27 research outputs found
Non-minimality of corners in subriemannian geometry
We give a short solution to one of the main open problems in subriemannian
geometry. Namely, we prove that length minimizers do not have corner-type
singularities. With this result we solve Problem II of Agrachev's list, and
provide the first general result toward the 30-year-old open problem of
regularity of subriemannian geodesics.Comment: 11 pages, final versio
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 laminations of a crystal near an anti-continuum limit
The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N - 1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit
Analogy and invention. Some remarks on Poincar\ue9\u2019s Analysis situs papers
The primary role played by analogy in Henri Poincar\ue9\u2019s work, and in
particular in his \u201canalysis situs\u201d papers, is emphasized. Poincar\ue9\u2019s \u201csixth example\u201d
(showing that Betti numbers do not suffice to classify 3-manifolds) and his construction of the homology sphere are discussed in detail