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

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**Notation 1.1.**

**Notation 1.1**).

**Notation 1.2.**

**Remark 1.1.**

- (i)
- Of course, the assignment of the N coefficients of a polynomial defines uniquely the corresponding unordered set of its N zeros, but generally it only allows to compute explicitly these N zeros for $N\le 4$.
- (ii)
- Moreover—and quite relevantly in our context (see below)—if a polynomial features a dependence on an additional variable (as, for instance, the dependence of the polynomial ${P}_{N}\left(z;\underline{z}\left(t\right);\overrightarrow{w}\left(t\right)\right)$ on the real variable t ("time") (see (4b)), then the unordered character of the set of its N zeros ${z}_{n}\left(t\right)$ 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 ${z}_{n}\left(t\right)$ evolve continuously over time. This prescription then fixes, for all time, the ordering of the zeros ${z}_{n}\left(t\right)$—i.e., the assignment of the value n of its index to each zero ${z}_{n}\left(t\right)$—as long as the coefficients ${w}_{m}\left(t\right)$ evolve themselves continuously and unambiguously over time and moreover no "collision" of two or more zeros occurs over the time evolution, i.e., for all time ${z}_{n}\left(t\right)\ne {z}_{\ell}\left(t\right)$ if $n\ne \ell $ (since clearly such collisions imply a loss of identity of the coinciding zeros). However, this identification requires an analysis of the time evolution of the N zeros ${z}_{n}\left(t\right)$ not only in the complex z-plane, but in fact over the N-sheeted Riemann surface associated to the N roots of the polynomial ${P}_{N}\left(z;\underline{z}\left(t\right);\overrightarrow{w}\left(t\right)\right),$ and/or over the evolution of each coefficient ${w}_{m}\left(t\right)$ if its time evolution takes itself place on a Riemann surface (as it indeed happens in the cases discussed below). ■

## 2. Results

**Notation 1.2**)

**Proposition 2.1.**

**Proposition 2.2**for which the system of evolution equations (11) runs into a singularity at a finite time.

**Proposition 2.1**implies that all generic solutions of the N-body model characterized by the Newtonian equations of motion (11a)—excluding the nongeneric solutions which are singular (see below

**Proposition 2.2**)—are completely periodic with the same period ${T}_{MAX}=\left(p\phantom{\rule{3.33333pt}{0ex}}N!\right)\phantom{\rule{3.33333pt}{0ex}}{T}_{0}$. 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.**

**Notation 1.2**). Note that this might be considered the special case of the first class of models (see above) with $r=0$, ${a}_{m}=2$ and ${b}_{m}=0$, which was previously excluded because it requires a special treatment (see Section 3).

**Proposition 2.3.**

**Remark 3.1**)—iff the initial data satisfy the inequality (19b), while iff instead the initial data satisfy the opposite inequality (20a) at least one of the particle coordinates ${z}_{n}\left(t\right)$ comes from or escapes to infinity in the remote past and future: see, for instance, the relevant discussion in Appendix G (“Asymptotic behavior of the zeros of a polynomial whose coefficients diverge exponentially”) of the book [9]. In addition, of course, if the (nongeneric) initial data imply validity of the equality (21a), the equations of motion run into a singularity at $t={\stackrel{\u02c7}{t}}_{m}^{\left(2\right)}$ (see (21b)). Other nongeneric initial data causing the equations of motion (17) to run into a singularity at a finite time are those leading to particle collisions. Note that generally the nongeneric initial data causing singularities are also those that separate the regions of initial data associated to different behaviors of the model, including the emergence of the higher periodicities associated to values of K larger than unity as well as the periodic and nonperiodic cases.

**Remark 2.1.**

**Proposition 2.4.**

**Remark 3.1**), the N coordinates ${z}_{n}\left(t\right)$ are periodic with period $Q\phantom{\rule{3.33333pt}{0ex}}T$, where Q is a positive integer in the range from 1 to $N!$.

## 3. Proofs

**Proposition 2.1**.

**Proposition 2.1**we ascertain, to begin with, the periodicity properties as functions of the time variable t of the coefficients ${\gamma}_{m}\left(t\right)\equiv {\gamma}_{m}\left(\tau \left(t\right)\right)$ (see (12b)). The starting point is the observation that $\tau \left(t\right)$—see (2) or (12a)—is a periodic function of t with period ${T}_{0}$, rotating in the complex τ-plane on the circle $\tilde{C}$ centered at the point $\mathbf{i}/\omega $ and having radius $1/\left|\omega \right|$. Hence, any holomorphic function of τ is as well periodic in t with period ${T}_{0}$; this clearly is (for all values of m in its range from 1 to N) the case of the functions ${\gamma}_{m}\left(\tau \right)$ (see (12b)), if r is a negative integer. If instead r is a positive integer, the functions ${\gamma}_{m}\left(\tau \right)$ are meromorphic in τ, featuring a polar singularity at $\tau ={\stackrel{\u02c7}{\tau}}_{m}^{\left(1\right)}$ (see (12b)). In this case, ${\gamma}_{m}\left(t\right)\equiv {\gamma}_{m}\left(\tau \left(t\right)\right)$ is again generally periodic in t with period ${T}_{0},$ ${\gamma}_{m}\left(t+{T}_{0}\right)={\gamma}_{m}\left(t\right)$, but for the nongeneric assignments of the initial data such that ${\stackrel{\u02c7}{\tau}}_{m}^{\left(1\right)}$ falls on the circle $\tilde{C}$—note that ${\stackrel{\u02c7}{\tau}}_{m}^{\left(1\right)}$ does depend on the initial data (see (12c) and (12e)), and that the condition for this to happen is validity of the equality

**Remark 3.1**is relevant.

**Remark 3.1.**

**Propositions 2.1**and

**2.2**.

**Propositions 2.1**and

**2.2**.

**Proposition 2.3**that we do report is the derivation of the formula (18). The starting point is the ODE

**Remark 2.1**, it is based on the observation that the decoupled nonlinear system of N ODEs

**Proposition 2.4**.

## 4. Outlook

## Conflicts of Interest

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Calogero, F.
Three New Classes of *Solvable N*-Body Problems of Goldfish Type with *Many Arbitrary* Coupling Constants. *Symmetry* **2016**, *8*, 53.
https://doi.org/10.3390/sym8070053

**AMA Style**

Calogero F.
Three New Classes of *Solvable N*-Body Problems of Goldfish Type with *Many Arbitrary* Coupling Constants. *Symmetry*. 2016; 8(7):53.
https://doi.org/10.3390/sym8070053

**Chicago/Turabian Style**

Calogero, Francesco.
2016. "Three New Classes of *Solvable N*-Body Problems of Goldfish Type with *Many Arbitrary* Coupling Constants" *Symmetry* 8, no. 7: 53.
https://doi.org/10.3390/sym8070053