Spatial bi-stacked central configurations formed by two dual regular polyhedra

In this paper we prove the existence of two new families of spatial stacked central configurations, one consisting of eight equal masses on the vertices of a cube and six equal masses on the vertices of a regular octahedron, and the other one consisting of twenty masses at the vertices of a regular dodecahedron and twelve masses at the vertices of a regular icosahedron. The masses on the two different polyhedra are in general different. We note that the cube and the octahedron, the dodecahedron and the icosahedron are dual regular polyhedra. The tetrahedron is itself dual. There are also spatial stacked central configurations formed by two tetrahedra, one and its dual.     

Global symplectic regularization for some restricted 3-body problems on

We develop a regularization for binary collisions in some restricted 3-body problems moving in planar one-dimensional spaces with constant curvature. The main characteristic of the regularization is that it preserves the Hamiltonian structure of the equations and it regularizes all the binary collisions with just one transformation. We apply this global symplectic regularization to the 2-body problem on the unit circle and we show the global dynamics. Also, we tackle the restricted 3-body problem with one fixed center in the unit circle and we give the global dynamics for the case when it has two fixed centers.     

Regularization of the Circular Restricted Three Body Problem on Surfaces of Constant Curvature

We consider a restricted three body problem on surfaces of constant curvature. As in the classical Newtonian case the collision singularities occur when the position particle with infinitesimal mass coincides with the position of one of the primaries. We prove that the singularities due to collision can be locally (each one separately) and globally (both as the same time) regularized through the construction of Levi-Civita and Birkhoff type transformations respectively. As an application we study some general properties of the Hill’s regions and we present some ejection–collision orbits for the symmetrical problem.     

The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part II: Singularities

We analyze the singularities of the equations of motion and several types of singular solutions of the n-body problem in spaces of positive constant curvature. Apart from collisions, the equations encounter noncollision singularities, which occur when two or more bodies are antipodal. This conclusion leads, on the one hand, to hybrid solution singularities for as few as three bodies, whose orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces nonsingularity collisions, characterized by finite velocities and forces at the collision instant.     

Stratospheric ozone dynamics according to the Chapman mechanism

The photochemical mechanisms behind the ozone dynamics in the stratosphere have been studied for a long time. Though the differential equations describing the kinetics associated with them are well known, their mathematical properties have not been fully explored. In this work some of these mathematical properties as well as their physical implications are investigated in the simplest photochemical mechanism, namely, Chapman’s one.     

Dynamics in the Schwarzschild Isosceles Three Body Problem

The Schwarzschild potential, defined as \(U(r)=-A/r-B/r^3\) , where \(r\) is the relative distance between two mass points and \(A,B>0\) , models astrophysical and stellar dynamics systems in a classical context. In this paper we present a qualitative study of a three mass point system with mutual Schwarzschild interaction where the motion is restricted to isosceles configurations at all times. We retrieve the relative equilibria and provide the energy–momentum diagram. We further employ appropriate regularization transformations to analyze the behavior of the flow near triple collision. We emphasize the distinct features of the Schwarzschild model when compared to its Newtonian counterpart. We prove that, in contrast to the Newtonian case, on any level of energy the measure of the set on initial conditions leading to triple collision is positive. Further, whereas in the Newtonian problem triple collision is asymptotically reached only for zero angular momentum, in the Schwarzschild problem the triple collision is possible for nonzero total angular momenta (e.g., when two of the mass points spin infinitely many times around the center of mass). This phenomenon is known in celestial mechanics as the black-hole effect and is understood as an analog in the classical context of behavior near a Schwarzschild black hole. Also, while in the Newtonian problem all triple collision orbits are necessarily homothetic, in the Schwarzschild problem this is not necessarily true. In fact, in the Schwarzschild problem there exist triple collision orbits that are neither homothetic nor homographic.     

Periodic orbits for anisotropic perturbations of the Kepler problem

We study the Kepler problem perturbed by an anisotropic term, that is a potential conformed by a Newtonian term, 1/r$1/r$, plus an anisotropic term, b/(r²[1+?cos²θ])^β/2$b/{\left(r^2[1+?{cos}^2\theta ]\right)}^{\beta /2}$. Because of the anisotropic term, although the system is conservative the angular momentum is not a constant of motion.     

Saari's conjecture for the collinear -body problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian -body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.