597 research outputs found
Stability of Relative Equilibria in the Planar N-Vortex Problem
We study the linear and nonlinear stability of relative equilibria in the
planar N-vortex problem, adapting the approach of Moeckel from the
corresponding problem in celestial mechanics. After establishing some general
theory, a topological approach is taken to show that for the case of positive
circulations, a relative equilibrium is linearly stable if and only if it is a
nondegenerate minimum of the Hamiltonian restricted to a level surface of the
angular impulse (moment of inertia). Using a criterion of Dirichlet's, this
implies that any linearly stable relative equilibrium with positive vorticities
is also nonlinearly stable. Two symmetric families, the rhombus and the
isosceles trapezoid, are analyzed in detail, with stable solutions found in
each case.Comment: 23 pages, 3 figure
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
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