2024/08/29 by Anna Gierzkiewicz, Gierzkiewicz, Anna, Rodrigo G. Schaefer +3 · 1 citation
Physics and Astronomy · #37N05 #70F10 #70F15 #70F16 #70G40 #70G60 #Classical Analysis and ODEs (math.CA) #Cold Atom Physics and Bose-Einstein Condensates #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum and Classical Electrodynamics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2408.16409
openalex publication_date 2024/08/29 · openalex created_date 2024/09/29 · openalex updated_date 2026/07/28
In the n-body problem, when a~cluster of bodies tends to a collision, then its normalized shape curve converges to the set of normalized central configurations, which has SO(2) symmetry in the planar case. This leaves a possibility that the normalized shape curve tends to the circle obtained by rotation of some central configuration instead of a particular point on it. This is the infinite spin problem which concerns the rotational behavior of total collision orbits in the n-body problem. The question also makes sense for partial collision. We show that the infinite spin is not possible if the limiting circle is isolated from other connected components of the set of normalized central configurations. Our approach extends the method from recent work for total collision by Moeckel and Montgomery, which was based on a combination of the center manifold theorem with Łojasiewicz inequality. To that we add a shadowing result for pseudo-orbits near normally hyperbolic manifold and careful estimates on the influence of other bodies on the cluster of colliding bodies.