Periods of the Goldfish Many-Body Problem

Abstract

Calogero's goldfish N -body problem describes the motion of N point particles subject to mutual interaction with velocity-dependent forces under the action of a constant magnetic field transverse to the plane of motion. When all coupling constants are equal to one, the model has the property that for generic initial data, all motions of the system are periodic. In this paper we investigate which are the possible periods of the system for fixed N , and we show that there exist initial data that realize each of these possible periods. We then discuss the asymptotic behaviour of the maximal period for large particle number N .

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

“…However, there are lots of solutions that are completely periodic with periods which are integer submultiples of T MAX . The detailed identification of these solutions and their periods is a nontrivial matter, as shown, for instance, by the discussion of this phenomenology in the paper [11]-that treats the "periodic goldfish model" (for this terminology, see [8]), which is in fact characterized by the same equations of motions (11a), but with all coupling constants vanishing, g m " 0-and by the detailed investigation of the structure of the Riemann surfaces associated with other analogous many-body models [21][22][23][24][25]. Proposition 2.2.…”

“…Let us complete our discussion of the first class of solvable N-body problems by displaying the equations of motion (11) in the (simplest) N " 2 case:…”

“…The N complex coordinates z n ptq providing the solution at time t of the initial-values problem of the Newtonian equations of motion (11) are the N zeros of the monic polynomial (4b), the N coefficients w m ptq of which are given, in terms of the initial data z n p0q , 9 z n p0q , by the following formulas:…”

“…(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.…”

Diophantine Properties Associated to the Equilibrium Configurations of an Isochronous N-Body Problem

“…Hence (see (2.2b)) in this special case all the coordinates z n (t) also evolve in time periodically with the same period T (or possibly, some of them, with a period which is a, generally small, integer multiple of T , due to the possibility that over the time evolution some of the zeros of the polynomial ψ (z,t) exchange their roles; for a discussion of this possibility, including a justification of the assertion that the integer multiple in question is "generally small", see [5]). Anyway, in this special case the N-body problem (2.1) is isochronous.…”