Skip to main content
SpringerLink
Log in

Asymptotics of the Solution of a Variational Problem on a Large Interval

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The variational problem of minimizing the energy functional that results in a second- order nonlinear differential equation of pendulum type on a finite interval with natural boundary conditions is analyzed. It is shown that the number of solutions of the boundary-value problem depends on the length \(L\) of the interval and unboundedly increases as \(L\to\infty\). The solutions on which the energy minimum is realized converge as \(L\to\infty\) to the solution of a variational problem in the class of periodic functions.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 3.
Fig. 5.
Fig. 7.

Similar content being viewed by others

Notes

  1. Such conditions are usually said to be natural in contrast to the external conditions (sometimes called the principal conditions) which are imposed outside the variational problem, for example, by specifying the boundary values of the sought function.

  2. They are small compared to the sample dimensions.

References

  1. A. P. Pyatakov and A. K. Zvezdin, “Magnetoelectric and multiferroic media,” Phys. Usp. 55 (6), 557–581 (2012).

    Article  Google Scholar 

  2. C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1962).

    MATH  Google Scholar 

  3. É. M. Galeev and V. M. Tikhomirov, Optimization: Theory, Examples, Problems (Editorial URSS, Moscow, 2000) [in Russian].

    Google Scholar 

  4. L. A. Kalyakin, A. K. Zvezdin, and Z. V. Gareeva, “Asymptotic analysis of a multiferroic model,” Theoret. and Math. Phys. 203 (1), 457–468 (2020).

    Article  MathSciNet  Google Scholar 

  5. Z. V. Gareeva, L. A. Kalyakin, I. R. Kayumov, and A. K. Zvezdin, “Spin-reorientation transitions in multiferroics with cycloidal spin ordering,” Physics of Metals and Metallography (PMM) 121 (4), 310–315 (2020).

    Article  Google Scholar 

  6. N. A. Spaldin and R. Ramesh, “Advances in magnetoelectric multiferroics,” Nat. Mater 18 (3), 203–212 (2019).

    Article  Google Scholar 

  7. N. E. Kulagin, A. F. Popkov, and A. K. Zvezdin, “Spatially modulated antiferromagnetic structures in a light-plane multiferroic,” Fizika Tverdogo Tela 53 (5), 912–918 (2011).

    Google Scholar 

  8. N. E. Kulagin, A. F. Popkov, S. V. Solov’ev, and A. K. Zvezdin, “Spin-modulation transitions induced by magnetic field in epitaxial BiFeO3 films with orientation (001),” Fizika Tverdogo Tela 61 (2), 248–256 (2019).

    Google Scholar 

  9. L. A. Kalyakin, “Asymptotics of solutions of the nonlinear oscillator model with natural boundary conditions,” J. Math. Sci., New York 247 (6), 877–887 (2020).

    Article  MathSciNet  Google Scholar 

  10. L. A. Kalyakin, “On the frequency of a nonlinear oscillator,” in Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., Vol. 163: Differential Equations (VINITI, Russian Academy of Sciences, Moscow, 2019), pp. 15–24 [in Russian].

    Google Scholar 

  11. L. A. Kalyakin, “Lyapunov functions in justification theorems for asymptotics,” Math. Notes 98 (5), 752–764 (2015).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Subdivision of the Ufa Federal Research Center of Russian Academy of Sciences, Ufa, 450008, Russia

    L. A. Kalyakin

Authors
  1. L. A. Kalyakin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. A. Kalyakin.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 688–703 https://doi.org/10.4213/mzm12723.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kalyakin, L.A. Asymptotics of the Solution of a Variational Problem on a Large Interval. Math Notes 110, 687–699 (2021). https://doi.org/10.1134/S0001434621110055

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434621110055

Keywords

Search

Navigation