Skip to main content
SpringerLink
Log in

Laplace series expansion of the potential of a homogeneous circular torus

  • Theoretical and Mathematical Physics
  • Published:
Technical Physics Aims and scope Submit manuscript

Abstract

The external potential of a homogeneous circular torus is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas are derived for the coefficients of this series, which can be expressed in terms of Legendre polynomials depending only on the geometrical parameter of the torus. The convergence of the series is proved and the radius of convergence is determined. The resultant expressions are verified numerically.

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

Similar content being viewed by others

References

  1. B. Riemann, On the Torus Potential (OGIZ, Moscow, 1948), pp. 367–372, translated from G. F. B. Riemann, Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, hrsg. unter Mitwirkung von Richard Dedekind, von Heinrich Weber (Dover, New York, 1953).

    Google Scholar 

  2. A. Züge, J. Reine Angew. Math. 104, 89 (1889).

    Google Scholar 

  3. B. P. Kondrat’ev, Potential Theory and Figures of Equilibrium (RKHD, Moscow-Izhevsk, 2003).

    Google Scholar 

  4. B. P. Kondrat’ev, Potential Theory: New Methods and Problems with Solutions (Mir, Moscow, 2007) [in Russian].

    Google Scholar 

  5. G. N. Duboshin, Celestial Mechanics: Basic Problems and Methods (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  6. N. N. Lebedev, Special Functions and Their Applications (Fizmatgiz, Moscow, 1963; Prentice-Hall, New York, 1965).

    Google Scholar 

  7. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968; Nauka, Moscow, 1984).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Udmurtia State University, Izhevsk, 426034, Russia

    B. P. Kondrat’ev, A. S. Dubrovskii, N. G. Trubitsyna & É. Sh. Mukhametshina

Authors
  1. B. P. Kondrat’ev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. S. Dubrovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. G. Trubitsyna
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. É. Sh. Mukhametshina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. P. Kondrat’ev.

Additional information

Original Russian Text © B.P. Kondrat’ev, A.S. Dubrovskii, N.G. Trubitsyna, É.Sh. Mukhametshina, 2009, published in Zhurnal Tekhnicheskoĭ Fiziki, 2009, Vol. 79, No. 2, pp. 17–21.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kondrat’ev, B.P., Dubrovskii, A.S., Trubitsyna, N.G. et al. Laplace series expansion of the potential of a homogeneous circular torus. Tech. Phys. 54, 176–181 (2009). https://doi.org/10.1134/S1063784209020042

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063784209020042

PACS numbers

Search

Navigation