A solvable many-body problem in the plane

A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (``acceleration equal force'') equations of motion, featuring one-body (``external'') and pair (``interparticle'') forces. The former depend quadratically on the velocity, and nonlinearly on the coordinate, of the moving particle. The latter depend linearly on the coordinate of the moving particle, and linearly respectively nonlinearly on the velocity respectively the coordinate of the other particle. The model contains $2n^2$ arbitrary coupling constants, $n$ being the number of particles. The behaviour of the solutions is outlined; special cases in which the motion is confined (multiply periodic), or even completely periodic, are identified

Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions

Various solutions are displayed and analyzed (both analytically and numerically) of arecently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in particular the origin of certain periodic behaviors is explained. The light thereby shone on the connection among \textit{integrability} and \textit{analyticity} in (complex) time, as well as on the emergence of a \textit{chaotic} behavior (in the guise of a sensitive dependance on the initial data) not associated with any local exponential divergence of trajectories in phase space, might illuminate interesting phenomena of more general validity than for the particular model considered herein.Comment: Published by JNMP at http://www.sm.luth.se/math/JNMP

Goldfish geodesics and Hamiltonian reduction of matrix dynamics

We relate free vector dynamics to the eigenvalue motion of a time-dependent real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars Schneider models.Comment: 8 page

A discrete linearizability test based on multiscale analysis

In this paper we consider the classification of dispersive linearizable partial difference equations defined on a quad-graph by the multiple scale reduction around their harmonic solution. We show that the A1, A2 and A3 linearizability conditions restrain the number of the parameters which enter into the equation. A subclass of the equations which pass the A3 C-integrability conditions can be linearized by a Möbius transformation

Convenient parameterizations of matrices in terms of vectors

Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely reviewed, and new ones introduced

Exact Solution of a N-body Problem in One Dimension

Complete energy spectrum is obtained for the quantum mechanical problem of N one dimensional equal mass particles interacting via potential $$V(x_1,x_2,...,x_N) = g\sum^N_{i < j}{1\over (x_i-x_j)^2} - {\alpha\over \sqrt{\sum_{i < j} (x_i-x_j)^2}}$$ Further, it is shown that scattering configuration, characterized by initial momenta $p_i (i=1,2,...,N)$ goes over into a final configuration characterized uniquely by the final momenta $p'_i$ with $p'_i=p_{N+1-i}$.Comment: 8 pages, tex file, no figures, sign in the first term on the right hand side of eq.3 correcte

Two novel classes of solvable many-body problems of goldfish type with constraints

Two novel classes of many-body models with nonlinear interactions "of goldfish type" are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints): i. e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited: i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.Comment: 30 pages, 2 figure

Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians.Comment: 13 pages, LaTeX, no figures, to be published in J. Phys.

Remarkable matrices and trigonometric identities

Abstract Two (n × n)-matrices are exhibited, which have a simple expression in terms of trigonometric functions of n arbitrary angles and possess remarkably neat spectral properties, such as integral eigenvalues. Several related trigonometric identities are also exhibited

G.R. Khan, Eur. Phys. J. D 53, 123–125 (2009)

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