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

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

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

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 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

Geometric methods for efficient planar swimming of copepod nauplii

Copepod nauplii are larval crustaceans with important ecological functions. Due to their small size, they experience an environment of low Reynolds number within their aquatic habitat. Here we provide a mathematical model of a swimming copepod nauplius with two legs moving in a plane. This model allows for both rotation and two-dimensional displacement by periodic deformation of the swimmer's body. The system is studied from the framework of optimal control theory, with a simple cost function designed to approximate the mechanical energy expended by the copepod. We find that this model is sufficiently realistic to recreate behavior similar to those of observed copepod nauplii, yet much of the mathematical analysis is tractable. In particular, we show that the system is controllable, but there exist singular configurations where the degree of non-holonomy is non-generic. We also partially characterize the abnormal extremals and provide explicit examples of families of abnormal curves. Finally, we numerically simulate normal extremals and observe some interesting and surprising phenomena.Comment: 17 pages, 9 figure