Related papers: Probabilistic properties of the elliptic motion
Orbits in triaxial ellipsoidal potentials precess about either the major or minor axis of the ellipsoid. In standard perturbation theory it can be shown that a circular orbit will precess about the minor axis if its angular momentum vector…
We consider a stationary Poisson hyperplane process with given directional distribution and intensity in $d$-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body $K$ and consider the intersection…
We consider the integrable dynamics of a Kepler billiard in the plane bounded by a branch of a conic section focused at the Kepler center. We show that in this case, for non-zero-energy orbits, the lines of consecutive second orbital foci…
Random flights in $\mathbb{R}^d,d\geq 2,$ with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position $\underline{\bf X}_d(t),\,t>0,$ when the number of…
The probability of a spin flip of an electron is calculated. It is assumed that the electron resides in a uniform magnetic field and interacts with an incoming electromagnetic pulse. The scattering matrix is constructed and the time needed…
Elliptic curves arise in many important areas of modern number theory. One way to study them is take local data, the number of solutions modulo $p$, and create an $L$-function. The behavior of this global object is related to two of the…
We analytically compute the orbital effects induced on the motion of a spinning particle geodesically traveling around a central rotating body by the general relativistic two-body spin-spin and spin-orbit interactions. We use the weak-field…
For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and…
We model the shape and spin characteristics of an object population when there are not enough data to model its single members. The data are random projection areas of the members. We construct a mapping $f(x)\rightarrow C(y)$,…
Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki (2006) generalized the elliptically contoured…
Interesting phenomena occur when an eccentric rigid body rolls on an inclined or horizontal plane. For example, a variety of motions between rolling and sliding are exhibited until suddenly a jump occurs. We provide a detailed theoretical…
We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating…
H${\acute{e}}$non [8] used an inclined billiard to investigate aspects of chaotic scattering which occur in satellite encounters and in other situations. His model consisted of a piecewise mapping which described the motion of a point…
A beer bottle or soda can on a table, when slightly tipped and released, falls to an upright position and then rocks up to a somewhat opposite tilt. Superficially this rocking motion involves a collision when the flat circular base of the…
Poisson plane and sphere --- homogeneous spaces of Poisson groups E(2) and SU(2) (resp.) --- have phase spaces (corresponding symplectic groupoids), in which a free Hamiltonian is naturally defined. We solve the equations of motion and…
In this article, we investigate the convergence behavior of two classes of gathering protocols with fixed circulant topologies using tools from dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we…
We study six pedal-like curves associated with the ellipse which are area-invariant for pedal points lying on one of two shapes: (i) a circle concentric with the ellipse, or (ii) the ellipse boundary itself. Case (i) is a corollary to…
Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the…
Circles of a single size can pack together densely in a hexagonal lattice, but adding in size variety disrupts the order of those packings. We conduct simulations which generate dense random packings of circles with specified size…
We prove the following variant of the Falconer conjecture in the plane. If the dimension of a compact planar set is greater than one, then the distance set with respect to almost every ellipse has positive Lebesgue measure.