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Dynamical Hierarchies
<Guest Editor's Introduction>
Integrable discretizations of the spin Ruijsenaars-Schneider models
Integrable discretizations are introduced for the rational and hyperbolic
spin Ruijsenaars--Schneider models. These discrete dynamical systems are
demonstrated to belong to the same integrable hierarchies as their
continuous--time counterparts. Explicit solutions are obtained for arbitrary
flows of the hierarchies, including the discrete time ones.Comment: LaTeX fil
Why are the rational and hyperbolic Ruijsenaars-Schneider hierarchies governed by the same R-operators as the Calogero-Moser ones?
We demonstrate that in a certain gauge the Lax matrices of the rational and
hyperbolic Ruijsenaars--Schneider models have a quadratic -matrix Poisson
bracket which is an exact quadratization of the linear --matrix Poisson
bracket of the Calogero--Moser models. This phenomenon is explained by a
geometric derivation of Lax equations for arbitrary flows of both hierarchies,
which turn out to be governed by the same dynamical --operator.Comment: LaTeX, 18pp, a revised versio
Stationary problems for equation of the KdV type and dynamical -matrices.
We study a quite general family of dynamical -matrices for an auxiliary
loop algebra related to restricted flows for equations of
the KdV type. This underlying -matrix structure allows to reconstruct Lax
representations and to find variables of separation for a wide set of the
integrable natural Hamiltonian systems. As an example, we discuss the
Henon-Heiles system and a quartic system of two degrees of freedom in detail.Comment: 25pp, LaTe
A hierarchical model for aging
We present a one dimensional model for diffusion on a hierarchical tree
structure. It is shown that this model exhibits aging phenomena although no
disorder is present. The origin of aging in this model is therefore the
hierarchical structure of phase space.Comment: 10 pages LaTeX, 4 postscript-figures include
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