Abstract
A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, has attracted full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skillful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables w, x 1, x 2 as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to the n = 3 case of the Great Poncelet Theorem for pencils of conics.
Similar content being viewed by others
References
Appel’rot, G.G.: Some suplements to the memoir of N. B. Delone. Tr. otd. fiz. nauk, 6 (1893)
Audin, M.: Spinning Tops. An introduction to integrable systems. Cambridge studies in advanced mathematics 51, Cambridge: Cambridge Univ. Press, 1999
Berger, M.: Geometry. Berlin: Springer-Verlag, 1987
Bobenko A.I., Reyman A.G., Semenov-Tian-Shansky M.A.: The Kowalevski top 99 years later: a Lax pair, generalizations and explicit solutions. Commun. Math. Phys. 122, 321–354 (1989)
Buchstaber, V.M., Novikov, S.P.: Formal groups, power systems and Adams operators. Mat. Sb. (N. S) 84 (126), 81–118 (1971) (in Russian)
Buchstaber V.M., Rees E.G.: Multivalued groups, their representations and Hopf algebras. Transform. Groups 2, 325–349 (1997)
Buchstaber V.M., Veselov A.P.: Integrable correspondences and algebraic representations of multivalued groups. Internat. Math. Res. Notices 1996, 381–400 (1996)
Buchstaber V.: n-valued groups: theory and applications. Moscow Math. J. 6, 57–84 (2006)
Darboux, G.: Principes de géométrie analytique. Paris: Gauthier-Villars, 1917, 519 p
Darboux, G.: Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitesimal. Volumes 2 and 3, Paris: Gauthier-Villars, 1887, 1889
Delone, N.B.: Algebraic integrals of motion of a heavy rigid body around a fixed point. Petersburg, 1892
Dragović V.: Multi-valued hyperelliptic continuous fractions of generalized Halphen type. Internat. Math. Res. Notices 2009, 1891–1932 (2009)
Dragović, V.: Marden theorem and Poncelet-Darboux curves. http://arXiv./org/abs/0812.4829v1[math.CA], 2008
Dragović V., Gajić B.: Systems of Hess-Appel’rot type. Commun. Math. Phys. 265, 397–435 (2006)
Dragović V., Radnović M.: Geometry of integrable billiards and pencils of quadrics. J. Math. Pures Appl. 85, 758–790 (2006)
Dragović V., Radnović M.: Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms. Adv. Math. 219, 1577–1607 (2008)
Dubrovin B.: Theta - functions and nonlinear equations. Usp. Math. Nauk 36, 11–80 (1981)
Dullin H.R., Richter P.H., Veselov A.P.: Action variables of the Kowalevski top. Reg. Chaotic Dynam. 3, 18–26 (1998)
Euler L.: Evolutio generalior formularum comparationi curvarum inservientium. Opera Omnia Ser 1 20, 318–356 (1765)
Golubev, V.V.: Lectures on the integration of motion of a heavy rigid body around a fixed point. Moscow: Gostechizdat, 1953 [in Russian], English translations: Israel Program for Scientific washington, DC: US Dept. of Commerce, Off, of Tech. Serv., 1960
Hirota, R.: The direct mthod in soliton theory. Cambridge Tracts in Mathematics 155, Cambridge: Cambridge Univ. Press, 2004
Horozov E., van Moerbeke P.: The full geometry of Kowalevski’s top and (1,2)-abelian surfaces. Comm. Pure Appl. Math. 42, 357–407 (1989)
Jurdjevic V.: Integrable Hamiltonian systems on Lie Groups: Kowalevski type. Ann. Math. 150, 605–644 (1999)
Kotter F.: Sur le cas traite par M-me Kowalevski de rotation d’un corps solide autour d’un point fixe. Acta Math. 17, 209–263 (1893)
Kowalevski S.: Sur la probleme de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12, 177–232 (1889)
Kowalevski S.: Sur une propriete du systeme d’equations differentielles qui definit la rotation d’un corps solide autour d’un point fixe. Acta Math. 14, 81–93 (1889)
Kuznetsov, V.B.: Kowalevski top revisted. CRM Proc. Lecture Notes 32, Providence, RI: Amer. Math. Soc., 2002, pp. 181–196
Markushevich D.: Kowalevski top and genus-2 curves. J. Phys. A 34(11), 2125–2135 (2001)
Mlodzeevskii, B.K.: About a case of motion of a heavy rigid body around a fixed point. Mat. Sb. 18 (1895)
Poncelet, J.V.: Traité des propriétés projectives des figures. Paris: Mett, 1822
Vein, R., Dale, P.: Determinants and their applications in Mathematical Physics. Appl. Math. Sciences 134, Berlin-Heidelberg-New York: Springer, 1999
Veselov A.P., Novikov S.P.: Poisson brackets and complex tori. Trudy Mat. Inst. Steklov 165, 49–61 (1984)
Weil, A.: Euler and the Jacobians of elliptic curves. In: Arithmetics and Geometry, Vol. 1, Progr. Math. 35, Boston, MA: Birkhauser, 1983, pp. 353–359
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Aizenman
Dedicated to my teacher Boris Anatol’evich Dubrovin on the occasion of his sixtieth birthday
Rights and permissions
About this article
Cite this article
Dragović, V. Geometrization and Generalization of the Kowalevski Top. Commun. Math. Phys. 298, 37–64 (2010). https://doi.org/10.1007/s00220-010-1066-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-010-1066-z