arXiv subjects

Florin Diacu

Publications and source records attributed to Florin Diacu.

23 records · Page 2Linked to original sources

arXiv2009

Saari's Homographic Conjecture of the Three-Body Problem

Saari's homographic conjecture, which extends a classical statement proposed by Donald Saari in 1970, claims that solutions of the Newtonian $n$-body problem with constant configurational measure are homographic. In other words, if the mutual distances satisfy a certain relationship, the configuration of the particle system may change size and position but not shape. We prove this conjecture for large sets of initial conditions in three-body problems given by homogeneous potentials, including the Newtonian one. Some of our results are true for $n\ge 3$.

math-ph↗

arXiv2009

The Kepler Problem with Anisotropic Perturbations

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-β$, $β\ge 2$. For $β>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $β>2$ and $β\ne 3$, the flow on the zero-energy manifold is chaotic. For $β=2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.

math-ph↗

arXiv2009

Saari's Conjecture for the Collinear $n$-Body Problem

In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.

math-ph↗

arXiv2008

The n-body problem in spaces of constant curvature

We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several classes of relative equilibria, including the hyperbolic rotations for k<0. In the end, we prove Saari's conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically. We also emphasize that fixed points are specific to the case k>0, hyperbolic relative equilibria to k<0, and Lagrangian orbits of arbitrary masses to k=0--results that provide new criteria towards understanding the large-scale geometry of the physical space.

math.DS↗

arXiv2000

Nonintegrability and Chaos in the Anisotropic Manev Problem

The anisotropic Manev problem, which lies at the intersection of classical, quantum, and relativity physics, describes the motion of two point masses in an anisotropic space under the influence of a Newtonian force-law with a relativistic correction term. Using an extension of the Poincare'-Melnikov method, we first prove that for weak anisotropy, chaos shows up on the zero-energy manifold. Then we put into the evidence a class of isolated periodic orbits and show that the system is nonintegrable. Finally, using the geodesic deviation approach, we prove the existence of a large non-chaotic set of uniformly bounded and collisionless solutions.

nlin.CD↗