9,320 research outputs found
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
arbitrary coupling constants, 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 : 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
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
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
of
the number of (S-wave) bound states provides a (rigorous) lower limit:
, where the double braces denote the integer part.
Higher partial waves can be included via the standard replacement of the
potential with the effective -wave potential
. An analogous upper
limit is also provided for a different class of potentials, which is however
quite severely restricted.Comment: 9 page
N Fermion Ground State of Calogero-Sutherland Type Models in Two and Higher Dimensions
I obtain the exact ground state of -fermions in -dimensions in case the 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 -fermion ground state in case the
oscillator potential is replaced by an -body Coulomb-like interaction.Comment: 10 pages, Latex fil
Hidden algebra of the -body Calogero problem
A certain generalization of the algebra of first-order
differential operators acting on a space of inhomogeneous polynomials in is constructed. The generators of this (non)Lie algebra depend on
permutation operators. It is shown that the Hamiltonian of the -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
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