2,064 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
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
Further, it is shown that scattering
configuration, characterized by initial momenta goes over
into a final configuration characterized uniquely by the final momenta
with .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
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
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.
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