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The Kowalewski top 99 years later: A Lax pair, generalizations and explicit solutions

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A “natural” Lax pair for the Kowalewski top is derived by using a general group-theoretic approach. This gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite-band integration theory.

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References

  1. Kowalewski, S.: Sur le problème de la rotation d'un corps solide autour d'un point fixe. Acta Math.12, 177–232 (1889)

    Google Scholar 

  2. Kötter, F.: Sur le cas traité par Mme Kowalewski de rotation d'un corps solide autour d'un point fixe. Acta Math.17, 209–264 (1893)

    Google Scholar 

  3. Dubrovin, B.A., Krichever, J.M., Novikov, S.P.: Integrable systems. I. In: Modern problems in mathematics. Fundamental developments. Vol. 4 (Dynamical systems 4) Itogi Nauki i Tekhniki. Moscow: VINITI 1985, pp. 179–285

    Google Scholar 

  4. Perelomov, A.M.: Lax representations for the systems of S. Kowalewskaya type. Commun. Math. Phys.81, 239–243 (1981)

    Google Scholar 

  5. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Lax representation with a spectral parameter for the Kowalewski top and its generalizations. Lett. Math. Phys.14, 55–62 (1987)

    Google Scholar 

  6. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations I, II. Invent. Math.54, 81–100 (1979);63, 423–432 (1981)

    Google Scholar 

  7. Reyman, A.G., Semenov-Tian-Shansky, M.A.: A new integrable case of the motion of the 4-dimensional rigid body. Commun. Math. Phys.105, 461–472 (1986)

    Google Scholar 

  8. Haine, L., Horozov, E.: A Lax pair for Lowalewski's top. Physica29 D, 173–180 (1987)

    Google Scholar 

  9. Adler, M., van Moerbeke, P.: The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows onSO(4) — A two-dimensional family of Lax pairs. Commun. Math. Phys.113, 659–700 (1988)

    Google Scholar 

  10. Horozov, E., van Moerbeke, P.: Abelian surfaces of polarization (1.2) and Kowalewski's top. Commun. Pure Appl. Math. (to appear) 1988

  11. Dubrovin, B.A., Matveev, V.B., Novikov, S.P.: Nonlinear equations of KdV type, finite-zone linear operators and abelian varieties. Usp. Mat. Nauk31, 55–136 (1976) [Russ. Math. Surv.31, 59–146 (1976)]

    Google Scholar 

  12. Veselov, A.P., Novikov, S.P.: Poisson brakcets and complex tori. Trudy Mos. Mat. O.-va165, 49–61 (1984)

    Google Scholar 

  13. Bobenko, A.I.: Solutions of the classical integrable tops via the inverse scattering method. LOMI Preprint P-10-87, Leningrad 1987

  14. Bogoyavlensky, O.I.: Euler equations on finite-dimensional Lie algebras arising in physical problems. Commun. Math. Phys.95, 307–315 (1984)

    Google Scholar 

  15. Komarov, I.V.: A generalization of the Kowalewski top. Phys. Lett.123 A, 14–15 (1987)

    Google Scholar 

  16. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Group-theoretical methods in the theory of integrable systems. In: Modern Problems in Mathematics. Fundamental Developments, Vol. 16 (Dynamical systems 7) Itogi Nauki i Tekhniki. Moscow: VINITI 1987, pp. 119–195

    Google Scholar 

  17. Reyman, A.G.: Integrable Hamiltonian systems connected with graded Lie algebras. In: Differential geometry. Lie groups and mechanics II. Zap. Nauchn. Semin. LOMI95, 3–54 (1980) [J. Soviet Math.19, 1507–1545 (1982)]

    Google Scholar 

  18. Guillemin, V., Sternberg, S.: The moment map and collective motion. Ann. Phys.127, 220–253 (1980)

    Google Scholar 

  19. Griffiths, P., Harris, J.: Principles of algebraic geometry. New York: Wiley 1978

    Google Scholar 

  20. Fay, J.D.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, vol.352. Berlin, Heidelberg, New York: Springer 1973

    Google Scholar 

  21. Bobenko, A.I., Kuznetsov, V.B.: Lax representation for the Goryachev-Chaplygin top and new formulae for its solutions. J. Phys. A21, 1999–2006 (1988)

    Google Scholar 

  22. Reyman, A.G., Semenov-Tian-Shansky, M.A.: Lax representation with a spectral parameter for the Kowalewski top and its generalizations. In: Plasma theory and nonlinear and turbulent processes in physics (Proceedings of the 1987 Kiev conference), pp. 135–152. Singapore: World Scientific 1988

    Google Scholar 

  23. Reyman, A.G.: New results on the Kowalewski top. In: Nonlinear Evolutions (Proceedings of the IV Workshop on Nonlinear Evolution Equations and Dynamical Systems, Balaruc-les-Bains, 1987). Singapore: World Scientific 1988

    Google Scholar 

  24. Yehia H.: New integrable cases of the motion of a gyrostat. Vestn. Mosk. Univ., Ser. I, No.4, 88–90 (1987)

    Google Scholar 

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Authors and Affiliations

  1. Leningrad Branch of V. A. Steklov Mathematical Institute, Fontanka 27, SU-191011, Leningrad, USSR

    A. I. Bobenko, A. G. Reyman & M. A. Semenov-Tian-Shansky

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  1. A. I. Bobenko
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  2. A. G. Reyman
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  3. M. A. Semenov-Tian-Shansky
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Communicated by Ya. G. Sinai

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Bobenko, A.I., Reyman, A.G. & Semenov-Tian-Shansky, M.A. The Kowalewski top 99 years later: A Lax pair, generalizations and explicit solutions. Commun.Math. Phys. 122, 321–354 (1989). https://doi.org/10.1007/BF01257419

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  • DOI: https://doi.org/10.1007/BF01257419

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