Some Problems on the Classical N-Body Problem

TL;DR

This paper presents a curated list of fundamental and challenging open problems in the classical n-body problem within celestial mechanics, inspired by Smale's list, emphasizing mathematical clarity and significance.

Contribution

It introduces a set of well-defined, significant open problems in the n-body problem, focusing on various aspects like stability, singularities, and configurations, with criteria for selection based on clarity, difficulty, and importance.

Findings

Identification of key open problems in celestial mechanics

Criteria for selecting mathematically significant problems

Focus on problems with potential impact on the field

Abstract

Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.