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Completely integrable systems: Jacobi’s heritage. (English) Zbl 0937.37046

Integrable systems are studied from the point of view of algebraic geometry, inverse spectral problems and mechanics from the point of view of Lie groups. The theoretical Lie algebra method leading to completely integrable systems based on the Kostant-Kirillov coadjoint action is studied. It is shown that algebraic integrability means that the system is completely integrable in the sense of the phase space being foliated by tori. Adler-van Moerbeke’s method turns out to be an effective tool to characterize and describe the algebraic nature of the invariant tori for the algebraic completely integrable systems. Certain integrable systems, such as Korteweg-de Vries equation, Toda lattice, Euler rigid body motion, Kowalewski’s top, Manakov’s geodesic flow on \(SO(4)\) etc. are treated.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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[1] Adler, M., On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equation, Invent. Math., 50, 219-248 (1979) · Zbl 0393.35058
[2] Adler, M.; van Moerbeke, P., Completely integrable systems, Euclidean Lie algebras and curves, Adv. Math., 38, 267-317 (1980) · Zbl 0455.58017
[3] Adler, M.; van Moerbeke, P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. Math., 38, 318-379 (1980) · Zbl 0455.58010
[4] Adler, M.; van Moerbeke, P., Kowalewski’s asymptotic method, Kac-Moody Lie algebras and regularization, Comm. Math. Phys., 83, 83-106 (1982) · Zbl 0491.58017
[5] Adler, M.; van Moerbeke, P., The algebraic complete integrability of geodesic flow on \(SO (4)\), Invent. Math., 67, 297-331 (1982) · Zbl 0539.58012
[6] Adler, M.; van Moerbeke, P., The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows on \(SO(4)\) — A two-dimensional family of Lax pairs, Comm. Math. Phys., 113, 659-700 (1988) · Zbl 0647.58022
[7] Arnold, V. I., Mathematical Methods in Classical Mechanics (1978), Springer: Springer Berlin · Zbl 0386.70001
[8] Clebsch, A., Der Bewegung eines starren Körpers in einen Flüssigkeit, Math. Ann., 3, 238-268 (1971)
[9] Dubrovin, B. A.; Novikov, S. P., Periodic and conditionally periodic analogues of multi-soliton solutions of the KdV equation, Dokl. Akad. Nauk URSS, 6, 2131-2144 (1974)
[10] Flaschka, H., The Toda lattice I, Phys. Rev. B, 9, 1924-1925 (1974) · Zbl 0942.37504
[11] Flaschka, H., The Toda lattice II, Progr. Theoret. Phys., 51, 703-716 (1974) · Zbl 0942.37505
[12] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Koteweg-de Vries equation, Phys. Rev. Lett., 10, 1095-1097 (1967) · Zbl 1061.35520
[13] Griffiths, P. A., Linearizing flows and a comological interpretation of Lax equations, Amer. J. Math., 107, 1445-1483 (1985) · Zbl 0585.58028
[14] Haine, L., Geodesic flow on \(SO(4)\) and Abelian surfaces, Math. Ann., 263, 435-472 (1983) · Zbl 0521.58042
[15] Hitchin, N. J., On the construction of monopoles, Comm. Math. Phys., 89, 145-190 (1983) · Zbl 0517.58014
[16] Jacobi, C. G.J., Vorlesungen über Dynamik, (Gesammelte Werke, Supplementband. Gesammelte Werke, Supplementband, Berlin (1980))
[17] Knörrer, H., Geodesics on the ellipsoid, Invent. Math., 59, 119-144 (1980) · Zbl 0431.53003
[18] Kostant, B., The solution to a generalized Toda lattice and representation theory, Adv. Math., 34, 195-338 (1979) · Zbl 0433.22008
[19] Kötter, F., Uber die Bewegung eines festen Körpers in einer Flüssigkeit I, II, J. Riene Angew. Math., 109, 89-110 (1892) · JFM 24.0908.01
[20] Kötter, F., Die von Steklow und Lyapunov entdeckten integralen Fälle der Bewegung eines Körpers in einen Flüssigkeit Sitzungsber, Königlich Preussische Akad. d. Wiss., Berlin, vol. 6, 79-87 (1990) · JFM 31.0726.02
[21] Kowalewski, S., Sur le problème de la rotation d’un corps solide autour d’un point fixe, Acta Math., 12, 177-232 (1889) · JFM 21.0935.01
[22] Kozlov, V. V., Russian Math. Surveys, 38, 1-76 (1983), Transl 1 · Zbl 0525.70023
[23] Krichever, M., Algebro-geometrical construction of the Zakharov Shabat equations and their periodic solutions, Sov. Math. Dokl., 17, 394-397 (1976) · Zbl 0361.35007
[24] Lax, P., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math., 21, 467-490 (1968) · Zbl 0162.41103
[25] Lebedev, D. R.; Manim, Yu. I., The Gelfand-Dikii Hamiltonian operator and the coadjoint representation of the Volterra group, Func. Anal. Appl., 13, 40-46 (1979)
[26] Lesfari, A., Equation de Korteweg-de Vries, (Mem. Dpt. Maths. Fac. Sci. (1982), U.C.L: U.C.L Louvainla-Neuve, Belgique), 1-113
[27] Lesfari, A., Une approache systématique à la résolution due corps solide de Kowalewski, C.R. Acad. Sc. Paris, t. 302, 347-350 (1986), Série I · Zbl 0606.34012
[28] Lesfari, A., Une approche systématique à la résolution des systémes intégrables, (Proc. de la \(2^e\) école de gémétrie- analyse (1987), E.H.T.P: E.H.T.P Casablanca, Maroc), 83, (Conférences organisées en l’honneur du Pr. A. Lichnerowicz)
[29] Lesfari, A., Abelian surfaces and Kowalewski’s top, Ann. Scient. Ec. Norm. Sup., Paris, t. 21, 193-223 (1988), \(4^e\) série · Zbl 0667.58019
[30] Lesfari, A., Systèmes Hamiltoniens algébriquement complètement intégrables, Preprint, Dpt. Maths. Fac. Sc., 1-38 (1990), El-Jadida, Maroc
[31] Lesfari, A., Eléments d’analyse (1991), Sochepress-Université: Sochepress-Université Casablanca, Maroc
[32] Lesfari, A., On affine surface that can be completed by a smooth curve, Results Math., 35, 107-118 (1999) · Zbl 0947.14022
[33] Lesfari, A., Geodesic flow on SO(4), Kac-Moody Lie algebra and its singularities in the complex \(t\)-plane, Publications Matemàtiques, 43 (1999), in press · Zbl 0968.35010
[34] Lesfari, A., Une méthode de compactification de variétés liées aux systèmes dynamiques, Cahiers de la recherche, Rectorat-Université Hassan II Aïn Chock, 1, 1, 147-157 (1999)
[35] Lyapunov, A. M., Report of Kharkov. Math. Soc. Ser. 2, 4, 81-85 (1893), (Gesammelte Werke, vol. 1, 320-324)
[36] Mc Kean, H. P.; van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30, 217-274 (1975) · Zbl 0319.34024
[37] Manakov, S. V., Remarks on the integrals of the Euler equations of the \(n\)-dimensional heavy top, Func. Anal. Appl., 10, 93-94 (1976) · Zbl 0343.70003
[38] Melnikov, V. K., On the stability of the center for time-periodic perturbations, Trans. Mosc. Math. Soc., 12, 1-57 (1963)
[39] van Moerbeke, P., The spectrum of Jacobi matrices, Invent. Math., 37, 45-81 (1976) · Zbl 0361.15010
[40] van Moerbeke, P., Algebraic complete integrability of Hamiltonian systems and Kac-Moody Lie algebras, (Proc. Int. Cong. Math. (1983)), Warzawa · Zbl 0566.58019
[41] van Moerbeke, P.; Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math., 143, 93-154 (1979) · Zbl 0502.58032
[42] Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math., 16, 197-219 (1975) · Zbl 0303.34019
[43] Moser, J., Geometry of quadrics and spectral theory, (Lecture delivered at the symposium in honor of S.S. Chern. Lecture delivered at the symposium in honor of S.S. Chern, Berkeley, 1979 (1980), Springer: Springer Berlin) · Zbl 0455.58018
[44] Mumford, D., (Tata lectures on theta. II. Progress in Math (1982), Birkhaüser: Birkhaüser Boston) · Zbl 0744.14033
[45] Neumann, C., De problemate quodam mechanics, quod ad primam integralium ultraellipticorum classem revocatur, Reine und Angew. Math. J., 56, 44-63 (1859) · ERAM 056.1472cj
[46] Steklov, V., On the motion of a solid in a liquid, Math. Ann., 42, 273-394 (1893)
[47] Symes, W., Systems of Toda type, Inverse spectral problems and representation theory, Invent. Math., 59, 13-53 (1980) · Zbl 0474.58009
[48] Yoshida, H., Necessary conditions for the existence of algebraic first integrals. I. Kowalewski’s exponents. II. Conditions for algebraic integrability, Celestial Mech., 31, 381-399 (1983) · Zbl 0556.70015
[49] Zaharov, V.; Shabat, A., A schema for integrating the non-linear equations of Math. Physics by the method of the inverse scattering problem I, Funct. Anal. Appl., 8 (1974)
[50] Ziglin, S. L., Splitting of sepatrices, branching of solutions and non-existence of an integral in the dynamics of a solid body, Trans. Moscow Math. Soc., 41, 287-303 (1980), (Transl. 1 (1982) 283-298) · Zbl 0466.70009
[51] Ziglin, S. L., Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. 1, Funct. Anal. Appl., 16, 30-41 (1982)
[52] Ziglin, S. L., Branching of solutions and non-existence of first integrals in Hamiltonian mechanics. 2, Func. Anal. Appl., 17, 8-23 (1983) · Zbl 0524.58015
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