Related papers: New coordinates for the Four-Body problem
We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum $N$-rotor problem has been used to model chains of coupled Josephson junctions and also arises via a partial…
The article formulates the classical three-body problem in conformal-Euclidean space (Riemannian manifold), and its equivalence to the Newton three-body problem is mathematically rigorously proved. It is shown that a curved space with a…
We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the…
In this paper,we study spatial central configurations where N bodies are at the vertices of a regular N-gon $T$ and the other 4 bodies are symmetrically located on the straight line that is perpendicular to the plane that contains $T$ and…
Consider four point particles with equal masses in the euclidean space, subject to the following symmetry constraint: at each instant they are symmetric with respect to the dihedral group $D_2$, that is the group generated by two rotations…
We study the bifurcations of central configurations of the Newtonian four-body problem when some of the masses are equal. First, we continue numerically the solutions for the equal mass case, and we find values of the mass parameter at…
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…
A new coordinate system on the tangent space to planar configurations is introduced to simplify some calculations on central configurations and relative equilibria in the $N$-body problem with a homogeneous potential, which includes the…
The three-body problem is a fundamental long-standing open problem, with applications in all branches of physics, including astrophysics, nuclear physics and particle physics. In general, conserved quantities allow to reduce the formulation…
In this article the problem for the using of uniformly rotating coordinate system is considered. The correct relations between the kinematical quantities characterizing the motion of a body relative to uniformly rotating system and static…
The relative equilibria of planar Newtonian $N$-body problem become coorbital around a central mass in the limit when all but one of the masses becomes zero. We prove a variety of results about the coorbital relative equilibria, with an…
In this paper we consider equations of motion for 2-body problem according to an observer close to one of the gravitational bodies. The influence of the Thomas precession of the observer's frame has an important role. The equations of…
We consider the classical 3-body system with $d$ degrees of freedom $(d>1)$ at zero total angular momentum. The study is restricted to potentials $V$ that depend solely on relative (mutual) distances $r_{ij}=\mid {\bf r}_i - {\bf r}_j\mid$…
Why would anyone wish to generalize the already unappetizing subject of rigid body motion to an arbitrary number of dimensions? At first sight, the subject seems to be both repellent and superfluous. The author will try to argue that an…
An approach is developed to find approximate solutions to the classical Newtonian problem of N bodies. Sets of N gravitating bodies having spherically symmetric mass distributions, small angular velocities (< 1 rad/s) and bounded position…
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…
Newton introduced the concept of mass in his {\it Principia} and gave an intuitive explanation for what it meant. Centuries have passed and physicists as well as philosophers still argue over its meaning. Three types of mass are generally…
Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type ("acceleration equals forces") which determine the motion of points in the complex plane. These…
The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of…
The most general coordinates transformations that allow for the exact separation of the kinetic energy operator of a quantum many-body system into total center of mass kinetic energy and internal kinetic energy are found and discussed. We…