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A new coordinate system is defined for the Four-Body dynamical problem with general masses, having as its origin of coordinates the center of mass. The transformation from the inertial coordinate system involves a combination of a rotation…
For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration…
We provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k…
We consider the $n$--body problem defined on surfaces of constant negative curvature. For the case of $n$--equal masses we prove that the hyperbolic relative equilibria with a regular polygonal shape do not exist. In particular the…
We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations…
Despite the huge number of research into the three-body problem in physics and mathematics, the study of this problem still remains relevant both from the point of view of its broad application and taking into account its fundamental…
We study periodic solutions of the planar Newtonian $N$-body problem with equal masses. Each periodic solution traces out a braid with $N$ strands in 3-dimensional space. When the braid is of pseudo-Anosov type, it has an associated stretch…
In this article, I discuss the motion of $N$ point masses in non-relativistic mechanics, when the interaction between them is purely the Newtonian gravitational interaction, with $N$ greater than or equal to 2. The dynamical equations of…
This research investigates centered co-circular central configurations in the general power-law potential $n$-body problem. Firstly, there are no such configurations when all masses are equal, except for two; secondly, unless all masses are…
We look for particular solutions to the restricted three-body problem where the bodies are allowed to either lose or gain mass to or from a static atmosphere. In the case that all the masses are proportional to the same function of time,we…
We establish the existence of a single-parameter family of the concave kite central configurations in the 4-body problem with two pairs of equal masses. In such configurations, one pair of the masses must lie on the base of an isosceles…
Let a number, N, of particles interact classically through Newton's Laws of Motion and Newton's inverse square Law of Gravitation. The resulting equations of motion provide an approximate mathematical model with numerous applications in…
Central configurations play an important role in the dynamics of the $n$-body problem: they occur as relative equilibria and as asymptotic configurations in colliding trajectories. We illustrate how they can be found as projective fixed…
We consider the critical points (equilibria) of a planar potential generated by $n$ Newtonian point masses augmented with a quadratic term (such as arises from a centrifugal effect). Particular cases of this problem have been considered…
In this paper, we consider the elliptic collinear solutions of the classical $n$-body problem, where the $n$ bodies always stay on a straight line, and each of them moves on its own elliptic orbit with the same eccentricity. Such a motion…
The figure eight is a remarkable solution to the Newtonian three-body problem in which the three equal masses chase each around a planar curve having the qualitative shape and symmetries of a figure eight. Here we prove that each lobe of…
The time-dependent restricted $(n+1)$-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by $n$ primary bodies following a periodic solution of the $n$-body problem. We…
Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle…
The restricted (equilateral) four-body problem consists of three bodies of masses m1, m2 and m3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral…
The positive mass theorem states that the total mass of a complete asymptotically flat manifold with non-negative scalar curvature is non-negative; moreover, the total mass equals zero if and only if the manifold is isometric to the…