Novel solvable extensions of the goldfish many-body model

Abstract

A novel solvable extension of the goldfish N-body problem is presented. Its Newtonian equations of motion read ζ̈n=2aζ̇nζn+2∑m=1,m≠nN(ζ̇n−aζn2)(ζ̇m−aζm2)∕(ζn−ζm), n=1,…,N, where a is an arbitrary (nonvanishing) constant and the rest of the notation is self-evident. The isochronous version of this model is characterized by the Newtonian equations of motion z̈n−3iωżn−2ω2zn=2a(żn−iωzn)zn+2∑m=1,m≠nN(żn−iωzn−azn2)(żm−iωzm−azm2)∕(zn−zm), n=1,…,N, where ω is an arbitrary positive constant and the points zn(t) move now necessarily in the complex z-plane. The generic solution of this second model is completely periodic with a period Tk=kT which is an integer multiple k (not larger than N!, indeed generally much smaller) of the basic period T=2π∕ω and which is independent of the initial data (for sufficiently small, but otherwise arbitrary, changes of such data). These many-body models have an intriguing variety of equilibrium configurations (genuine: with no two particles sitting at the same place), but only for small values of N (N=2,3,4 for the first model, N=2,3,4,5 for the second). Other versions of these models are also discussed. The study of the behavior of the second, isochronous model around its equilibrium configurations yields some amusing diophantine results.

Two New Solvable Dynamical Systems of Goldfish Type

“…Because this nonlinear matrix ODE is just -up to trivial notational changes -the point of departure of a previous treatment yielding a dynamical system of goldfish type (see [7] or, equivalently, Example 4.2.2-6 in the monograph [1]), one might question the validity of the assertion made above, that the solvable goldfish models identified in the present paper, see (3), are new. Yet this claim is in fact quite valid.…”

Goldfishing by gauge theory

“…In the special case γ = 0 a model is obtained whose solvability was already known [8]. Our new model reported in Section II (and already derived by our other method in the preceding Subsection A) is obtained for β = 0, γ = 1, and α = −a 2 .…”

“…Remark 2.5. The solvable character [10] of the matrix evolution equation (2) entails that all its solutions U (t) are meromorphic functions of the independent variable t. Hence (see (10)) all the nonsingular solutionsŨ (t) of the matrix evolution equation (8) are periodic with period 2 π,…”

“…relating the N × N matrixŨ (t) evolving according to (8) to the N × N matrix U (t) evolving according to (2) with a = 0. Remark 2.5.…”

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

“…(The original goldfish model is the special case of these equations of motion with r " 0 and f nˆ 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 (see, for instance, [9][10][11][12][13][14][15][16][17][18][19][20]). Notation 1.1.…”