1,539 research outputs found
Perturbed Three Vortex Dynamics
It is well known that the dynamics of three point vortices moving in an ideal
fluid in the plane can be expressed in Hamiltonian form, where the resulting
equations of motion are completely integrable in the sense of Liouville and
Arnold. The focus of this investigation is on the persistence of regular
behavior (especially periodic motion) associated to completely integrable
systems for certain (admissible) kinds of Hamiltonian perturbations of the
three vortex system in a plane. After a brief survey of the dynamics of the
integrable planar three vortex system, it is shown that the admissible class of
perturbed systems is broad enough to include three vortices in a half-plane,
three coaxial slender vortex rings in three-space, and `restricted' four vortex
dynamics in a plane. Included are two basic categories of results for
admissible perturbations: (i) general theorems for the persistence of invariant
tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff
type arguments; and (ii) more specific and quantitative conclusions of a
classical perturbation theory nature guaranteeing the existence of periodic
orbits of the perturbed system close to cycles of the unperturbed system, which
occur in abundance near centers. In addition, several numerical simulations are
provided to illustrate the validity of the theorems as well as indicating their
limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic
The Hess-Appelrot system and its nonholonomic analogs
This paper is concerned with the nonholonomic Suslov problem and its
generalization proposed by Chaplygin. The issue of the existence of an
invariant measure with singular density (having singularities at some points of
phase space) is discussed
2D String Theory as Normal Matrix Model
We show that the bosonic string theory at finite temperature has two
matrix-model realizations related by a kind of duality transformation. The
first realization is the standard one given by the compactified matrix quantum
mechanics in the inverted oscillator potential. The second realization, which
we derive here, is given by the normal matrix model. Both matrix models exhibit
the Toda integrable structure and are associated with two dual cycles (a
compact and a non-compact one) of a complex curve with the topology of a sphere
with two punctures. The equivalence of the two matrix models holds for an
arbitrary tachyon perturbation and in all orders in the string coupling
constant.Comment: lanlmac, 21 page
An Exactly Solvable Model for the Integrability-Chaos Transition in Rough Quantum Billiards
A central question of dynamics, largely open in the quantum case, is to what
extent it erases a system's memory of its initial properties. Here we present a
simple statistically solvable quantum model describing this memory loss across
an integrability-chaos transition under a perturbation obeying no selection
rules. From the perspective of quantum localization-delocalization on the
lattice of quantum numbers, we are dealing with a situation where every lattice
site is coupled to every other site with the same strength, on average. The
model also rigorously justifies a similar set of relationships recently
proposed in the context of two short-range-interacting ultracold atoms in a
harmonic waveguide. Application of our model to an ensemble of uncorrelated
impurities on a rectangular lattice gives good agreement with ab initio
numerics.Comment: 29 pages, 5 figure
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