Skip to main content
SpringerLink
Log in

Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems

  • Chapter
Book cover Dynamical Systems VII

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 16))

Abstract

The present survey is devoted to a general group-theoretic scheme which allows to construct integrable Hamiltonian systems and their solutions in a systematic way. This scheme originates from the works of Kostant [1979a] and Adler [1979] where some special but very instructive examples were studied. Some years later a link was established between this scheme and the so-called classical R-matrix method (Faddeev [1984], Semenov-Tian-Shansky [1983]). One of the advantages of this approach is that it unveils the intimate relationship between the Hamiltonian structure of an integrable system and the specific Riemann problem (or, more generally, factorization problem) that is used to find its solutions. This shows, in particular, that the Hamiltonian structure is completely determined by the Riemann problem. The simplest system which may be studied in this way is the open Toda lattice already described in Chapter 1 by Olshanetsky and Perelomov. (The Toda lattices will be considered here again in a more general framework.) However, the most interesting examples are related to infinite-dimensional Lie algebras. In fact, it can be shown that the solutions of Hamiltonian systems associated with finite-dimensional Lie algebras have a too simple time dependence (roughly speaking, like trigonometric polinomials). By contrast, genuine mechanical problems often lead to more sophisticated (e.g. elliptic or abelian) functions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abraham, R., Marsden, J.E. [1978]: Foundations of Mechanics. (2nd ed.), Benjamin: New York, Zbl.393.70001

    MATH  Google Scholar 

  • Adams, M.R., Harnad, J., Previato, E. [1988]: Isospectral Hamiltonian flows in finite and infinite dimensions. I. Commun. Math. Phys. 117, 451–500, Zbl.659.58021

    MathSciNet  MATH  Google Scholar 

  • Adler, M. [1979]: On a trace functional for formal pseudodifferential operators and the symplectic structure for Korteweg-de Vries type equations. Invent. Math. 50, 219–248, Zbl.393.35058

    MATH  Google Scholar 

  • Adler, M., van Moerbeke, P. [1980a]: Completely integrable systems, Euclidean Lie algebras and curves. Adv. Math. 38, 267–317, Zbl.455.58017

    MATH  Google Scholar 

  • Adler, M., van Moerbeke, P. [1980b]: Linearizations of Hamiltonian systems, Jacobi varieties, and representation theory. Adv. Math. 38, 318–379, Zbl.455.58010

    MATH  Google Scholar 

  • Adler, M., van Moerbeke, P. [1982]: Kowalewski’s asymptotic method, Kac-Moody Lie algebras, and regularization. Commun. Math. Phys. 83, 83–106, Zbl.491.58017

    MATH  Google Scholar 

  • Adler, M., van Moerbeke, P. [1984]: Geodesic flow on SO(4) and intersection of quadrics. Proc. Natl. Acad. Sci. USA 81, 4613–4616, Zbl.545.58027

    MATH  Google Scholar 

  • Adler, M., van Moerbeke, P. [1988]: The Kowalewski and Henon-Heiles motions as Manakov geodesic flows on SO(4): a two-dimensional family of Lax pairs. Commun. Math. Phys. 113, 659–700, Zbl.647.58022

    Google Scholar 

  • Arnol’d, V.I. [1966]: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaites. Ann. Inst. Fourier 16, No. 1, 319–361, Zbl. 148, 453

    MATH  Google Scholar 

  • Arnol’d, V.I. [1974]: Mathematical Methods of Classical Mechanics. Nauka: Moscow. English transl.: Graduate Texts in Math. 60, Springer-Verlag: New York-Heidelberg-Berlin, 1978, 2nd ed. 1989, Zbl.386.70001

    MATH  Google Scholar 

  • Baxter, R.J. [1982]: Exactly Solved Models in Statistical Mechanics. Academic Press: New York, Zbl.538.60093

    MATH  Google Scholar 

  • Belavin, A. A. [1980]: Discrete groups and integrability of quantum systems. Funkts. Anal. Prilozh. 14, No. 4, 18–26. English transl.: Funct. Anal. Appl. 14, 260–267 (1980), Zbl.454.22012

