Florin Diacu

The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.

Relative Equilibria of the Curved N-Body Problem

Introduction.- Preliminary developments.- Equations of motion.- Isometric rotations.- Relative equilibria (RE).- Fixed Points (FP).- Existence criteria.- Qualitative behavior.- Positive elliptic RE.- Positive elliptic-elliptic RE.- Negative RE.- Polygonal RE.- Lagrangian and Eulerian RE.- Saaris conjecture.

ON THE SINGULARITIES OF THE CURVED n-BODY PROBLEM

We study singularities of the n-body problem in spaces of constant curvature and generalize certain results due to Painleve, Weierstrass, and Sundman. For positive curvature, some of our proofs use the correspondence between total collision solutions of the original system and their orthogonal projection—a property that offers a new method of approaching the problem in this par- ticular case.

On the anisotropic Manev problem

We consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positive-measure set of collision orbits. Besides frontal homothetic, frontal nonhomothetic, and spiraling collisions and ejections, we put into the evidence the surprising class of oscillatory collision and ejection orbits. Using the infinity manifold, we further tackle capture and escape solutions in the zero-energy case. By finding the connection orbits between equilibria and/or cycles at impact and at infinity, we describe a large class of capture-collision and ejection-escape solutions.

An intrinsic approach in the curved n-body problem: The negative curvature case

Abstract We consider the motion of n point particles of positive masses that interact gravitationally on the 2-dimensional hyperbolic sphere, which has negative constant Gaussian curvature. Using the stereographic projection, we derive the equations of motion of this curved n-body problem in the Poincare disk, where we study the elliptic relative equilibria. Then we obtain the equations of motion in the Poincare upper half plane in order to analyze the hyperbolic and parabolic relative equilibria. Using techniques of Riemannian geometry, we characterize each of the above classes of periodic orbits. For n = 2 and n = 3 we recover some previously known results and find new qualitative results about relative equilibria that were not apparent in an extrinsic setting.

The global flow of the Manev problem

The Manev problem (a two‐body problem given by a potential of the form A/r+B/r2, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinates and topological methods, and offer the physical interpretation of all solutions. We prove that if the energy constant is negative, the orbits are, generically, precessional ellipses, except for a zero‐measure set of initial data, for which they are ellipses. For zero energy, the orbits are precessional parabolas, and for positive energy they are precessional hyperbolas. In all these cases, the set of initial data leading to collisions has positive measure.

Polygonal homographic orbits of the curved

In the

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Relative equilibria in the 3-dimensional curved -body problem

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