Josep Maria Cors

Four-body co-circular central configurations

We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.

Hip-hop solutions of the 2N-body problem

We show the existence of families of hip–hop solutions in the equal–mass 2N–body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non–harmonic oscillations. By introducing a parameter 2, the homographic motion and the small amplitude oscillations can be uncoupled into a purely Keplerian homographic motion of fixed period and a vertical oscillation described by a Hill type equation. Small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise, via a Bolzano argument, to resonant periodic solutions of the uncoupled system in a rotating frame. For small 2 6= 0, the topological transversality persists and Brouwer’s fixed point theorem shows the existence of this kind of solutions in the full system.

Periodic solutions in the spatial elliptic restricted three-body problem

We show the existence of a new class of periodic orbits in the three-dimensional elliptic restricted three-body problem in the case of equal masses of the primaries. The doubly symmetric periodic solutions found are perturbations of very large circular Keplerian orbits lying in a plane perpendicular to that of the primaries. They exist for a discrete sequence of values of the mean motion, irrespective of the values of the eccentricity of the primaries orbit.

The global flow of the hyperbolic restricted three-body problem

Two mass points of equal masses m1 = m2 > 0 move under Newtons law of attraction in a non-collision hyperbolic orbit while their center of mass is at rest. We consider a third mass point, of mass m3 = 0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their center of mass. Since m3 = 0, the motion of m1 and m2 is not affected by the third, and from the symmetry of the motion it is clear that m3 remains on the line L. The hyperbolic restricted 3-body problem is to describe the moton of m3. Our main result is the characterization of the global flow of this problem.

Coorbital periodic orbits in the three body problem

We consider the dynamics of coorbital motion of two small moons about a large planet which have nearly circular orbits with almost equal radii. These moons avoid collision because they switch orbits during each close encounter. We approach the problem as a perturbation of decoupled Kepler problems as in Poincares periodic orbits of the first kind. The perturbation is large but only in a small region in the phase space. We discuss the relationship required among the small quantities (radial separation, mass, and minimum angular separation). Persistence of the orbits is discussed.

Central configurations, periodic orbits, and Hamiltonian systems

1 The Averaging Theory for Computing Periodic Orbits.- Introduction: the classical theory.- Averaging theory for arbitrary order and dimension.- Three applications of Theorem.- 2 Lectures on Central Configurations.- The n-body problem.- Symmetries and integrals.- Central configurations and self-similar solutions.- Matrix equations of motion.- Homographic motions of central configurations in Rd.- Albouy-Chenciner reduction and relative equilibria in Rd.- Homographic motions in Rd.- Central configurations as critical points.- Collinear central configurations.- Morse indices of non-collinear central configurations.- Morse theory for CCs and SBCs.- Dziobek configurations.- Convex Dziobek central configurations.- Generic finiteness for Dziobek central configurations.- Some open problems.- 3 Dynamical Properties of Hamiltonian Systems.- Introduction.- Low dimension.- Some theoretical results, their implementation and practical tools.- Applications to Celestial Mechanics.

Qualitative study of the hyperbolic collision restricted three-body problem

We have two mass points of equal masses moving under Newtons law of gravitational attraction in a collision hyperbolic orbit while their centre of mass is at rest. We consider a third mass point, of mass , moving on the straight line L perpendicular to the line of motion of the first two mass points and passing through their centre of mass. Since , the motion of the masses and is not affected by the third mass and from the symmetry of the motion it is clear that will remain on the line L. The hyperbolic collision rectricted three-body problem consists in describing the motion of . Our main result is the characterization of the global flow of this problem.

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.

A Limit Case of the “Ring Problem”: The Planar Circular Restricted

We study the dynamics of an extremely idealized model of a planetary ring. In particular, we study the motion of an infinitesimal particle moving under the gravitational influence of a large central body and a regular n-gon of smaller bodies as n tends to infinity. Our goal is to gain insight into the structure of thin, isolated rings.

1+n

We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem.