Related papers: Partially rigid motions in the n-body problem
In contrast to the well-known solution of the two-body problem through the use of the concept of reduced mass, a solution is proposed involving separation of potentials. It is shown that each of the two point bodies moves in its own…
We consider the gauging of space translations with time-dependent gauge functions. Using fixed time gauge of relativistic theory, we consider the gauge-invariant model describing the motion of nonrelativistic particles. When we use…
We show the existence of some infinite families of periodic solutions of the planar Newtonian n-body problem --with positive masses-- which are symmetric with respect to suitable actions of finite groups (under a strong--force assumption,…
Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants…
We solve the N-body problems in which the total potential energy is any function of the mass-weighted root-mean-square radius of the system of N point masses. The fundamental breathing mode of such systems vibrates non-linearly for ever. If…
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…
The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation, a no-relativistic particle with a time explicitly depending force, a no-relativistic particle with a constant force and time…
In this paper, we consider the elliptic relative equilibria of the restricted $N$-body problems, where the $N-1$ primaries form an Euler-Moulton collinear central configuration or a $(1+n)$-gon central configuration. We obtain the…
The problem of describing the free motions of three gravitating bodies under each others gravitational influence is one of the oldest of unsolved problems of classical mechanics. Henry Poincare proved in his dictum that due to the nature of…
In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-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…
This paper investigates the Newton's problem of minimal resistance for a body moving through a fluid whose density decreases exponentially with altitude. We prove the local existence and regularity of radial solutions $u(r)$ satisfying the…
Central configurations play a fundamental role in the Newtonian $n$-body problem, as they give rise to motions in which the configuration evolves while preserving its shape up to rotation and scaling. These include relative equilibria,…
We consider the 3-body problem in relativistic lineal gravity and obtain an exact expression for its Hamiltonian and equations of motion. While general-relativistic effects yield more tightly-bound orbits of higher frequency compared to…
We obtain a new exact equilibrium solution to the N-body problem in a one-dimensional relativistic self-gravitating system. It corresponds to an expanding/contracting spacetime of a circle with N bodies at equal proper separations from one…
We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three…
The internal space for a molecule, atom, or other n-body system can be conveniently parameterised by 3n-9 kinematic angles and three kinematic invariants. For a fixed set of kinematic invariants, the kinematic angles parameterise a…
We revisit polygonal positive elliptic rotopulsator solutions and polygonal negative elliptic rotopulsator solutions of the $n$-body problem in $\mathbb{H}^{3}$ and $\mathbb{S}^{3}$ and prove existence of these solutions, prove that the…
This paper introduces a new difference scheme to the difference equations for N-body type problems. To find the non-collision periodic solutions and generalized periodic solutions in multi-radial symmetric constraint for the N-body type…
The restricted planar four body problem describes the motion of a massless body under the Newtonian gravitational force of other three bodies (the primaries), of which the motion gives us general solutions of the three body problem. A…
In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration.…