Skip to main content
Log in

Relativistic Toda systems

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We present and study Poincaré-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Ruijsenaars, S. N. M., Schneider, H.: A new class of integrable systems and its relation to solitons. Ann. Phys. (NY)170, 370–405 (1986)

    Article  Google Scholar 

  2. Ruijsenaars, S. N. M.: Complete integrability of relativistic Calogero-Moser systems and elliptic function identities. Commun. Math. Phys.110, 191–213 (1987)

    Article  Google Scholar 

  3. Toda, M.: Theory of nonlinear lattices. Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  4. Olshanetsky, M. A., Perelomov, A. M.: Classical integrable finite-dimensional systems related to Lie algebras. Phys. Reps.71, 313–400 (1981)

    Article  Google Scholar 

  5. Olshanetsky, M. A., Perelomov, A. M.: Quantum integrable systems related to Lie algebras. Phys. Reps.94, 313–404 (1983)

    Article  Google Scholar 

  6. Moser, J.: Three integrable Hamiltonian systems connected with isospectral deformations. Adv. Math.16, 197–220 (1975)

    Article  Google Scholar 

  7. Moser, J.: Finitely many mass points on the line under the influence of an exponential potential—An integrable system, pp. 467–497. In: Dynamical systems, theory and applications. Moser, J. (ed.). Lecture Notes in Physics, vol.38, Berlin, Heidelberg, New York: Springer 1975

    Google Scholar 

  8. Olshanetsky, M. A., Perelomov, A. M.: Explicit solutions of classical generalized Toda models. Invent. Math.54, 261–269 (1979)

    Article  Google Scholar 

  9. Kostant, B.: The solution to a generalized Toda, lattice and representation theory. Adv. Math.34, 195–338 (1979)

    Article  Google Scholar 

  10. Symes, W. W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math.59, 13–51 (1980)

    Article  Google Scholar 

  11. Goodman, R., Wallach, N. R.: Classical and quantum mechanical systems of Toda-lattice type II. Solutions of the classical flows. Commun. Math. Phys.94, 177–217 (1984)

    Article  Google Scholar 

  12. Ruijsenaars, S. N. M.: unpublished manuscript, 1985

  13. Bruschi, M., Ragnisco, O.: Recursion operator and Bäcklund transformations for the Ruijsenaars-Toda lattice. Phys. Lett.A129, 21–25 (1988)

    Article  Google Scholar 

  14. Bruschi, M., Ragnisco, O.: Lax representation and complete integrability for the periodic relativistic Toda lattice. Phys. Lett.A134, 365–370 (1989)

    Article  Google Scholar 

  15. Ragnisco, O., Bruschi, M.: The periodic relativistic Toda lattice: Direct and inverse problem. Inverse Problems5, 389–405 (1989)

    Article  Google Scholar 

  16. Oevel, W., Fuchssteiner, B., Zhang, H.: Mastersymmetries, angle variables and recursion operator of the relativistic Toda lattice. J. Math. Phys.30, 2664–2670 (1989)

    Article  Google Scholar 

  17. Oevel, W., Ragnisco, O.:R-matrices and higher Poisson brackets for integrable systems. PhysicaA161, 181–220 (1989)

    Google Scholar 

  18. Bruschi, M., Ragnisco, O.: On a new integrable Hamiltonian system with nearest neighbours interaction. Inverse Problems5, 983–998 (1989)

    Article  Google Scholar 

  19. Alber, S. J.: On finite-zone solutions of relativistic Toda, lattices. Lett. Math. Phys.17, 149–155 (1989)

    Article  Google Scholar 

  20. Mikhailov, A. V.: Integrability of a two-dimensional generalization of the Toda chain. JETP Lett.30, 414–418 (1979)

    Google Scholar 

  21. Leznov, A. N., Saveliev, M. V.: Representation of zero curvature for the system of nonlinear partial differential equations\(x_{\alpha ,\bar zz} = \exp (kx)_\alpha \) and its integrability. Lett. Math. Phys.3, 489–494 (1979)

    Article  Google Scholar 

  22. Ruijsenaars, S. N. M.: Action-angle maps and scattering theory for some finite-dimensional integrable systems. I The pure soliton case. Commun. Math. Phys.115, 127–165 (1988)

    Article  Google Scholar 

  23. Flaschka, H.: The Toda lattice. II Existence of integrals. Phys. Rev.B9 1924–1925 (1974)

    Article  Google Scholar 

  24. Suris, Y. B.: Discrete time generalized Toda lattices: complete integrability and relation with relativistic Toda lattices. Phys. Lett.A145, 113–119 (1990)

    Article  Google Scholar 

  25. Ruijsenaars, S. N. M.: Finite-dimensional solition systems. In: Integrable and super-integrable systems. Kupershmidt, B. (ed.) Singapore: World Scientific (to appear)

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by J.-P. Eckmann

Work supported by the Netherlands Organisation for the Advancement of Research (NWO)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ruijsenaars, S.N.M. Relativistic Toda systems. Commun.Math. Phys. 133, 217–247 (1990). https://doi.org/10.1007/BF02097366

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02097366

Keywords

Navigation