Skip to main content
SpringerLink
Log in

The Goryachev-Chaplygin top and the Toda lattice

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The Goryachev-Chaplygin top is a rigid body rotating about a fixed point with principal moments of inertiaA, B, C satisfyingA=B=4C and with center of mass lying in the equatorial plane. The problem is algebraically completely integrable as a linear flow on a hyperelliptic Jacobian, only upon putting the principal angular momentum in the horizontal plane.

This system admits asymptotic solutions with fractional powers int and depending on 4 degrees of freedom. As a consequence, the affine invariant surfaces of the Chaplygin top are double covers of the hyperelliptic Jacobian above, ramified along two translates of the theta divisor, touching in one point. This system is an instance of a (master) system of differential equations in 7 unknowns having 5 quadratic constants of motion; a careful analysis of this system reveals an intimate (rational) relationship with the 3-body periodic Toda lattice.

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.

Similar content being viewed by others

References

  1. Adler, M., van Moerbeke, P.: The algebraic integrability of geodesic flow onSO(4). Invent. math.67, 297–326 (1982), with an appendix by D. Mumford

    Google Scholar 

  2. Adler, M., van Moerbeke, P.: The intersection of four quadrics in IP6 and its moduli. Math. Ann. (to appear in 1987)

  3. Adler, M., van Moerbeke, P.: Kowalewski's asymptotic method, Kac-Moody Lie algebras and regularization. Commun. Math. Phys.83, 83–106 (1982)

    Google Scholar 

  4. Adler, M., van Moerbeke, P.: A systematic approach towards solving integrable systems. Perspectives in Mathematics. New York: Academic Press (to appear in 1987)

  5. Chaplygin, S.A.: A new case of rotation of a rigid body, supported at one point (Collected works), Vol. I, Moscow: Gostekhizdat 1948, pp. 118–124

    Google Scholar 

  6. Golubev, V.V.: Lectures on the integration of equations of motions of a heavy solid about a fixed point. Moscow: Gostekhizdat 1953 (English translation by Israeli Information agency)

    Google Scholar 

  7. Goryachev, D.: On the motion of a rigid material body about a fixed point in the caseA=B=4C. Mat. Sb.21, 3 (1900)

    Google Scholar 

  8. van Moerbeke, P.: The spectrum of Jacobi matrices. Invent. math.37, 45–81 (1976)

    Google Scholar 

  9. Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge, MA: Cambridge University Press 1984

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Louvain, B-1348, Louvain-la-Neuve, Belgium

    C. Bechlivanidis & P. van Moerbeke

  2. Department of Mathematics, Brandeis University, 02254, Waltham, MA, USA

    P. van Moerbeke

Authors
  1. C. Bechlivanidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. van Moerbeke
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

The support of National Science Foundation grant No. DMS-8403136 is gratefully acknowledged. I thank Victor Guillemin for mentioning this problem

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bechlivanidis, C., van Moerbeke, P. The Goryachev-Chaplygin top and the Toda lattice. Commun.Math. Phys. 110, 317–324 (1987). https://doi.org/10.1007/BF01207371

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01207371

Keywords

Search

Navigation