Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

Francesco Calogero

doi:10.3390/sym8070053

Abstract

Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer ( N ≥ 2 ). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods T / p with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future.

Highlights

A new technique to identify many-body problems solvable by algebraic operations has been introduced [1,2], and several examples of such models have been discussed [1,2,3,4,5,6]
Three new classes of such models are introduced and discussed
Is the special case of these equations of motion with r “ 0 and f n ~z, ~z “ 0; after its first identification as a solvable model [7], and its tentative recognition as a “goldfish” [8], this N-body problem and some of its extensions have been investigated in several publications

Summary

Introduction

A new technique to identify many-body problems solvable by algebraic operations has been introduced [1,2], and several examples of such models have been discussed [1,2,3,4,5,6]. ~ ptqq on the real variable t (“time”) (see (4b)), the unordered character of the set PN pz; z ptq ; w of its N zeros zn ptq is generally only relevant at one value of time, say at the “initial” time t “ 0, since, at other values of time, the ordering gets generally determined by the natural requirement that the functions zn ptq evolve continuously over time This prescription fixes, for all time, the ordering of the zeros zn ptq—i.e., the assignment of the value n of its index to each zero zn ptq—as long as the coefficients wm ptq evolve themselves continuously and unambiguously over time and no “collision” of two or more zeros occurs over the time evolution, i.e., for all time zn ptq ‰ zptq if n ‰ ` (since clearly such collisions imply a loss of identity of the coinciding zeros). A key formula for the identification and investigation of solvable Newtonian N-body problems is the following relation [1] among the τ-evolutions of the N zeros and the N coefficients of the monic polynomial p N pζ; τq (see 4a)) of degree N in its argument ζ and depending on the extra variable τ:. The corresponding solvable N-body models satisfied by the coordinates zn ” zn ptq are displayed—and their properties discussed— 2, with the corresponding proofs provided in Section 3, while the special case with am “ 2 and bm “ ́1 (and with gm an arbitrary rational number) is treated in [6]

Results

Proofs

Outlook