308 research outputs found
Continuous and Discrete (Classical) Heisenberg Spin Chain Revised
Most of the work done in the past on the integrability structure of the
Classical Heisenberg Spin Chain (CHSC) has been devoted to studying the
case, both at the continuous and at the discrete level. In this paper we
address the problem of constructing integrable generalized ''Spin Chains''
models, where the relevant field variable is represented by a
matrix whose eigenvalues are the roots of unity. To the best of our
knowledge, such an extension has never been systematically pursued. In this
paper, at first we obtain the continuous generalization of the CHSC
through the reduction technique for Poisson-Nijenhuis manifolds, and exhibit
some explicit, and hopefully interesting, examples for and matrices; then, we discuss the much more difficult discrete case, where a
few partial new results are derived and a conjecture is made for the general
case.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
B\"acklund Transformations for the Kirchhoff Top
We construct B\"acklund transformations (BTs) for the Kirchhoff top by taking
advantage of the common algebraic Poisson structure between this system and the
trigonometric Gaudin model. Our BTs are integrable maps providing an
exact time-discretization of the system, inasmuch as they preserve both its
Poisson structure and its invariants. Moreover, in some special cases we are
able to show that these maps can be explicitly integrated in terms of the
initial conditions and of the "iteration time" . Encouraged by these partial
results we make the conjecture that the maps are interpolated by a specific
one-parameter family of hamiltonian flows, and present the corresponding
solution. We enclose a few pictures where the orbits of the continuous and of
the discrete flow are depicted
B\"acklund Transformations for the Trigonometric Gaudin Magnet
We construct a Backlund transformation for the trigonometric classical Gaudin
magnet starting from the Lax representation of the model. The Darboux dressing
matrix obtained depends just on one set of variables because of the so-called
spectrality property introduced by E. Sklyanin and V. Kuznetsov. In the end we
mention some possibly interesting open problems.Comment: contribution to the Proc. of "Integrable Systems and Quantum
Symmetries 2009", Prague, June 18-20, 200
Dynamical R-Matrices for Integrable Maps
The integrability of two symplectic maps, that can be considered as
discrete-time analogs of the Garnier and Neumann systems is established in the
framework of the -matrix approach, starting from their Lax representation.
In contrast with the continuous case, the -matrix for such discrete systems
turns out to be of dynamical type; remarkably, the induced Poisson structure
appears as a linear combination of compatible ``more elementary" Poisson
structures. It is also shown that the Lax matrix naturally leads to define
separation variables, whose discrete and continuous dynamics is investigated.Comment: 16 plain tex page
From su(2) Gaudin Models to Integrable Tops
In the present paper we derive two well-known integrable cases of rigid body
dynamics (the Lagrange top and the Clebsch system) performing an algebraic
contraction on the two-body Lax matrices governing the (classical) su(2) Gaudin
models. The procedure preserves the linear r-matrix formulation of the ancestor
models. We give the Lax representation of the resulting integrable systems in
terms of su(2) Lax matrices with rational and elliptic dependencies on the
spectral parameter. We finally give some results about the many-body extensions
of the constructed systems.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Integrability of V. Adler's discretization of the Neumann system
We prove the integrability of the discretization of the Neumann system
recently proposed by V. Adler.Comment: 9 pp., LaTe
What is the relativistic Volterra lattice?
We develop a systematic procedure of finding integrable ''relativistic''
(regular one-parameter) deformations for integrable lattice systems. Our
procedure is based on the integrable time discretizations and consists of three
steps. First, for a given system one finds a local discretization living in the
same hierarchy. Second, one considers this discretization as a particular
Cauchy problem for a certain 2-dimensional lattice equation, and then looks for
another meaningful Cauchy problems, which can be, in turn, interpreted as new
discrete time systems. Third, one has to identify integrable hierarchies to
which these new discrete time systems belong. These novel hierarchies are
called then ''relativistic'', the small time step playing the role of
inverse speed of light. We apply this procedure to the Toda lattice (and
recover the well-known relativistic Toda lattice), as well as to the Volterra
lattice and a certain Bogoyavlensky lattice, for which the ''relativistic''
deformations were not known previously.Comment: 48 pp, LaTe
Algebraic extensions of Gaudin models
We perform a In\"on\"u--Wigner contraction on Gaudin models, showing how the
integrability property is preserved by this algebraic procedure. Starting from
Gaudin models we obtain new integrable chains, that we call Lagrange chains,
associated to the same linear -matrix structure. We give a general
construction involving rational, trigonometric and elliptic solutions of the
classical Yang-Baxter equation. Two particular examples are explicitly
considered: the rational Lagrange chain and the trigonometric one. In both
cases local variables of the models are the generators of the direct sum of
interacting tops.Comment: 15 pages, corrected formula
The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
In this paper, we investigate the main algebraic properties of the maximally
superintegrable system known as "Perlick system type I". All possible values of
the relevant parameters, and , are considered. In particular,
depending on the sign of the parameter entering in the metrics, the motion
will take place on compact or non compact Riemannian manifolds. To perform our
analysis we follow a classical variant of the so called factorization method.
Accordingly, we derive the full set of constants of motion and construct their
Poisson algebra. As it is expected for maximally superintegrable systems, the
algebraic structure will actually shed light also on the geometric features of
the trajectories, that will be depicted for different values of the initial
data and of the parameters. Especially, the crucial role played by the rational
parameter will be seen "in action".Comment: 16 pages, 7 figure
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