    MathSciNet  MATH  Google Scholar 

  • Belavin, A.A., Drinfel’d V.G. [1982]: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funkts. Anal. Prilozh. 16, No. 3, 1–29. English transl.: Funct. Anal. Appl. 16, 159–180 (1982), Zbl.504.22016

    MathSciNet  Google Scholar 

  • Belavin, A.A., Drinfel’d, V.G. [1984]: Triangle equations and simple Lie algebras. Sov. Sci. Rev., Sect. C., Math. Phys. Rev. 4, 93–165, Zbl.553.58040

    MathSciNet  MATH  Google Scholar 

  • Berezin, F.A. [1967]: Some remarks on the associative envelope of a Lie algebra. Funkts. Anal. Prilozh. 1, No. 2, 1–14. English transl.: Funct. Anal. Appl. 1, 91–102 (1968), Zbl.227.22020

    MathSciNet  Google Scholar 

  • Bobenko, A.I. [1986]: Euler equations in the algebras SO(4) and e(3). Isomorphisms of integrable cases. Funkts. Anal. Prilozh. 20, No. 1, 64–66. English transl: Funct. Anal. Appl. 20, 53–56 (1986), Zbl.622.58010

    MathSciNet  Google Scholar 

  • Bobenko, A.I., Reyman, A.G., Semenov-Tian-Shansky, M.A. [1989]: The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions. Commun. Math. Phys. 122, 321–354

    MathSciNet  MATH  Google Scholar 

  • Bogoyavlensky, O.I. (= Bogoyavlenskij, O.I.) [1976]: On perturbations of the periodic Toda lattice. Commun. Math. Phys. 51, 201–209

    MathSciNet  Google Scholar 

  • Bogoyavlensky, O.I. [1984]: Integrable equations on Lie algebras arising in problems of mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 883–938. English transl.: Math. USSR, Izv. 25, 207–257 (1985), Zbl.583.58012

    MathSciNet  Google Scholar 

  • Borovik, A.E., Robuk, V.N. [1981]: Linear pseudopotentials and conservation laws for the Landau-Lifshitz equation describing the nonlinear dynamics of a ferromagnetic with axial anisothropy. Teor. Mat. Fiz. 46, No. 3, 371–381. English transl.: Theor. Mat. Phys. 46, 242–248 (1981)

    Google Scholar 

  • Bourbaki, N. [1968]: Groupes et algèbres de Lie, Ch. IV–VI. Hermann: Paris, Zbl. 186,330

    Google Scholar 

  • Cherednik, I.V. [1983a]: Integrable differential equations and coverings of elliptic curves. Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 2, 384–406. English transl.: Math. USSR, Izv. 22, 357–377 (1984, Zbl.547.35109

    MathSciNet  Google Scholar 

  • Cherednik, I.V. [1983b]: Definition of τ-functions for generalized affine Lie algebras. Funkts. Anal. Prilozh. 17, No. 3, 93–95. English transl.: Funct. Anal. Appl. 17, 243–245 (1983), Zbl.528.17004

    MathSciNet  Google Scholar 

  • Deift, P.A., Li, L.C., Nanda, T., Tomei, C. [1986]: The Toda flow on a generic orbit is integrable. Commun. Pure Appl. Math. 39,183–232, Zbl.606.58020

    MathSciNet  MATH  Google Scholar 

  • Deift, P.A., Li, L.C. [1989]: Generalized affine Lie algebras and the solution of a class of flows associated with the QR eigenvalue algorithm. Commun. Pure Appl. Math. 42, 963–991

    MathSciNet  MATH  Google Scholar 

  • Drinfel’d, V.G. [1977]: On commutative subrings of certain noncommutative rings. Funkts. Anal. Prilozh. 11, No. 1, 11–14. English transl.: Funct. Anal. Appl. 11, 9–12 (1977), Zbl.359.14011

    MathSciNet  MATH  Google Scholar 

  • Drinfel’d V.G. [1983]: Hamiltonian structures on Lie groups, Lie Bialgebras and the geometrical meaning of classical Yang-Baxter equations. Dokl. Akad. Nauk SSSR 268, 285–287. English transl: Sov. Math. Dokl. 27, 68–71 (1983), Zbl.526.58017

