wirschafts und Werker ministeriums. Nardheim Westfalen. 1958. Nr. 415.
  47. Paul V. Electromagnetic traps for charged and neutral particles. Nobel lecture. Sov. Phys. Uspekhi. 1990. Vol. 160, Iss. 12. P. 109–127.
  48. Levi M. Geometry and physics of averaging with applications // Physica D. 1999. Vol. 132. P. 150–164.
  49. Levi M., Zehnder E. Boundedness of solutions for quasiperiodic potentials // SIAM J. Math. Anal. 1995. Vol. 26. P. 1233–1256.
  50. Gerving C.S., Hoang T.M. and oth. Non- equilibrium dinamics of un unstable quantum pendulum explored in a spin-1 Bose-Einstein condensate // Nature communication. School of physics, Georgia Ist. of Tech. 2012. P. 1–6.
  51. Citro R., Dalla Torre E. G., D’Alessio L., Polkovnikov A., Babadi M., Oka T., and Demler E. Dynamical stability of a many-body Kapitza pendulum // Ann. of Physics. 2015. Vol. 360. P. 694-710.
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