Numerical Methods and Closed Orbits in the Kepler-Heisenberg Problem

The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the fundamental solution to the sub-Laplacian. This system is known to admit closed orbits, which all lie within a fundamental integrable subsystem. Here, we develop a computer program which finds these closed orbits using Monte Carlo optimization with a shooting method, and applying a recently developed symplectic integrator for nonseparable Hamiltonians. Our main result is the discovery of a family of flower-like periodic orbits with previously unknown symmetry types. We encode these symmetry types as rational numbers and provide evidence that these periodic orbits densely populate a one-dimensional set of initial conditions parametrized by the orbit's angular momentum. We provide links to all code developed.Comment: 9 pages, 7 figures, completed in residence at MSRI; updated all images and some tex

Periodic Orbits in the Kepler-Heisenberg Problem

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.Comment: 19 pages, 3 figure

Bridges Between Subriemannian Geometry and Algebraic Geometry

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.Comment: 10 pages, 2 figures, Proceedings of 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Madrid 201

Complete spelling rules for the Monster tower over three-space

The Monster tower, also known as the Semple tower, is a sequence of manifolds with distributions of interest to both differential and algebraic geometers. Each manifold is a projective bundle over the previous. Moreover, each level is a fiber compactified jet bundle equipped with an action of finite jets of the diffeomorphism group. There is a correspondence between points in the tower and curves in the base manifold. These points admit a stratification which can be encoded by a word called the RVT code. Here, we derive the spelling rules for these words in the case of a three dimensional base. That is, we determine precisely which words are realized by points in the tower. To this end, we study the incidence relations between certain subtowers, called Baby Monsters, and present a general method for determining the level at which each Baby Monster is born. Here, we focus on the case where the base manifold is three dimensional, but all the methods presented generalize to bases of arbitrary dimension.Comment: 14 pages, 4 figures; new titl

The Puiseux Characteristic of a Goursat Germ

Germs of Goursat distributions can be classified according to a geometric coding called an RVT code. Jean (1996) and Mormul (2004) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and Zhitomirskii (2010) have shown that such germs correspond to finite jets of Legendrian curve germs, and that the RVT coding corresponds to the classical invariant in the singularity theory of planar curves: the Puiseux characteristic. Here we derive a simple formula for the Puiseux characteristic of the curve corresponding to a Goursat germ with given small growth vector. The simplicity of our theorem (compared with the more complex algorithms previously known) suggests a deeper connection between singularity theory and the theory of nonholonomic distributions.Comment: 13 pages; expanded backgroun

The structural invariants of Goursat distributions

This is the first of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to connections with the topic of singularities of curves on surfaces. Here we study those invariants of Goursat distributions akin to those of curves on surfaces, which we call structural invariants. In the subsequent paper we will relate these structural invariants to the small-growth invariants.Comment: 35 pages, 6 figure

Anti-microbial Activity of Urine after Ingestion of Cranberry: A Pilot Study

We explore the anti-microbial activity of urine specimens after the ingestion of a commercial cranberry preparation. Twenty subjects without urinary infection, off antibiotics and all supplements or vitamins were recruited. The study was conducted in two phases: in phase 1, subjects collected the first morning urine prior to ingesting 900 mg of cranberry and then at 2, 4 and 6 h. In phase 2, subjects collected urine on 2 consecutive days: on Day 1 no cranberry was ingested (control specimens), on Day 2, cranberry was ingested. The pH of all urine specimens were adjusted to the same pH as that of the first morning urine specimen. Aliquots of each specimen were independently inoculated with Escherichia coli, Klebsiella pneumoniae or Candida albicans. After incubation, colony forming units/ml (CFU ml−1) in the control specimen was compared with CFU ml−1 in specimens collected 2, 4 and 6 h later. Specimens showing ≥50% reduction in CFU ml−1 were considered as having ‘activity’ against the strains tested. In phase 1, 7/20 (35%) subjects had anti-microbial activity against E. coli, 13/20 (65%) against K. pneumoniae and 9/20 (45%) against C. albicans in specimens collected 2–6 h after ingestion of cranberry. In phase 2, 6/9 (67%) of the subjects had activity against K. pneumoniae. This pilot study demonstrates weak anti-microbial activity in urine specimens after ingestion of a single dose of commercial cranberry. Anti-microbial activity was noted only against K. pneumoniae 2–6 h after ingestion of the cranberry preparation