    MathSciNet  Google Scholar 

  • Drinfel’d, V.G. [1987]: Quantum groups. In: Proc. Int. Congr. Math., Berkeley 1986, Vol. 1, 798–820, Zbl.667.16003

    MathSciNet  Google Scholar 

  • Dubrovin, B.A. [1981]: Theta functions and nonlinear equations. Usp. Mat. Nauk 36, No. 2, 11–80. English transl: Russ. Math. Surv. 36, No. 2, 11–92 (1981), Zbl.478.58038

    MathSciNet  Google Scholar 

  • Dubrovin, B.A., Novikov, S.P., Matveev, V.B. [1976]: Nonlinear equations of the Korteweg-de Vries type, finite-band linear operators, and abelian varieties. Usp. Math. Nauk 31, No. 1, 55–136. English transl.: Russ. Math. Surv. 31, No. 1, 59–146 (1976), Zbl.326.35011

    MathSciNet  MATH  Google Scholar 

  • Faddeev, L.D. [1980]: Quantum completely integrable models in field theory. Sov. Sci. Rev. Sect. C., Math. Phys. Rev. 1, 107–155, Zbl.569.35064

    MATH  Google Scholar 

  • Faddeev, L.D. [1984]: Integrable models in 1 + 1-dimensional quantum field theory. In: Recent Advances in Field Theory and Statistical Mechanics. J.-B. Zuber and R. Stora (eds.) Les Houches, Elsevier Science Publishers: Amsterdam, 563–608

    Google Scholar 

  • Faddeev, L.D., Takhtajan, L.A. [1986]: Hamiltonian methods in the theory of solitons. Nauka: Moscow, English transl.: Springer-Verlag: Berlin-Heidelberg-New York (1987), Zbl.632.58003

    MATH  Google Scholar 

  • Flaschka, H. [1974a]: On the Toda lattice. I. Phys. Rev. B9, 1924–1925

    MathSciNet  Google Scholar 

  • Flaschka, H. [1974b]: On the Toda lattice. II. Progr. Theor. Phys. 51, 703–716

    MathSciNet  MATH  Google Scholar 

  • Fordy, A.P., Wojciechowski, S., Marshall, I. [1986]: A family of integrable quartic potentials related to symmetric spaces. Phys. Lett. A 113, 395–400

    MathSciNet  Google Scholar 

  • Gel’fand, I.M., Dikij, L.A. [1978]: A family of Hamiltonian structures related to integrable nonlinear differential equations. Preprint IPM AN SSSR, No. 136. Institute of Applied Math.: Moscow (Russian)

    Google Scholar 

  • Goddard, P., Olive, D. (Eds.) [1988]: Kac-Moody and Virasoro algebras. A reprint volume for physicists. Adv. Ser. Math. Phys. 3. World Scientific: Singapore, Zbl.661.17001

    Google Scholar 

  • Gohberg, I.Z., Fel’dman, I.A. [1971]: Convolution Equations and Projection Methods for Their Solution. Nauka: Moscow. English transl.: Transl. Math. Monogr. 41, Providence (1974), Zbl.214, 385

    Google Scholar 

  • Golod, P.I. [1984]: Hamiltonian systems associated with anisotropic affine Lie algebras and higher Landau-Lifshitz equations. Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 6–8 (Russian), Zbl.542, 58010

    MathSciNet  Google Scholar 

  • Goodman, R., Wallach, N.R. [1982]: Classical and quantum mechanical systems of Toda lattice type. I. Commun. Math. Phys. 83, 355–386, Zbl.503.22013

    MathSciNet  MATH  Google Scholar 

  • Goodman, R., Wallach, N.R. [1984]: Classical and quantum mechanical systems of Toda lattice type. II. Commun. Math. Phys. 94, 177–217, Zbl.592.58028

    MathSciNet  MATH  Google Scholar 

  • Guillemin, V., Sternberg, S. [1980]: The moment map and collective motion. Ann. Phys. 127, 220–253, Zbl.453.58015

