Solvable and/or integrable and/or linearizable N-body problems in ordinary (three-dimensional) space. I

TL;DR

This paper introduces specific N-body problems in three-dimensional space that are exactly solvable, integrable, or linearizable, highlighting their properties and explicit solutions, with a focus on techniques to identify such models.

Contribution

It presents new classes of N-body problems with exact solutions and discusses methods to find such models, emphasizing their solvability and periodic behaviors.

Findings

Several models with completely periodic or multiply periodic motions

Explicit solutions provided for some cases

Techniques for uncovering solvable N-body problems

Abstract

Several N-body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (``acceleration equal force;'' in most cases, the forces are velocity-dependent) and are amenable to exact treatment (``solvable'' and/or ``integrable'' and/or ``linearizable''). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider ``few-body problems'' (with, say, \textit{N}=1,2,3,4,6,8,12,16,...) as well as ``many-body problems'' (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N-body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases.