Completely integrable systems, Euclidean Lie algebras, and curves. (English) Zbl 0455.58017


37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53D50 Geometric quantization
70H05 Hamilton’s equations
70H99 Hamiltonian and Lagrangian mechanics
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B99 Lie algebras and Lie superalgebras
14H40 Jacobians, Prym varieties
Full Text: DOI


[1] Arnold, V. I., Mathematical Methods of Classical Mechanics (1978), Springer-Verlag: Springer-Verlag New York · Zbl 0386.70001
[2] Kazhdan, D.; Kostant, B.; Sternberg, S., Hamiltonian Group Actions and Dynamical Systems of Calogero Type (July 1978), CPAM · Zbl 0368.58008
[3] van Moerbeke, P.; Mumford, D., The spectrum of difference operators and algebraic curves, Acta Math. (1979) · Zbl 0502.58032
[4] Russian Math. Surveys, 31, No. 1 (1976)
[5] Adler, M., On a trace functional for pseudo-differential operators and the symplectic structure of the Korteweg-deVries Equation, Invent. Math., 50, No. 3 (1979) · Zbl 0393.35058
[6] van Moerbeke, P., The spectrum of Jacobi matrices, Invent. Math., 37 (1976) · Zbl 0361.15010
[7] J. Moser; J. Moser · Zbl 0455.58018
[8] Kostant, B., The solution to a generalized Toda lattice and representation theory, Advances in Math., 34, 195-338 (1979) · Zbl 0433.22008
[9] Mishchenko, A. S.; Fomenko, A. T., Euler’s equations on finite-dimensional Lie groups, Izv. Akad. Nauk. SSSR Ser. Mat., 42 (1978) · Zbl 0383.58006
[10] Dikii, L. A., Hamiltional systems connected with the rotation group, Funksional. Anal. i Prilozen., 6, No. 4, 83-84 (1972)
[11] Manakov, S. V., Remarks on the integrals of the Euler equations of the \(n\)-dimensional heavy top, Functional Anal. Appl., 10, 4 (1976) · Zbl 0343.70003
[12] Moser, J., Various aspects of integrable Hamiltonian systems, (Proceedings, CIME Conference. Proceedings, CIME Conference, Bressanone, Italy (June 1978)), in press · Zbl 0468.58011
[13] Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point (1960), Published for the NSF by the Israel Program for Scientific Translations, Haifa, Israel · Zbl 0122.18701
[14] Chevalley, C., Theory of Lie Groups (1946), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 0063.00842
[15] E Artin; E Artin
[16] Symes, W., Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math., 59, No. 1, 13-53 (1980) · Zbl 0474.58009
[17] Moody, R., A new class of Lie algebras, J. Algebra, 10, 211-230 (1968) · Zbl 0191.03005
[18] Gilmore, R., Lie Groups, Lie Algebras, and Some of Their Applications (1974), Wiley: Wiley New York · Zbl 0279.22001
[19] Humphreys, J., Introduction to Lie Algebras and Representation Theory (1972), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0254.17004
[20] Flaschka, H., On the Toda lattice, I, Phys. Rev. B, 9 (1974) · Zbl 0942.37504
[21] Bogoyavlensky, O. I., On perturbations of the periodic Toda lattices, Comm. Math. Phys., 51 (1976) · Zbl 0555.70003
[22] Coolidge, J., A Treatise on Algebraic Plane Curves, ((1931), Oxford Univ. Press (Clarendon): Oxford Univ. Press (Clarendon) London), 321
[23] Adler, M.; van Moerbeke, P., Linearization of Hamiltonian systems, Jacobi varieties, and representation theory, Advances in Math., 38, 318-379 (1980) · Zbl 0455.58010
[24] S. Sternberg and A. Jacob; S. Sternberg and A. Jacob
[25] T. Ratiu; T. Ratiu
[26] T. Ratiu and P. van Moerbeke; T. Ratiu and P. van Moerbeke · Zbl 0466.58020
[27] D. R. Lebedev. and Yu. I., Manin; D. R. Lebedev. and Yu. I., Manin
[28] Wilson, G., Commuting flows and conservation laws for Lax equations, (Math. Proc. Cambridge Philos. Soc., 86 (1979)), 131 · Zbl 0427.35024
[29] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, ((1978), Academic Press: Academic Press New York), 491 · Zbl 0451.53038
[30] Knörrer, H., Geodesics on the ellipsoid, Invent. Math., 59, No. 2, 119-144 (1980) · Zbl 0431.53003
[31] Frenkel, I.; Reiman, A.; Semenov-Tian-Shansky, M., Graded Lie algebras and completely integrable dynamical systems, Soviet Math. Dokl., 20, No. 4, 811-814 (1979) · Zbl 0437.58008
[32] Reiman, A.; Semenov-Tian-Shansky, M., Reduction of Hamiltonian systems, affine Lie algebras and Lax equations, Invent. Math., 54, No. 1, 81-101 (1979) · Zbl 0403.58004
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