    MathSciNet  MATH  Google Scholar 

  • Guillemin, V., Sternberg, S. [1984]: Symplectic Techniques in Physics. Cambridge University Press: Cambridge, Zbl.576.58012

    MATH  Google Scholar 

  • Haine, L., Horozov, E. [1987]: A Lax pair for Kowalevski’s top. Physica D29, 173–180, Zbl.627.58026

    MathSciNet  Google Scholar 

  • Helgason, S. [1978]: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press: New York, Zbl.451.53038

    MATH  Google Scholar 

  • Izergin, A.G., Korepin, V.E. [1981]: The lattice quantum Sine-Gordon model. Lett. Math. Phys. 5, 199–205

    MathSciNet  Google Scholar 

  • Izergin, A.G., Korepin, V.E. [1982]: Lattice versions of quantum field theory models in two dimensions. Nucl. Phys. B 205, 401–413

    MathSciNet  Google Scholar 

  • Kac, V.G. [1968]: Simple irreducible graded Lie algebras of finite growth. Ivz. Akad. Nauk SSSR, Ser. Mat. 32, No. 6, 1323–1367. English transl.: Math. USSR, Izv. 2, 1271–1311 (1968), Zbl.222.17007

    Google Scholar 

  • Kac, V.G. [1969]: Automorphisms of finite order of semisimple Lie algebras. Funkts. Anal. Prilozh. 3, No. 3, 94–96. English transl.: Funct. Anal. Appl. 3, 252–254 (1970), Zbl.274.17002

    Google Scholar 

  • Kac, V.G. [1984]: Infinite-Dimensional Lie Algebras. Progress in Math. 44. Birkhäuser: Basel-Boston-Stuttgart, Zbl.537.17001

    Google Scholar 

  • Karasev, M.V. [1981]: The Maslov quantization conditions in higher cohomology and the analogs of the objects of Lie theory for canonical fibre bundles of symplectic manifolds. MIEM: Moscow. VINITI, No. 1092–82, 1093–82. English transl.: Sel. Math. Sov. 8, No. 3, 213–234, 235–258 (1989), Zbl.704.58019

    Google Scholar 

  • Karasev, M.V. [1986]: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets. Izv. Akad. Nauk SSSR, Ser. Mat. 50, No. 3, 508–538. English transl.: Math. USSR, Izv. 28, 497–527 (1987), Zbl.608, 58023

    MathSciNet  MATH  Google Scholar 

  • Kazhdan, D., Kostant, B., Sternberg, S. [1978]: Hamiltonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31, No. 4, 491–508, Zbl.368.58008

    MathSciNet  Google Scholar 

  • Kirillov, A.A. [1972]: Elements of the Theory of Representations. Nauka: Moscow. English transl.: Grundlehren der Mathematischen Wissenschaften 220 Springer-Verlag: New York Berlin-Heidelberg. 1976. Zbl.264.22011

    Google Scholar 

  • Kirillov, A.A. [1976]: Local Lie algebras. Usp. Mat. Nauk 31, No. 4, 55–76. English transl.: Russ. Math. Surv. 31, No. 4, 55–75 (1976), Zbl.352.58014

    MathSciNet  MATH  Google Scholar 

  • Komarov, I.V. [1987]: A generalization of the Kowalewski top. Phys. Lett. A123, 14–15

    MathSciNet  Google Scholar 

  • Kosmann-Schwarzbach, Y., Magri, F. [1988]: Poisson-Lie groups and complete integrability. I. Ann. Inst. Henri Poincaré, Phys. Théor. 49, 433–460, Zbl.667.16005

    MathSciNet  MATH  Google Scholar 

  • Kostant, B. [1970]: Quantization and unitary representations, I: Prequantization. In: Lect. Notes Math. 170, 87–208, Zbl.223.53028

    MathSciNet  Google Scholar 

  • Kostant, B. [1979a]: Quantization and representation theory. In: Representation theory of Lie groups, Proc. SRC/LMS Res. Symp., Oxford 1977, Lond Math. Soc. Lect. Note Ser., 34 (Atiyah, M.F. (Ed.)), 287–316, Zbl.474.58010

