On Maxwell's (n+1)-Body Problem in the Manev-Type Field and on the Associated Restricted Problem

Abstract

We consider the polarization of unstable type-IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large-N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S 3 geometry to that of a fuzzy S 4 .

Finiteness of polygonal relative equilibria for generalised quasi-homogeneous n-body problems and n-body problems in spaces of constant curvature

“…This means that getting any information about polygonal relative equilibria that exist in spaces of positive constant curvature, zero curvature, or negative constant curvature can further our understanding about the geometry of the universe. Additionally, the ring problem, or a regular polygonal relative equilibrium with one mass at its center and all masses on the circle equal (see for example [31]) is a model that was originally formulated by Maxwell to describe the dynamics of particles orbiting Saturn (see [44]) and has since then been applied to describing other planetary rings, asteroid belts, planets orbiting stars, stellar formations, stars with an accretion ring, planetary nebula and motion of satellites (see [3], [4], [5], [31], [32], [34], [35], [45], [46], [47], [53]- [56]). In this context, considering the more general solutions of polygonal relative equilibria, proving the number of possible equilibria to be finite may be a very fruitful endeavour.…”

Symmetric planar central configurations of five bodies: Euler plus two

“…The ring problem studies the motion of (n + 1)-bodies where n bodies of equal masses are located at the vertices of a regular polygon centered at the remaining body, thus forming a central configuration. It was proposed by Maxwell in [17] as a model for the motion of the particles surrounding Saturn, and used more recently to model systems like planetary rings, asteroid belts, planets around a star, certain stellar formations, stars with accretion ring, planetary nebula, motion of an artificial satellite about a ring, (see [23,25,24,19,20,12,13,3,9,4,5]). We remark that the ring problem with four equal masses on the ring and a fifth mass at the center of the ring considered in [23] coincides with the special case found in Theorem 1.1 (a); such a configuration has been found to be locally unstable.…”

Homographic solutions in the planar n + 1 body problem with quasi-homogeneous potentials

“…For more general potentials the existence of homographic solutions is also analyzed, in particular Manev potentials (Mioc and Stavinschi 1999) and Schwarzschild potentials (Mioc and Stavinschi 1998). In this paper, we go a step ahead considering that the force between any two bodies is a generalized force which is a function of the mutual distance (Sect.…”