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The secular approximation of the hierarchical three body systems has been proven to be very useful in addressing many astrophysical systems, from planets, stars to black holes. In such a system two objects are on a tight orbit, and the…
We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the…
In this paper we define and study the billiard problem on bounded regions on surfaces of constant curvature. We show that this problem defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism,…
We study the spatial isosceles three body problem, which is a system with two degrees of freedom after modulo the rotation symmetry. For certain choices of energy and angular momentum, we find some disk-like global surfaces of section with…
We consider the configuration space of ordered points on the two-dimensional sphere that satisfy a specific system of quadratic equations. We construct periodic orbits in this configuration space using elliptic theta functions and show that…
The case of the planar circular restricted three-body problem where one of the two primaries is an oblate spheroid is investigated. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of…
A 3D-dynamical model is constructed for the study of motion in the central regions of a disk galaxy with a double nucleus. Using the results of the 2D-model, we find the regions of initial conditions in the (x,px,z,py)=EJ, (y=pz=0) phase…
In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the eight figure choreography, and we explore some symmetric periodic orbits of…
In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist -- at least -- theoretically -- two different types: the exchange-a orbits and the…
We consider autonomous Newtonian systems with Coriolis forces in two and three dimensions and study the existence of branches of periodic orbits emanating from equilibria. We investigate both degenerate and nondegenerate situations. While…
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…
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the…
The elliptic restricted three body problem has been well studied. However, the previous formulations of the problem have used a rotating coordinate system to keep the positions of the primary and secondary on the x-axis. This requires the…
We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces, and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of…
Relationship between quantum shell structure and classical periodic orbits is briefly reviewed on the basis of semi-classical trace formula. Using the spheroidal cavity model, it is shown that three-dimensional periodic orbits, which are…
Comet-type periodic orbits of the circular restricted three-body problem (CR3BP) are periodic solutions that are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries.…
We study the dynamics of two bodies moving on elliptic Keplerian orbits around a fixed center of attraction and interacting only by means of elastic or inelastic collisions. We show that there exists a bounded invariant region: for suitable…
The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of $\mu,…
The regularization of a new problem, namely the three-body problem, using 'similar' coordinate system is proposed. For this purpose we use the relation of 'similarity', which has been introduced as an equivalence relation in a previous…
The three-body problem, which describes three masses interacting through Newtonian gravity without any restrictions imposed on the initial positions and velocities of these masses, has attracted the attention of many scientists for more…