    Google Scholar 

  • Kostant, B. [1979b]: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34, 195–338, Zbl.433.22008

    MathSciNet  MATH  Google Scholar 

  • Kostant, B. [1982]: Poisson commutativity and the generalized periodic Toda lattice. In: Differential Geometric Methods in Physics, Proc. Int. Conf., Clausthal 1980, In: Lect. Notes Math. 905 (Doebner, H.D., Andersson, S.I., Petry, H.R. (Eds.)), 12–28, Zbl.485.58010

    Google Scholar 

  • Krichever, I.M. [1977a]: Integration of nonlinear equations by methods of algebraic geometry. Funkts. Anal. Prilozh. 11, No. 1, 15–31. English transl.: Funct. Anal. Appl. 11, 12–26 (1977), Zbl.346.35028

    MATH  Google Scholar 

  • Krichever, I.M. [1977b]: Methods of algebraic geometry in the theory of nonlinear equations. Usp. Mat. Nauk 32, No. 6, 183–208. English transl.: Russ. Math. Surv. 32, No. 6, 185–213 (1977), Zbl.372.35002

    MATH  Google Scholar 

  • Krichever, I.M. [1978]: Algebraic curves and nonlinear difference equations. Usp. Mat. Nauk 33, No. 4, 215–216. English transl.: Russ. Math. Surv. 33, No. 4, 255–256 (1978), Zbl.382.39003

    MathSciNet  MATH  Google Scholar 

  • Kulish, P.P., Reyman, A.G. [1983]: Hamiltonian structure of polynomial bundles. Zap. Nauchn. Semin. Leningr, Otd. Mat. Inst. Steklova 123, 61–16. English transl.: J. Sov. Math. 28, No. 4, 505–512 (1985), Zbl.511.47010

    Google Scholar 

  • Lebedev, D.R., Manin, Yu.I. [1979]: The Gel’fand-Dikij Hamilton’s operator and the coadjoint representation of the Volterra group. Funkts. Anal. Prilozh. 13, No. 4, 40–46. English transl.: Funct. Anal. Appl. 13, 268–273 (1980), Zbl.441.58007

    MathSciNet  MATH  Google Scholar 

  • Leznov, A.N., Savel’ev, M.V. [1979]: Representation of zero curvature for the system of non-linear partial differential equations X α,zz̄ = exp(KX) α and its integrability. Lett. Math. Phys. 3, 489–494, Zbl.415.35017

    MathSciNet  MATH  Google Scholar 

  • Leznov, A.N., Savel’ev, M.V. [1985]: Group-theoretical methods for the solution of nonlinear dynamical systems. Nauka: Moscow (Russian), Zbl.667.58020

    Google Scholar 

  • Li, L.-C., Parmentier, S. [1989]: Nonlinear Poisson structures and r-matrices. Commun. Math. Phys. 125, 545–563, Zbl.695.58011

    MathSciNet  MATH  Google Scholar 

  • Lie, S. (unter Mitwirkung von F. Engel) [1893]: Theorie der Transformationsgruppen, Abschn. III. Teubner: Leipzig

    MATH  Google Scholar 

  • Lu, J.-H. [1989]: Momentum mapping and reduction of Poisson actions. Preprint. Univ. of California: Berkeley

    Google Scholar 

  • Lu, J.-H., Weinstein, A. [1988]: Poisson Lie groups, dressing transformations and Bruhat decompositions. Preprint PAM-414. Univ. of California: Berkeley. Appeared in: J. Differ. Geom. 31, No. 2, 501–526 (1990), Zbl.673.58018

    MathSciNet  Google Scholar 

  • Mackey, G.W. [1958]: Unitary representations of group extensions. I. Acta Math. 99, 265–311, Zbl.82, 113

    MathSciNet  MATH  Google Scholar 

  • Magri, F. [1978]: A simple model of the integrable Hamiltonian equation. J. Math. Phys. 19, No. 5, 1156–1162, Zbl.383.35065

