Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable

Year 2020, Volume: 3 Issue: 1, 24 - 52, 25.03.2020

Lesfari Ahmed

https://doi.org/10.33434/cams.649612

https://izlik.org/JA82XH57NR

Abstract

The aim of this paper is to demonstrate the rich interaction between the Kowalewski-Painlev\'{e} analysis, the properties of algebraic completely integrable (a.c.i.) systems, the geometry of its Laurent series solutions, and the theory of Abelian varieties. We study the classification of metrics for which geodesic flow on the group $SO(n)$ is a.c.i. For $n=3$, the geodesic flow on $SO(3)$ is always a.c.i., and can be regarded as the Euler rigid body motion. For $n=4$, in the Adler-van Moerbeke's classification of metrics for which geodesic flow on $SO(4)$ is a.c.i., three cases come up; two are linearly equivalent to the Clebsch and Lyapunov-Steklov cases of rigid body motion in a perfect fluid, and there is a third new case namely the Kostant-Kirillov Hamiltonian flow on the dual of $so(4)$. Finally, as was shown by Haine, for $n\geq 5$ Manakov's metrics are the only left invariant diagonal metrics on $SO(n)$ for which the geodesic flow is a.c.i.

Keywords

Jacobians varieties , Prym varieties , Integrable systems , Topological structure of phase space , Methods of integration

