Generalizations of the St\"ormer Problem for Dust Grain Orbits

We consider the generalized St\"ormer Problem that includes the electromagnetic and gravitational forces on a charged dust grain near a planet. For dust grains a typical charge to mass ratio is such that neither force can be neglected. Including the gravitational force gives rise to stable circular orbits that encircle that plane entirely above/below the equatorial plane. The effects of the different forces are discussed in detail. A modified 3rd Kepler's law is found and analyzed for dust grains.Comment: 21 pages LaTeX, 12 figure

The Lie-Poisson structure of the reduced n-body problem

The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.Comment: 26 pages, 2 figure

Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$Comment: 7 pages, no figure

Vanishing Twist near Focus-Focus Points

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)Comment: revised version, 11p, accepted for publication in a sepecial volume in Regular and Chaotic Dynamics in honor of Richard Cushma

Extended phase diagram of the Lorenz model

The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz model equations for thermal convection in Rayleigh-B\'enard flow is studied. Its bifurcation structure has commonly been investigated as a function of r, the normalized Rayleigh number, at fixed Prandtl number \sigma. The present work extends the analysis to the entire (r,\sigma) parameter plane. An onion like periodic pattern is found which is due to the alternating stability of symmetric and non-symmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and \sigma. In addition to the limit which was previously analyzed by Sparrow, we identify two more distinct asymptotic regimes in which either \sigma/r or \sigma^2/r is constant. In both limits the dynamics is approximately described by Airy functions whence the periodicity in parameter space can be calculated analytically. Furthermore, some observations about sequences of bifurcations and coexistence of attractors, periodic as well as chaotic, are reported.Comment: 36 pages, 20 figure

Symmetry Reduction by Lifting for Maps

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure

Bifurcations of periodic and chaotic attractors in pinball billiards with focusing boundaries

We study the dynamics of billiard models with a modified collision rule: the outgoing angle from a collision is a uniform contraction, by a factor lambda, of the incident angle. These pinball billiards interpolate between a one-dimensional map when lambda=0 and the classical Hamiltonian case of elastic collisions when lambda=1. For all lambda<1, the dynamics is dissipative, and thus gives rise to attractors, which may be periodic or chaotic. Motivated by recent rigorous results of Markarian, Pujals and Sambarino, we numerically investigate and characterise the bifurcations of the resulting attractors as the contraction parameter is varied. Some billiards exhibit only periodic attractors, some only chaotic attractors, and others have coexistence of the two types.Comment: 30 pages, 17 figures. v2: Minor changes after referee comments. Version with some higher-quality figures available at http://sistemas.fciencias.unam.mx/~dsanders/publications.htm

X-ray based lung function measurement - a sensitive technique to quantify lung function in allergic airway inflammation mouse models

In mice, along with the assessment of eosinophils, lung function measurements, most commonly carried out by plethysmography, are essential to monitor the course of allergic airway inflammation, to examine therapy efficacy and to correlate animal with patient data. To date, plethysmography techniques either use intubation and/or restraining of the mice and are thus invasive, or are limited in their sensitivity. We present a novel unrestrained lung function method based on low-dose planar cinematic x-ray imaging (X-Ray Lung Function, XLF) and demonstrate its performance in monitoring OVA induced experimental allergic airway inflammation in mice and an improved assessment of the efficacy of the common treatment dexamethasone. We further show that XLF is more sensitive than unrestrained whole body plethysmography (UWBP) and that conventional broncho-alveolar lavage and histology provide only limited information of the efficacy of a treatment when compared to XLF. Our results highlight the fact that a multi-parametric imaging approach as delivered by XLF is needed to address the combined cellular, anatomical and functional effects that occur during the course of asthma and in response to therapy

Semitoric integrable systems on symplectic 4-manifolds

Let M be a symplectic 4-manifold. A semitoric integrable system on M is a pair of real-valued smooth functions J, H on M for which J generates a Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce