A solvable many-body problem in the plane

TL;DR

This paper introduces a new exactly solvable many-body problem in the plane with complex velocity and position-dependent forces, revealing special periodic solution cases and expanding understanding of nonlinear Newtonian systems.

Contribution

It presents a novel solvable many-body model with specific velocity and position-dependent forces, characterized by a large parameter set and special periodic solution cases.

Findings

Identified conditions for confined and periodic motions.

Derived explicit solution behaviors for the model.

Expanded the class of solvable nonlinear Newtonian systems.

Abstract

A solvable many-body problem in the plane is exhibited. It is characterized by rotation-invariant Newtonian (``acceleration equal force'') equations of motion, featuring one-body (``external'') and pair (``interparticle'') forces. The former depend quadratically on the velocity, and nonlinearly on the coordinate, of the moving particle. The latter depend linearly on the coordinate of the moving particle, and linearly respectively nonlinearly on the velocity respectively the coordinate of the other particle. The model contains $2n^2$ arbitrary coupling constants, $n$ being the number of particles. The behaviour of the solutions is outlined; special cases in which the motion is confined (multiply periodic), or even completely periodic, are identified.