References

• [1] M. Adler, P. van Moerbeke, Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. in Math., 38 (1980), 318-379.
• [2] M. Adler, P. van Moerbeke, The algebraic complete integrability of geodesic flow on SO(4), Invent. Math., 67 (1982), 297-331, with an appendix by D. Mumford.
• [3] M. Adler, P. van Moerbeke, Algebraic completely integrable systems : a systematic approach, I, II, III, Seminaire de Math´ematique, Rapport No 110, p.1-145, SC/MAPA - Institut de mathematique pure et appliqu´ee, UCL, 1985.
• [4] M. Adler, P. van Moerbeke, The Intersection of Four Quadrics in P6, Abelian Surfaces and their Moduli, Math. Ann., 279 (1987), 25-85.
• [5] M. Adler, P. van Moerbeke, The complex geometry of the Kowalewski-Painleve analysis, Invent. Math., 97 (1989), 3-51.
• [6] M. Adler, P. van Moerbeke, P. Vanhaecke, Algebraic integrability, Painleve geometry and Lie algebras, A series of modern surveys in mathematics, Volume 47, Springer-Verlag, 2004.
• [7] V.I.: Arnold, Mathematical methods in classical mechanics, Springer-Verlag, Berlin-Heidelberg- New York, 1978, 2nd edn., Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989.
• [8] A. Clebsch, Der Bewegung eines starren Körpers in einen Fl¨ussigkeit, Math. Ann., 3 (1871), 238-268.
• [9] R. Donagi, E. Markman, Spectral covers, algebraically completely integrable, hamiltonian systems, and moduli of bundles, Lecture Notes in Mathematics, 1620, Springer 1996.
• [10] B.A. Dubrovin, Theta functions and non-linear equations, Russian Math. Surv., 36, 2 (1981), 11-92.
• [11] L. Euler, Theoria motus corporum solidorum seu rigidorum, Rostock, 1765. Memoires Acad. Sc, Berlin, 1758.
• [12] P. A. Griffiths, J. Harris, Principles of algebraic geometry, Wiley-Interscience, New-York, 1978.
• [13] L. Haine, Geodesic flow on SO(4) and Abelian surfaces, Math. Ann., 263 (1983), 435-472.
• [14] L. Haine, The algebraic complete integrability of geodesic flow on SO(N) and Abelian surfaces, Comm. Math. Phys., 94(2) (1984), 271-287.
• [15] N. Hitchin, Stable bundles and integrable systems, Duke Mathematical Journal, 54,1 (1994), 91-114.
• [16] C. G. J. Jacobi, Vorlesungen über Dynamik, Königsberg lectures of 1842-1843, (reprinted by Chelsea Publishing Co., New York, 1969.
• [17] G. Kirchoff, Vorlesungen über Mathematische Physik, Vol. 1, Mechanik. Teubner, Leipzig, 1876.
• [18] H. Knörrer, Integrable Hamiltonsche Systeme und algebraische Geometrie, Jahresber. Deutsch. Math.- Verein., 88 (2) (1986), 82-103.
• [19] F. Kötter, Uber die Bewegung eines festen K¨orpers in einer Flüssigkeit I, II, Journal f¨ur die reine und angewandte Mathematik, 109 (1892), 51-81, 89-111.
• [20] F. Kötter, Die von Steklow und Lyapunov entdeckten intgralen Falle der Bewegung eines Körpers in einen Fl¨ussigkeit Sitzungsber, K¨oniglich Preussische Akad. d. Wiss., Berlin 6, 79-87 (1900).
• [21] S. Kowalewski, Sur le probleme de la rotation d’un corps solide autour d’un point fixe, Acta Math., 12 (1889), 177-232.
• [22] A. Lesfari, Geodesic flow on SO(4), Kac-Moody Lie algebra and singularities in the complex t-plane, Publ. Mat., Barc., 431 (1999), 261-279.
• [23] A. Lesfari, The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch’s case, NoDEA, Nonlinear diff.Equ. Appl., 81 (2001), 1-13.
• [24] A. Lesfari, Etude des solutions m´eromorphes d’´equations diff´erentielles, Ren. Semin. Mat. Univ. Politec. Torino, 654 (2007), 451-464.
• [25] A. Lesfari, Prym varieties and applications, J. Geom. Phys., 58 (2008), 1063-1079.
• [26] A. Lesfari, Algebraic integrability : the Adler-van Moerbeke approach, Regul. Chaotic Dyn., 16 (3-4) (2011), 187-209.
• [27] A. Lesfari, Introduction a la geometrie algebrique complexe, Editions Hermann, Paris, 2015.
• [28] A. Lyapunov, On a property of the differential equations of the problem of motion of a rigid body having a fixed point, Soobshceniya Kharkovskogo matematicheskog obshchestva (Transactions of the Kharkov Mathematical Society), Vol. IV (1894) and Reports of Kharkov. Math. Soc. Ser. 24, No 1-2 (1893), 81-85, Gesammelte Werke, Vol.1, 320-324.
• [29] S. V. Manakov, Remarks on the Integrals of the Euler Equations of the n-Dimensional Heavy Top, Fund. Anal. Appl., 10(4) (1976), 93-94.
• [30] B.G. Moishezon, On n-dimensional compact varieties with n algebraically independent meromorphic functions, Amer. Math. Soc. Transl., 63 (1967), 51-177.
• [31] J. Moser, Geometry of quadrics and spectral theory, Lecture delivred at the symposium in honor of S.S. Chern, Berkeley, 1979. Springer, Berlin, Heidelberg, New-York, 1980.
• [32] D. Mumford, On the equations defining Abelian varieties I, II, III, Invent. Math., 1 (1966), 287-354, 3 (1967), 75-135, 3 (1967), 215-244.
• [33] D. Mumford, Algebraic geometry I: complex projective varieties, Springer-Verlag, 1975.
• [34] D. Mumford, Tata lectures on theta I, II, Progress in Math., Birkhaüser, Boston, 1983.
• [35] P. Painleve, Oeuvres, tomes 1,2,3. ´ Edition du C.N.R.S., 1975.
• [36] L. Poinsot, Theorie nouvelle de la rotation des corps, Journal de Liouville, Volume 16, (1851).
• [37] V.A. Steklov, Über die Bewegung eines festen Körper in einer Flüssigkeit, Math. Ann., 42 (1893), 273-374.
• [38] P. Vanhaecke, Integrable systems in the realm of algebraic geometry, Lecture Notes in Math., 1638, Springer-Verlag, Berlin, 2001.