The complex geometry of an integrable system
Lesfari, Ahmed
Archivum Mathematicum, Tome 039 (2003), p. 257-270 / Harvested from Czech Digital Mathematics Library
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In this paper, a finite dimensional algebraic completely integrable system is considered. We show that the intersection of levels of integrals completes into an abelian surface (a two dimensional complex algebraic torus) of polarization $\left( 2,8\right) $ and that the flow of the system can be linearized on it.

Publié le : 2003-01-01
Classification:  14H70,  37J35,  70G55,  70H06 
@article{107873,
     author = {Ahmed Lesfari},
     title = {The complex geometry of an integrable system},
     journal = {Archivum Mathematicum},
     volume = {039},
     year = {2003},
     pages = {257-270},
     zbl = {1110.70022},
     mrnumber = {2028736},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107873}
}

Lesfari, Ahmed. The complex geometry of an integrable system. Archivum Mathematicum, Tome 039 (2003) pp. 257-270. http://gdmltest.u-ga.fr/item/107873/

Adler M.; Van Moerbeke P. The algebraic complete integrability of geodesic flow on $SO(4)$, Invent. Math. 67 (1982), 297–331. (1982) | MR 0665159

Arbarello E.; Cornalba M.; Griffiths P. A.; Harris J. Geometry of algebraic curves I, Springer-Verlag, 1994. (1994)

Arnold V. I. Mathematical methods in classical mechanics, Springer-Verlag, Berlin-Heidelberg-New York, 1978. (1978) | MR 0690288

Belokolos E. D.; Bobenko A. I.; Enolskii V. Z.; Its A. R.; Matveev V. B. Algebro-Geometric approach to nonlinear integrable equations, Springer-Verlag, 1994. (1994)

Christiansen P. L.; Eilbeck J. C.; Enolskii V. Z.; Kostov N. A. Quasi-periodic solutions of the coupled nonlinear Schrödinger equations, Proc. Roy. Soc. London Ser. A 451 (1995), 685–700. (1995) | MR 1369055

Griffiths P. A.; Harris J. Principles of algebraic geometry, Wiley-Interscience, 1978. (1978) | MR 0507725 | Zbl 0408.14001

Haine L. Geodesic flow on $SO(4)$ and Abelian surfaces, Math. Ann. 263 (1983), 435–472. (1983) | MR 0707241 | Zbl 0521.58042

Lesfari A. Une approche systématique à la résolution du corps solide de Kowalewski, C. R. Acad. Sc. Paris, série I, t. 302 (1986), 347–350. (1986) | MR 0837502

Lesfari A. Abelian surfaces and Kowalewski’s top, Ann. Scient. École Norm. Sup. 4, 21 (1988), 193–223. (1988) | MR 0956766 | Zbl 0667.58019

Lesfari A. On affine surface that can be completed by a smooth curve, Results Math. 35 (1999), 107–118. (1999) | MR 1678068 | Zbl 0947.14022

Lesfari A. Une méthode de compactification de variétés liées aux systèmes dynamiques, Les cahiers de la recherche, Rectorat-Université Hassan II-Aïn Chock, Casablanca, Maroc, Vol. I, No. 1, (1999), 147–157. (1999)

Lesfari A. Geodesic flow on $SO(4)$, Kac-Moody Lie algebra and singularities in the complex t-plane, Publ. Mat. 43 (1999), 261–279. (1999) | MR 1697525

Lesfari A. Completely integrable systems: Jacobi’s heritage, J. Geom. Phys. 31 (1999), 265–286. (1999) | MR 1711527 | Zbl 0937.37046

Lesfari A. The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch’s case, Nonlinear Differential Equations Appl. 8 (2001), 1–13. | MR 1828945 | Zbl 0982.35085

Lesfari A. The generalized Hénon-Heiles system, Abelian surfaces and algebraic complete integrability, Rep. Math. Phys. 47 (2001), 9–20. | MR 1823005 | Zbl 1054.37038

Lesfari A. A new class of integrable systems, Arch. Math. 77 (2001), 347–353. | MR 1853551 | Zbl 0996.70014

Lesfari A. The Hénon-Heiles system via the Kowalewski-Painlevé analysis, Int. J. Theor. Phys. Group Theory Nonlinear Opt. 9, N${{}^{\circ }}$4 (2003), 305–330. | MR 2128205

Lesfari A. Le théorème d’Arnold-Liouville et ses conséquences, Elem. Math. 58, Issue 1 (2003), 6–20. | MR 1961831 | Zbl 1112.37043

Lesfari A. Le système différentiel de Hénon-Heiles et les variétés Prym, Pacific J. Math. 212, No. 1 (2003), 125–132. | MR 2016973

Mumford D. Tata lectures on theta II, Progr. Math., Birkhaüser, Boston, 1982. (1982)

Mumford D. On the equations defining Abelian varieties, Invent. Math. 1 (1966), 287–354. (1966) | MR 0204427 | Zbl 0219.14024

Perelomov A. M. Integrable systems of classical mechanics and Lie algebras, Birkhäuser, 1990. (1990) | MR 1048350 | Zbl 0717.70003