Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2=νn−1,n∈Z\mu^2=\nu^n-1, n\in{\Bbb Z}: ergodicity, isochrony, periodicity and fractals

We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically periodic with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behaviour. We suggest a possible theoretical explanation of these different behaviours. We also introduce a one-parameter family of 2-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such mapping the center map. Computer experiments for the center map show a typical multi-fractal behaviour with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.Comment: LaTex, 28 pages, 10 figure

Spin generalization of the Ruijsenaars-Schneider model, non-abelian 2D Toda chain and representations of Sklyanin algebra

Action-angle type variables for spin generalizations of the elliptic Ruijsenaars-Schneider system are constructed. The equations of motion of these systems are solved in terms of Riemann theta-functions. It is proved that these systems are isomorphic to special elliptic solutions of the non-abelian 2D Toda chain. A connection between the finite gap solutions of solitonic equations and representations of the Sklyanin algebra is revealed and discrete analogs of the Lame operators are introduced. A simple way to construct representations of the Sklyanin algebra by difference operators is suggested.Comment: 38 pages, latex, no figure

Secants of Abelian Varieties, Theta functions, and Soliton Equations

This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the analytic theory of theta functions is also given.Comment: 69 pages, accepted by Russian Mathematical Survey