Skip to main content
Log in

General method for the solution of some problems of stabilization and destabilization of motion

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We demonstrate a complete mathematical analogy between the description of motion of an electron in a periodic field and the phenomenon of parametric resonance. A band approach to the analysis of the phenomenon of parametric resonance is formulated. For an oscillator under the action of an external force described by the Weierstrass function, we calculate the increments of increase in oscillations and formulate a condition for parametric resonance. For the known problem of a pendulum with vibrating point of suspension, we find exact conditions for the stabilization of the pendulum in the upper (unstable) equilibrium position by using the Lamé equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. D. Courant and H. S. Snyder, “Theory of alternating-gradient synchrotron,” Ann. Phys., 3, 1–123 (1958).

    Article  MATH  Google Scholar 

  2. A. A. Kolomenskii and A. N. Lebedev, Theory of Cyclic Accelerators [in Russian], Fizmatgiz, Moscow (1962).

    Google Scholar 

  3. A. N. Lebedev and A. V. Shal’nov, Foundations of Physics and Technology of Accelerators [in Russian], Énergoatomizdat, Moscow (1991).

    Google Scholar 

  4. L. I. Mandel’shtam, “Lectures on Oscillations. Lecture 19,” in: Complete Collection of Works [in Russian], Vol. 4, Izd. Akad. Nauk SSSR (1955).

  5. N. D. Papaleksi, “Evolution of the notion of resonance,” Usp. Fiz. Nauk, 31, Issue 4, 447–460 (1947).

    Google Scholar 

  6. N. N. Bogolyubov, “Perturbation theory in nonlinear mechanics,” Sb. Inst. Stroit. Mekh. Akad. Nauk Ukr. SSR, 14, 9–34 (1950).

    Google Scholar 

  7. P. L. Kapitsa, “Dynamical stability of a pendulum with vibrating suspension point,” Zh. Éksp. Teor. Fiz., 21, Issue 5, 588–598 (1951).

    Google Scholar 

  8. Yu. A. Mitropol’skii, Averaging Method in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1971).

    Google Scholar 

  9. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge (1927).

    MATH  Google Scholar 

  10. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. Inverse Scattering Method [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  11. L. A. Takhtadzhyan and L. D. Faddeev, Hamiltonian Approach in Soliton Theory [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  12. V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).

    MATH  Google Scholar 

  13. A. A. Abrikosov, Foundations of the Theory of Metals [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  14. I. M. Lifshits, M. Ya. Azbel’, and M. I. Kaganov, Electron Theory of Metals [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  15. V. G. Baryakhtar, E. D. Belokolos, and A. M. Korostil, “A new method for calculating the electron spectrum of solids. Application to high-temperature superconductivity,” Phys. Stat. Sol. (b), 169, 105–114 (1992).

    Article  Google Scholar 

  16. V. G. Bar’yakhtar, I. V. Bar’yakhtar, and A. V. Samar, “An exactly solvable model of parametric resonance,” in: Proceedings of the International Conference Dedicated to M. A. Krivoglaz [in Russian], June 16–18, Institute of Metal Physics, Ukrainian Academy of Sciences, Kiev (2004).

    Google Scholar 

  17. V. G. Bar’yakhtar, “Common equation for the motion of an electron in a crystal and parametric resonance. Exactly solvable model,” Metallofiz. Noveish. Tekhnol., 27, No. 1, 119–134 (2005).

    Google Scholar 

  18. V. G. Bar’yakhtar, I. V. Bar’yakhtar, and A. V. Samar, “Common equation for the motion of an electron in a crystal and parametric resonance. Exactly solvable model,” in: Abstracts of the Bogolyubov Conference “Contemporary Problems of Mathematics and Theoretical Physics” [in Russian], Institute for Theoretical Physics, Ukrainian Academy of Sciences, Kiev (2004).

    Google Scholar 

  19. E. D. Belokolos, Mathematical Foundations of the Theory of Solids with Quasiperiodic Structure [in Russian], Preprint, Institute for Theoretical Physics. Ukrainian Academy of Sciences, Kiev (1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 152–161, February, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bar’yakhtar, V.G., Samar, A.V. General method for the solution of some problems of stabilization and destabilization of motion. Ukr Math J 59, 158–168 (2007). https://doi.org/10.1007/s11253-007-0013-4

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-007-0013-4

Keywords

Navigation