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

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

Exact solutions of the 3-wave resonant interaction equation

The Darboux--Dressing Transformations are applied to the Lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1+1 dimensions. We display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary plus and minus infinity, and which are not pure soliton solutions. These solutions depend on an arbitrary function and allow to deal with collisions of waves with various profiles.Comment: 15 pages, 9 figures, standard LaTeX2e, submitted for publication to Physica

Understanding complex dynamics by means of an associated Riemann surface

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.Comment: 25 pages, 14 figures, typed in AMS-LaTe

Cosmological models with fluid matter undergoing velocity diffusion

A new type of fluid matter model in general relativity is introduced, in which the fluid particles are subject to velocity diffusion without friction. In order to compensate for the energy gained by the fluid particles due to diffusion, a cosmological scalar field term is added to the left hand side of the Einstein equations. This hypothesis promotes diffusion to a new mechanism for accelerated expansion in cosmology. It is shown that diffusion alters not only quantitatively, but also qualitatively the global dynamical properties of the standard cosmological models.Comment: 11 Pages, 4 Figures. Version in pres

Lower limit in semiclassical form for the number of bound states in a central potential

We identify a class of potentials for which the semiclassical estimate $N^{\text{(semi)}}=\frac{1}{\pi}\int_0^\infty dr\sqrt{-V(r)\theta[-V(r)]}$ of the number $N$ of (S-wave) bound states provides a (rigorous) lower limit: $N\ge {{N^{\text{(semi)}}}}$, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential $V(r)$ with the effective $\ell$-wave potential $V_\ell^{\text{(eff)}}(r)=V(r)+\frac{\ell(\ell+1)}{r^2}$. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted.Comment: 9 page

Integrable Systems for Particles with Internal Degrees of Freedom

We show that a class of models for particles with internal degrees of freedom are integrable. These systems are basically generalizations of the models of Calogero and Sutherland. The proofs of integrability are based on a recently developed exchange operator formalism. We calculate the wave-functions for the Calogero-like models and find the ground-state wave-function for a Calogero-like model in a position dependent magnetic field. This last model might have some relevance for matrix models of open strings.Comment: 10 pages, UVA-92-04, CU-TP-56

N Fermion Ground State of Calogero-Sutherland Type Models in Two and Higher Dimensions

I obtain the exact ground state of $N$-fermions in $D$-dimensions $(D \geq 2)$ in case the $N$ particles are interacting via long-ranged two-body and three-body interactions and further they are also interacting via the harmonic oscillator potential. I also obtain the $N$-fermion ground state in case the oscillator potential is replaced by an $N$-body Coulomb-like interaction.Comment: 10 pages, Latex fil

Hidden algebra of the NN-body Calogero problem

A certain generalization of the algebra $gl(N,{\bf R})$ of first-order differential operators acting on a space of inhomogeneous polynomials in ${\bf R}^{N-1}$ is constructed. The generators of this (non)Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the $N$-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. Given representation implies that the Calogero Hamiltonian possesses infinitely-many, finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators.Comment: 10pp., CWRU-Math, October 199