Abstract
A novel impetus to the construction of integrable discretisations of given integrable continuous-time hamiltonian systems has been given in recent years by a number of relevant findings: we mention the successful application to differential-difference integrable hierarchies of the stationary flow or restricted flow approach [1], the results obtained toward the identification of integrable mappings of the standard type [2], the discovery of Backlund transformations for Calogero-Moser and Rujsenaars systems [3], including relativistic Toda [4], and finally the construction of non-autonomous mappings which are endowed with a proper discrete analog of the Painleve’ property [5]. Of course, time-discretisation is a highly non-unique procedure, even if we restrict considerations to integrability-preserving difference schemes. One may just ask to get an integrable Poisson map such that the discrete dynamics it describes goes into the continuous one in a suitable asymptotic limit, together with integrals of motion and Poisson structure, or require that Poisson structure and integrals of motion be exactly preserved by the discretisation. Stationary or restricted flow technique typically lead to discretisation of the former type, while Backlund transformations provide an example of the latter one. In the present paper, we compare two integrable discretisations of the Neumann system, both belonging to the first family; indeed, one of them is obtained by applying the restricted flow technique to the Toda lattice hierarchy.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G.R.W. Quispel, J.A.Roberts, C.J.Thompson. - Physica D, 1990, v.34, p. 183-192. M.Bruschi, O.Ragnisco, P.Santini, and G.Tu. Integrable symplectic maps. - Physica D, 1991, v. 49, p.273-294. O. Ragnisco. A simple method to generate integrable symplectic maps - in: Solitons and chaos, I.Antoniou, F.J.Lambert (Springer, 1991 ), p. 227 - 231;
Yu.B.Suris.-Funct. Anal. Appl. 1989, v. 23, p. 84 - 85
F. Nijhoff, G.D. Pang. Discrete-time Calogero-Moser model and lattice KP equation.- in: Symmetries and Integrability of Difference Equations, D. Levi, P. Winternitz and L. Vinet, CRM Montreal 1995; F.Nijhoff, O.Ragnisco, V.Kuzsnetsov. Integrable time-discretization of the Rujisenaars-Schneider model -Comm. Math. Phys. (to appear);
Yu. B.Suris. A discrete-time relativistic Toda lattice.- Preprint University of Bremen, Centre for Complex Systems and Visualization, 1995.
B. Grammaticos, A.Ramani. Discrete Painleve equations: coalescences, limits and degeneracies. -Preprint Universite’ de Paris VII, October 1995, solv-int/9510011
C. Neumann. De problemate quodam mechanica, quod ad primam inte- gralium ultraellipticorum classem revocatur. - J. Reine Angew. Math., 1859, v.56, p.46-69.
M. Adler, P. van Moerbecke. Completely integrable systems, Euclidean Lie algebras, and curves. - Adv. Math., 1980, v.38, p.267-317.
J. Moser. Geometry of quadrics and spectral theory, - In: Chern Symposium 1979. Springer, 1981, p. 147-188.
T. Ratiu. The C.Neumann problem as a completely integrable system on an adjoint orbit. - Trans. Amer. Math. Soc., 1981, v.264, P-321- 329.
H. Flashka. Towards an algebro-geometric interpretation of the Neumann problem. - Tohoku Math. J., 1984, v.36, p.407-426.
M. Adams, J.Harnad, and E.Previato. Isospectral Hamiltonian flows in finite and infinite dimensions. - Commun. Math. Phys., 1988, v. 117, p. 451-500.
J. Avan, M.Talon. Integrability and Poisson brackets for the Neumann- Moser-Uhlenbeck model. - Int. J. Mod. Phys. A, 1990, v.5, p.4477- 4485; Alternative Lax structures for the classical and quantum Neumann model. - Phys. Lett. B, 1991, v. 268, p. 209-216.
A.G. Reyman, M.A.Semenov-Tian-Shansky. Group-theoretical method in the theory of finite-dimensional integrable systems. - In: Encyclopaedia of Math. Sciences, V.16, Dynamical systems VII. Springer, 1993.
Yu.B. Suris. On the bi-Hamiltonian structure of Toda and relativistic Toda lattices. - Physics Letters A, 1993, v. 180, p.419-429.
A discrete Neumann system. - Physics Lett. A, 1992, v. 167, p. 165- 171; Dynamical r-matrices for integrable maps.- Physics Lett. A, 1995, v.198, p. 295-305; O.Ragnisco, S.Rauch-Wojciechowski. Integrable maps for the Gamier and for the Neumann systems. - Preprint Linkoping Univ., 1994; O.Ragnisco, C.Cao, Y.Wu. On the relation of the stationary Toda equation and the symplectic maps. - J. Phys. A: Math, and Gen., 1995, v.28, p.573-588.
A.P. Veselov. Integrable systems with discrete time and difference operators. - Funct. Anal. Appl, 1988, v.22, p. 1-13; J.Moser, A.P.Veselov. Discrete versions of some classical integrable systems and factorization of matrix polynomials. - Commun. Math. Phys., 1991, v.139, p.217-243; P.Deift, L.-Ch. Li, C.Tomei. Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions. - Mem. Amer. Math. Soc., 1992, No 479.
Yu.B. Suris. A discrete-time Gamier system. - Physics Letters A, 1994, v.189, p. 281-289; A family of integrable standard-like maps related to symmetric spaces. - Physics Letters A, 1994, v. 192, p.9-16; Discrete-time analogs of some nonlinear oscillators in an inverse- square potential. - J. Phys. A: Math, and Gen., 1994, v.27, p.8161-8169
E.K. Sklyanin. Separation of variables in the classical integrable SL(3) magnetic chain. - Commun. Math. Phys., 1992, v. 142, p. 123-132.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Birkhäuser Boston
About this chapter
Cite this chapter
Ragnisco, O., Suris, Y.B. (1997). On the r-Matrix Structure of the Neumann System and its Discretizations. In: Fokas, A.S., Gelfand, I.M. (eds) Algebraic Aspects of Integrable Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 26. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2434-1_14
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2434-1_14
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-7535-0
Online ISBN: 978-1-4612-2434-1
eBook Packages: Springer Book Archive