Another New Solvable Many-Body Model of Goldfish Type

TL;DR

This paper introduces a new exactly solvable many-body model of goldfish type with velocity-dependent forces, exhibiting properties like isochrony and asymptotic isochrony, and explores its mathematical implications.

Contribution

It presents a novel solvable many-body system with explicit solutions, including variants with isochronous behavior and alternative formulations, expanding the class of integrable models.

Findings

Model characterized by nonlinear Newtonian equations with velocity-dependent forces.

Existence of isochronous and asymptotically isochronous variants.

Explicit solutions via eigenvalues of a time-dependent matrix.

Abstract

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion ("acceleration equal force") featuring one-body and two-body velocity-dependent forces "of goldfish type" which determine the motion of an arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ matrix $U(t)$ explicitly known in terms of the 2N initial data $z_{n}(0)$ and $\dot{z}_{n}(0)$. This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data ("isochrony"); for other special values of these parameters this property holds up to corrections vanishing exponentially as $t\rightarrow \infty$ ("asymptotic isochrony"). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical findings implied by some of these results - such as Diophantine properties of the zeros of certain polynomials - are outlined, but their analysis is postponed to a separate paper.