    MathSciNet  MATH  Google Scholar 

  • Manakov, S.V. [1974]: Complete integrability and stochastization of discrete dynamical systems. Zh. Exp. Teor. Fiz. 40, 269–274. English transl.: Sov. Phys. ZETP 40, 269–274 (1975)

    Google Scholar 

  • Manakov, S.V. [1976]: Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body. Funkts. Anal. Prilozh. 10, No. 4, 93–94. English transl: Funct. Anal. Appl. 10, 328–329 (1977), Zbl.343.70003

    MathSciNet  MATH  Google Scholar 

  • Marsden, J., Ratiu, T., Weinstein, A. [1984]: Semidirect products and reduction in mechanics. Trans. Am. Math. Soc. 281, No. 1, 147–177, Zbl.529.58011

    MathSciNet  MATH  Google Scholar 

  • Marsden, J., Weinstein, A. [1974]: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121–130, Zbl.327.58005

    MathSciNet  MATH  Google Scholar 

  • Mikhailov, A.V. [1981]: The reduction problem in the inverse scattering method. Physica D 3, 73–117

    MATH  Google Scholar 

  • Mishchenko, A.S., Fomenko, A.T. [1978]: Euler equations on finite-dimensional Lie groups. Izv. Akad. Nauk SSSR, Ser. Mat. 42, 396–415. English transl.: Math. USSR., Izv. 12, 371–389. (1978), Zbl.383.58006

    MathSciNet  MATH  Google Scholar 

  • Moerbeke, P. van, Mumford, D. [1979]: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154, Zbl.502.58032

    MathSciNet  MATH  Google Scholar 

  • Moody, R.V. [1968]: A new class of Lie algebras. J. Algebra 10, 221–230, Zbl.191, 30

    MathSciNet  MATH  Google Scholar 

  • Moser, J. [1975]: Three integrable Hamiltonian systems, connected with isospectral deformations. Adv. Math. 16, 197–220, Zbl.303.34019

    MATH  Google Scholar 

  • Moser, J. [1980a]: Geometry of quadrics and spectral theory. In: Differential Geometry. Proc. Int. Chem. Symp., Berkeley 1979, 147–188, Zbl.455.58018

    Google Scholar 

  • Moser, J. [1980b]: Various aspects of integrable Hamiltonian systems. In: Dynamical Systems, CIME Lect., Bressanone 1978, Prog. Math. 8, 233–290, Zbl.468.58011

    Google Scholar 

  • Mumford, D. [1978]: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equations, Korteweg-de Vries equation and related non-linear equations. In: Proc. Int. Symp. on Algebraic Geometry, Kyoto 1977, 115–153, Zbl.423.14007

    Google Scholar 

  • Mumford, D. [1984]: Tata Lectures on Theta. II. Prog. Math. 43, Zbl.549.14014

    Google Scholar 

  • Novikov, S.P., Shmeltser, I. [1981]: Periodic solutions of the Kirchhoff equations for the free motion of a rigid body in an ideal fluid and the generalized Lyusternik-Shnirel’man-Morse theory. Funkts. Anal. Prilozh. 15, No. 3, 54–66. English translation: Funct. Anal. Appl., 15, No. 3, 197–207 (1982), Zbl.571, 58009

    MathSciNet  Google Scholar 

  • Ol’shanetsky, M.A. (= Ol’shanetskij, M.A.), Perelomov, A.M. [1979]: Explicit solutions of the classical generalized Toda models. Invent. Math. 54, 261–269, Zbl.419, 58008

    MathSciNet  Google Scholar 

  • Ol’shanetsky, M.A., Perelomov, A.M. [1980]: The Toda lattice as a reduced system. Teor. Mat. Fiz. 45, No. 1, 3–18. English transl.: Theor. Math. Phys. 45, 843–854 (1981)

    Google Scholar 

  • Perelomov, A.M. [1981a]: Several remarks on the integrability of the equations of motion of a rigid body in an ideal fluid. Funkts. Anal. Prilozh. 15, No. 2, 83–85. English transl.: Funct. Anal. Appl. 15, 144–146 (1981), Zbl.495.70016

