Classical Integrable Systems and Billiards Related to Generalized Jacobians

Acta Applicandae Mathematicae

Volumn 55, Issue 3, 1999, Pages 251-301

  Fedorov, Yu ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Billiards; Integrable systems; Jacobian varieties; Theta functions

Indexed keywords

EID: 0042877956     PISSN: 01678019     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1006178224117     Document Type: Article

References (33)

1. Adler, M. and van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras and curves, Adv. Math. 38 (1980), 267-317.
2. Alber, M. S., Camassa, R., Holm, D. and Marsden, J.: On the link between umbilic geodesics and soliton solutions of nonlinear PDEs, Proc. Roy. Soc. London Ser. A 450 (1995), 677-692.
3. Alber, M. S. and Marsden, J.: On geometric phases for soliton equations, Comm. Math. Phys. 149(2) (1992), 217-240.
4. Baker, H. F.: Abels Theorem and the Allied Theory Including the Theory of Theta Functions, Cambridge Univ. Press, Cambridge, 1897.
5. Belokolos, E. D., Bobenko, A. I., Enol'skii, V. Z., Its, A. R. and Matveev, V. B.: Algebro-Geometric Approach to Nonlinear Integrable Equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, 1994.
6. von Braunmühl, A.: Geodatische Linien und ihre Enveloppen auf dreiaxigen Flachen zweiten Grades, Math. Ann. 20 (1882), 557-586.
7. Buchstaber, V. M., Enolskii, V. Z. and Leikin, D. V.: Kleinian Functions, Hyperelliptic Jacobians and Applications, Amer. Math. Soc. Transl. Ser. 2 179, Providence, 1997.
8. Clebsch, A. and Gordan, P.: Theorie der abelschen Funktionen, Teubner, Leipzig, 1866.
9. Deift, P., Li, L-C. and Tomei, C.: Loop groups, discrete versions of some integrable systems and rank 2 extensions, Mem. Amer. Math. Soc. 479 (1992).
10. Delshams, A. and Ramirez-Ros, R.: Poincaré-Melnikov-Arnold method for analytic planar maps, Nonlinearity 9 (1996), 1-26.
11. Dubrovin, B. A.: Theta-functions and nonlinear equations, Russian Math. Surveys 36(2) (1981), 11-92.
12. Jacobi, C. G. J.: Sur la rotation d'un corps, in: Gesammelten Werke 2, 1884, pp. 139-172.
13. Jacobi, C. G. J.: Vorlesungen uber Dynamik, in: Gesammelten Werke, Supplementband, Berlin, 1884.
14. Jacobi, C. G. J.: Solution nouvelle d'un probleme fondamental de geodesie, in: Gesammelten Werke 2, 1884.
15. Ercolani, N.: Generalized theta-functions and homoclinic varieties, in: Proceedings of Symposia in Pure Math. 49, 1889.
16. Fay, J.: Theta-Functions of Riemann Manifolds, Lecture Notes in Math. 352, 1973.
17. Gagnon, L., Harnad, J., Hurtubise, J. and Winternitz, P.: Abelian integrals and the reduction method for an integrable Hamiltonian system, J. Math. Phys. 26 (1885), 1605-1612.
18. Gavrilov, L. and Zivkov, A.: The complex geometry of Lagrange top, Preprint Laboratorie de Topologie et Géométrie, Université Paul Sabatier, 1995. (To appear in Enseignment Mathematique.)
19. Königsberger, L.: Zur Transformation der Abelschen Functionen erster Ordnung, J. Reine Angew. Math. 64, 3-42.
20. Kötter, F.: Ueber eine Darstellung der Richtungcosinus zweier orthogonalen Coordinatesys-teme durch Thetafunctionen zweier Argumente, welche die Losungen mehrerer Probleme der Mechanik als Specialfalle umfasst, Sitzungsber. König, preuss Akad. Wiss., Berlin, 1895, pp. 807-814.
21. Kozlov, V. V. and Treschev, D. V.: Billiards. A Genetic Introduction to Systems with Impacts, Amer. Math. Soc. Publications, 1994.
22. Levallois, P.: Non-integrabilite des billiars definis par certaines perturbations algebriques d'une ellipse et du flot geodesique de certaines perturbations algebriques d'une ellipsoide, Thése, l'Université Paris VII, 1993.
23. Matveev, V. B.: Generalized Wronskian formula for solutions of the KdV equations: First applications, Phys. Lett. A 166 (1992), 205-208.
24. McKean, H. P.: Integrable Systems and Algebraic Curves, Lecture Notes in Math. 755, Springer-Verlag, 1979.
25. Moser, J. and Veselov, A.: Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys. 139 (1991), 217-243.
26. Mumford, D.: Tata Lectures on Theta II, Progress in Math., Birkhäuser, Basel, 1984.
27. Previato, E.: Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math. J. 52(1) (1985), 329-377.
28. Rosenlicht, M.: Generalized Jacobian varieties, Ann. of Math. 59 (1954), 505-530.
29. Soucaris, A.: The ampleness of the theta divisor on the compactified Jacobian of a proper and integral curve, Comput. Math. 93 (1994), 231-242.
30. Serre, J.-P.: Groupes algébriques et corps de classes, Hermann, Paris, 1959.
31. Weierstrass, K.: Über die geodätischen Linien auf dem dreiachsigen Ellipsoid, in: Mathematische Werke I, 1886, pp. 257-266.
32. Whittaker, E. T.: A Treatise on Analytical Dynamics, 3rd edn, Cambridge Univ. Press, Cambridge, 1927.
33. Veselov, A. P.: Integrable systems with discrete time and difference operators, Funktsional. Anal, i Prilozhen. 22 (1988), 1-13 (in Russian).