Related papers: Relative equilibria in quasi-homogeneous three bod…
In this paper, we consider the elliptic relative equilibria of four-body problem with two infinitesimal masses. The most interesting case is when the two small masses tend to the same Lagrangian point $L_4$ (or $L_5$). In \cite{Xia}, Z. Xia…
Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in $\mathbb{R}^4$ with a general pairwise potential. We investigate the central force problem, set up the n-body problem and discuss certain properties…
We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological…
An improved hyperspherical harmonic method for the quantum three-body problem is presented to separate three rotational degrees of freedom completely from the internal ones. In this method, the Schr\"{o}dinger equation of three-body problem…
We introduce a circular restricted charged three-body problem on the plane. In this model, the gravitational and Coulomb forces, due to the primary bodies, act on a test particle; the net force exerted by some primary body on the test…
For the three-body problem, we consider the Lagrange stability. To analyze the stability, along with integrals of energy and angular momentum, we use relations by the author from Sosnitskii (2005), which band together separately squared…
The classical three-body harmonic system in $\mathbb{R}^d$ ($d>1$) with finite rest lengths and zero total angular momentum $L=0$ is considered. This model describes the dynamics of the $L=0$ near-equilibrium configurations of three point…
Stable assemblages of localized vortices exist which have particle-like properties, such as mass, and which can interact with one another when they closely approach. In this article I calculate the mass of these localized states and…
It is often assumed that few- and many-body systems can be accurately described by considering only pairwise two-body interactions of the constituents. We illustrate that three- and higher-body forces enter naturally in effective field…
Three-dimensional central symmetric bodies different from spheres that can float in all orientations are considered. For relative density rho=1/2 there are solutions, if holes in the body are allowed. For rho different from 1/2 the body is…
We consider the sticky collision between two sphere of equal rest mass, moving with equal but opposite speeds in a horizontal plane. The collision takes place on the pan of a high sensitivity balance. The conditions of mechanical and…
In this work it is shown that there are symmetries beyond the Euclidean group $E\left(3\right)$ in 3-body problem, and by extension in many-body problem, with inverse squared distance inter particle force. The symmetries in 3-body problem…
We investigate the secular dynamics of a planetary system composed of the parent star and two massive planets in mutually inclined orbits. The dynamics are investigated in wide ranges of semi-major axes ratios (0.1-0.667), and planetary…
A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (``acceleration equal force'') equations of motion, featuring one-body (``external'') and pair (``interparticle'') forces. The…
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…
Consider the planar three-body problem with masses positive $m_1,m_2,m_3$ position vector $q(t) = (q_1(t),q_2(t),q_3(t))\in\mathbb{R}^6$. Let $$U(q) = \frac{m_1m_2}{r_{12}}+\frac{m_1m_3}{r_{13}}+\frac{m_2m_3}{r_{23}}$$ where…
We prove that if the masses of the $n$-body problem of celestial mechanics are given, the number of classes of relative equilibria that solve the $n$-body problem is finite.
We investigate systems of three mutually interacting particles with masses of which the inner is much bigger than the intermediate and the latter is much bigger than the outer. Then the three-body problem reduces to the two-body scattering…
We study self-similar solutions of the point-vortex system. The explicit formula for self-similar solutions has been obtained for the three point-vortex problem and for a specific example of the four and five point-vortex problems. We see…
Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level…