    MathSciNet  Google Scholar 

  • Perelomov, A.M. [1981b]: Lax representation for systems of S. Kowalevskaya type. Commun. Math. Phys. 81, 239–241, Zbl.478.70005

    MathSciNet  MATH  Google Scholar 

  • Perelomov, A.M., Ragnisco, P., Wojciechowski, S. [1986]: Integrability of two interacting n-dimensional rigid bodies. Commun. Math. Phys. 102, 573–583, Zbl.596.58019

    MathSciNet  MATH  Google Scholar 

  • Pressley, A., and Segal, G. [1986]: Loop Groups. Clarendon Press: Oxford, Zbl.618.22011

    MATH  Google Scholar 

  • Ratiu, T. [1982]: Euler-Poisson equations on Lie algebras and the n-dimensional heavy rigid body. Am. J. Math. 104, 409–448, Zbl.509.58026

    MathSciNet  MATH  Google Scholar 

  • Rawnsley, J.H. [1975]: Representations of a semi-direct product by quantization. Math. Proc. Camb. Philos. Soc. 78, No. 2, 345–350, Zbl.313.22014

    MathSciNet  MATH  Google Scholar 

  • Reyman, A.G. [1980]: Integrable Hamiltonian systems connected with graded Lie algebras. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 95, 3–54. English transl.: J. Sov. Math. 19, 1507–1545 (1982), Zbl.488.70013

    MATH  Google Scholar 

  • Reyman, A.G. [1986]: Orbit interpretation of Hamiltonian systems of the type of an unharmonic oscillator. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 155, 187–189. English transl.: J. Sov. Math. 41, 999–1001 (1988), Zbl.619.58022

    Google Scholar 

  • Reyman, A.G. [1988]: New results on the Kowalewski top. In: Nonlinear Evolutions, Proc. of the IV Workshop on Nonlinear Evolution Equations and Dynamical Systems, Balaruc-Les-Bains 1987. World Scientific: Singapore

    Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1979]: Reduction of Hamiltonian systems, affine Lie algebras and Lax equations. I. Invent. Math. 54, 81–100, Zbl.403.58004

    MathSciNet  MATH  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1980]: Current algebras and nonlinear partial differential equations. Dokl. Akad. Nauk SSSR 251, No. 6, 1310–1314. English transl.: Sov. Math., Dokl. 21, 630–634 (1980), Zbl.501.58018

    MathSciNet  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1981]: Reduction of Hamiltonian systems, Affine Lie algebras and Lax equations II. Invent. Math. 63, 423–432, Zbl.442.58016

    MathSciNet  MATH  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1986a]: Lie algebras and Lax equations with spectral parameter on an elliptic curve. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 150, 104–118. English transl.: J. Sov. Math. 46, No. 1, 1631–1640 (1989), Zbl.603.35083

    MATH  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1986b]: A new integrable case of the motion of the four-dimensional rigid body. Commun. Math. Phys. 105, 461–472, Zbl.606.58029

    MathSciNet  MATH  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1987]: Lax representation with a spectral parameter for the Kowalewski top and its generalizations. Lett. Math. Phys. 14, 55–61, Zbl.627.58027

    MathSciNet  MATH  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A. [1988]: Compatible Poisson structures for Lax equations: an r-matrix approach. Phys. Lett. A 130, 456–460

    MathSciNet  Google Scholar 

  • Reyman, A.G., Semenov-Tian-Shansky, M.A., Frenkel, I.B. [1979]: Graded Lie algebras and completely integrable systems. Dokl. Akad. Nauk SSSR 247, 802–805. English transl.: Sov. Math., Dokl. 20, 811–814 (1979), Zbl.437, 58008

    MathSciNet  Google Scholar 

  • Semenov-Tian-Shansky, M.A. [1982]: Group-theoretical aspects of completely integrable systems. In: Twistor Geometry and Nonlinear Systems, 4th. Bulg. Summer Sch., Primorsko 1980, Lect. Notes Math. 970 (Doebner, H.D., Palev, T.D. (Eds.)), 173–185, Zbl.507.58028

