Related papers: New coordinates for the Four-Body problem
It is well known that a rotation of a free generic three-dimensional rigid body is stationary if and only if it is a rotation around one of three principal axes of inertia. As it was noted by many authors, the analogous result is true for a…
The dynamics of compact binary systems at the fourth post-Newtonian (4PN) approximation of general relativity has been recently completed in a self-consistent way. In this paper, we compute the ten Poincar\'e constants of the motion and…
The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in…
In this paper we find the families of relative equilibria for the three body problem in the plane, when the interaction between the bodies is given by a quasi-homogeneous potential, which is the sum of two homogeneous functions. The number…
The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of…
We extract all the invariants (i.e. all the functions which do not depend on the choice of phase-space coordinates) of the dynamics of two point-masses, at the third post-Newtonian (3PN) approximation of general relativity. We start by…
We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We…
An approach is developed to find approximate solutions to the restricted circular three body problem. The solution is useful in approximately describing the position vectors of three spherically symmetric masses, one of which has a much…
We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the Brutus N-body code to include Post-Newtonian pairwise terms…
We introduce a model of multi-agent dynamics for self-organised motion; individuals travel at a constant speed while trying to adopt the averaged body attitude of their neighbours. The body attitudes are represented through unitary…
The Lagrangian, the Hamiltonian and the constant of motion of the gravitational attraction of two bodies when one of them has variable mass is considered. The relative and center of mass coordinates are not separated, and choosing the…
Our concept of mass has evolved considerably over the centuries, most notably from Newton to Einstein, and then even more vigorously with the establishment of the standard model and the subsequent discovery of the Higgs boson. Mass is now…
The exact analytic solution is introduced for the rotational motion of a rigid body having three equal principal moments of inertia and subjected to an external torque vector which is constant for an observer fixed with the body, and to…
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…
We construct a highly-symmetric periodic orbit of six bodies in three dimensions. In this orbit, binary collisions occur at the origin in a regular periodic fashion, rotating between pairs of bodies located on the coordinate axes.…
In Einstein's equation we suggest a geometrical object substituting the tensor of energy of impulse and tension. The obtained equation, together with the equation for external field, makes up the complete problem of mathematical equations…
We discuss several conditions for four points to lie on a plane, and we use them to find new equations for four-body central configurations that use angles as variables. We use these equations to give novel proofs of some results for…
The motion of three interacting point vortices in the plane can be thought of as the motion of three geometrical points endowed with a dynamics. This motion can therefore be re-formulated in terms of dynamically evolving geometric…
It has been shown that the extension of the elasticity theory in more than three dimensions allows a description of space-time as a properly stressed medium, even recovering the Minkowski metric in the case of uniaxial stress. The…
We calculate the motion of binary mass systems in gravity up to the fourth post--Newtonian order. We use momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates by applying…