Related papers: Periodic oscillations in a 2N-body problem
We prove the existence of periodic solutions of the N=(n+1)-body problem starting with n bodies whose reduced motion is close to a non-degenerate central configuration and replacing one of them by the center of mass of a pair of bodies…
The present work studies the continuation class of the regular $n$-gon solution of the $n$-body problem. For odd numbers of bodies between $n = 3$ and $n = 15$ we apply one parameter numerical continuation algorithms to the energy/frequency…
Trivial choreographies are special periodic solutions of the planar three-body problem. In this work we use a modified Newton's method based on the continuous analog of Newton's method and a high precision arithmetic for a specialized…
We consider the $N$-vortex problem on the sphere assuming that all vortices have equal strength. We develop a theoretical framework to analyse solutions of the equations of motion with prescribed symmetries. Our construction relies on the…
Motivated by the recent works on the stability of symmetric periodic orbits of the elliptic Sitnikov problem, for time-periodic Newtonian equations with symmetries, we will study symmetric periodic solutions which are emanated from…
For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have…
We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only…
We present a proof of the existence of a periodic orbit for the Newtonian six-body problem with equal masses. This orbit has three double collisions each period and no multiple collisions. Our proof is based on the minimization of the…
This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision…
We extend our previous analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized fully symmetric equal mass four-body problem to the analytic existence of a symmetric periodic simultaneous binary…
We study the existence of non-collision periodic solutions with Newtonian potentials for the following planar restricted 4-body problems: Assume that the given positive masses $m_{1},m_{2},m_{3}$ in a Lagrange configuration move in circular…
The partial case of the planar $N+1$ body problem, $N\ge2$, of the type of planetary system with satellites is studied. One of the bodies (the Sun) is assumed to be much heavier than the other bodies ("planets" and "satellites"), moreover…
We consider an $N$--body problem under a harmonic potential of the form $\frac{1}{2}\sum \kappa_{jl} |q_j-q_l|^2$. A $p$-lima\c{c}on curve is a planar curve parametrized by $t$ given by $a(\cos t,\sin t)+b(\cos pt, \sin pt)$, where $a,b\in…
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…
We prove existence and multiplicity of periodic motions for the forced 2-body problem under conditions of topological character. In the different cases, the lower bounds obtained for the number of solutions are related to the winding number…
For variational problems with $O(N)$-symmetry the existence of several geometrically distinct solutions had been shown by use of group theoretic approach in previous articles. It was done by a crafty choice of a family $H_i \subset O(N)$…
We study a classical model for the atom that considers the movement of $n$ charged particles of charge $-1$ (electrons) interacting with a fixed nucleus of charge $\mu >0$. We show that two global branches of spatial relative equilibria…
This paper is devoted to the study of periodic solutions for a radially symmetric semilinear wave equation in an $n$-dimensional ball. By combining the variational methods and saddle point reduction technique, we prove there exist at least…
A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion ("acceleration equal force") featuring one-body and two-body velocity-dependent forces "of goldfish type" which determine the…
A very few three-dimensional (3D) periodic orbits of general three-body problem (with three finite masses) have been discovered since Newton mentioned it in 1680s. Using a high-accuracy numerical strategy we discovered 10,059…