    Google Scholar 

  • Semenov-Tian-Shansky, M.A. [1983]: What is a classical r-matrix. Funkts. Anal. Prilozh. 17, No. 4, 17–33. English transl.: Funct. Anal. Appl. 17, 259–272 (1983), Zbl.535.58031

    Google Scholar 

  • Semenov-Tian-Shansky, M.A. [1985]: Dressing transformations and Poisson group actions. Publ. Res. Inst. Math. Sci 21, No. 6, 1237–1260, Zbl.673.58019

    MathSciNet  Google Scholar 

  • Semenov-Tian-Shansky, M.A. [1987]: Classical r-matrices, Lax equations, Poisson Lie groups and dressing transformations. In: Field Theory, Quantum Gravity and Strings. II. Proc. Semin., Mendon and Paris 1985/86, Lect. Notes Phys. 280, (de Vega, H.J., Sanchez, N. (Eds.)), 174–214, Zbl.666.22010

    Google Scholar 

  • Sklyanin, E.K. [1979]: On complete integrability of the Landau-Lifschitz equation. Preprint Leningr. Otd. Mat. Inst. E-3–79: Leningrad, Zbl.449.35089

    Google Scholar 

  • Sklyanin, E.K. [1982]: Algebraic structures connected with the Yang-Baxter equation. Funkts. Anal Prilozh. 16, No. 4, 27–34. English transl.: Funct. Anal. Appl. 16, 263–270 (1982), Zbl.513.58028

    MathSciNet  Google Scholar 

  • Sklyanin, E.K. [1983]: Algebraic structures connected with the Yang-Baxter equation. II. Representations of the quantum algebra. Funkts. Anal. Prilozh. 17, No. 4, 34–48. English transl.: Funct. Anal. Appl. 17, 273–284 (1983), Zbl.536.58007

    MathSciNet  Google Scholar 

  • Sklyanin, E.K. [1987]: Boundary conditions for integrable equations. Funkts. Anal. Prilozh. 21, No. 2, 86–87. English transl.: Funct. Anal. Appl. 21, 164–166 (1987)., Zbl.643.35093

    MathSciNet  Google Scholar 

  • Souriau, J.-M. [1970]: Structure des systèmes dynamiques. Dunod: Paris, Zbl. 186, 580

    MATH  Google Scholar 

  • Symes, W. [1980a]: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 59, 13–51, Zbl.474.58009

    MathSciNet  MATH  Google Scholar 

  • Symes, W. [1980b]: Hamiltonian group actions and integrable systems. Physica D1, 339–374

    MathSciNet  Google Scholar 

  • Veselov, A.P. [1984]: Cnoidal solutions of the Landau-Lifshits equations. Dokl. Akad. Nauk SSSR 276, No. 3, 590–593 (Russian)

    MathSciNet  Google Scholar 

  • Weinstein, A. [1978]: A universal phase space for particles in Yang-Mills fields. Lett. Math. Phys. 2, 417–420, Zbl.388.58010

    MATH  Google Scholar 

  • Weinstein, A. [1983]: Local structure of Poisson manifolds. J. Differ. Geom. 18, No. 3, 523–557, Zbl.524.58011

    MATH  Google Scholar 

  • Weinstein, A. [1988]: Some remarks on dressing transformations. J. Fac. Sci., Univ. Tokyo, Sect. 1 A, 35, No. 1, 163–167, Zbl.653.58012

    MATH  Google Scholar 

  • Yakh’ya, H. [1987]: New integrable cases of the motion of a gyrostat. Vestn. Mosk. Univ., Ser. I. 1987, No. 4, 88–90. English transl.: Mosc. Univ. Mech. Bull. 42, No. 4, 29–31 (1987), Zbl.661.70012

    Google Scholar 

Download references

Authors
  1. A. G. Reyman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Semenov-Tian-Shansky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

V. I. Arnol’d S. P. Novikov

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Reyman, A.G., Semenov-Tian-Shansky, M.A. (1994). Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems. In: Arnol’d, V.I., Novikov, S.P. (eds) Dynamical Systems VII. Encyclopaedia of Mathematical Sciences, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06796-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06796-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-05738-0

  • Online ISBN: 978-3-662-06796-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation