On the Problem of Infinite Spin in Total Collisions of the Planar N-Body Problem

TL;DR

This paper introduces a new moving frame approach for the planar N-body problem to analyze infinite spin phenomena near total collision orbits, providing comprehensive solutions for various degeneracy cases and initial condition measures.

Contribution

It develops a novel moving frame method tailored for near-collision orbits, enabling the resolution of the infinite spin problem across multiple degeneracy scenarios in the planar N-body problem.

Findings

Full solution for nondegenerate central configurations.

Partial solutions for degenerate configurations using normal forms.

Measure estimates for initial conditions leading to total collisions.

Abstract

For the planar $N$-body problem, we first introduce a class of moving frame suitable for orbits near central configurations, especially for total collision orbits, which is the main new ingredient of this paper. The moving frame allows us to reduce the degeneracy of the problem according to intrinsic symmetrical characteristic of the $N$-Body problem. First, we give a full answer to the infinite spin or $Painlev\acute{e}$-$Wintner$ problem in the case corresponding to nondegenerate central configurations. Then following some original ideas of C.L. Siegel, especially the idea of normal forms, and applying the theory of central manifolds, we give a partial answer to the problem in the case corresponding to degenerate central configurations. We completely answer the problem in the case corresponding to central configurations with one degree of degeneracy. Combining some results on the planar nonhyperbolic equilibrium point, we give a criterion for the case corresponding to central configurations with two degrees of degeneracy. We further answer the problem in the case corresponding to all known degenerate central configurations of four-body. Therefore, we solve the problem for almost every choice of the masses of the four-body problem. Finally, we give a measure of the set of initial conditions leading to total collisions.