Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces

The class of solvable many-body problems ''of goldfish type'' is extended by including (the additional presence of) three-body forces. The solvable N-body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitrary ''coupling co...

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Дата:	2013
Автори:	Bihun, O., Calogero, F.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149349
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Цитувати:	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces / O. Bihun, F. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1493492019-02-22T01:22:48Z Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces Bihun, O. Calogero, F. The class of solvable many-body problems ''of goldfish type'' is extended by including (the additional presence of) three-body forces. The solvable N-body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitrary ''coupling constants''. Restrictions on these constants are identified which cause these systems – or appropriate variants of them – to be isochronous or asymptotically isochronous, i.e. all their solutions to be periodic with a fixed period (independent of the initial data) or to have this property up to contributions vanishing exponentially as t→ ∞. 2013 Article Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces / O. Bihun, F. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70F10; 70H06; 37J35; 37K10 DOI: http://dx.doi.org/10.3842/SIGMA.2013.059 http://dspace.nbuv.gov.ua/handle/123456789/149349 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The class of solvable many-body problems ''of goldfish type'' is extended by including (the additional presence of) three-body forces. The solvable N-body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitrary ''coupling constants''. Restrictions on these constants are identified which cause these systems – or appropriate variants of them – to be isochronous or asymptotically isochronous, i.e. all their solutions to be periodic with a fixed period (independent of the initial data) or to have this property up to contributions vanishing exponentially as t→ ∞.
format	Article
author	Bihun, O. Calogero, F.
spellingShingle	Bihun, O. Calogero, F. Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Bihun, O. Calogero, F.
author_sort	Bihun, O.
title	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
title_short	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
title_full	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
title_fullStr	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
title_full_unstemmed	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces
title_sort	solvable many-body models of goldfish type with one-, two- and three-body forces
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149349
citation_txt	Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces / O. Bihun, F. Calogero // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 18 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT bihuno solvablemanybodymodelsofgoldfishtypewithonetwoandthreebodyforces AT calogerof solvablemanybodymodelsofgoldfishtypewithonetwoandthreebodyforces
first_indexed	2025-07-12T21:54:16Z
last_indexed	2025-07-12T21:54:16Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 059, 18 pages Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces Oksana BIHUN † and Francesco CALOGERO ‡ † Department of Mathematics, Concordia College at Moorhead, MN, USA E-mail: obihun@cord.edu URL: http://wwwp.cord.edu/faculty/obihun/Oksana%20Bihun.htm ‡ Physics Department, University of Rome “La Sapienza”, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Italy E-mail: francesco.calogero@roma1.infn.it, francesco.calogero@uniroma1.it Received June 07, 2013, in final form October 02, 2013; Published online October 09, 2013 http://dx.doi.org/10.3842/SIGMA.2013.059 Abstract. The class of solvable many-body problems “of goldfish type” is extended by including (the additional presence of) three-body forces. The solvable N -body problems thereby identified are characterized by Newtonian equations of motion featuring 19 arbitra- ry “coupling constants”. Restrictions on these constants are identified which cause these systems – or appropriate variants of them – to be isochronous or asymptotically isochronous, i.e. all their solutions to be periodic with a fixed period (independent of the initial data) or to have this property up to contributions vanishing exponentially as t→∞. Key words: many-body problems; N -body problems; partial differential equations; isochro- nous systems 2010 Mathematics Subject Classification: 70F10; 70H06; 37J35; 37K10 1 Introduction Over three decades ago [3] a class of solvable N -body problems featuring several free parame- ters (“coupling constants”) was introduced by identifying the coordinates zn(t) of the moving particles with the N zeros of the time-dependent (monic) polynomial ψ(z, t) (of degree N in the independent variable z), ψ(z, t) = N∏ n=1 [z − zn(t)] = zN + N∑ m=1 cm(t)zN−m, (1.1) itself evolving according to a linear partial differential equation (PDE) – suitably restricted to guarantee that it feature a polynomial solution of degree N in z. The simplest dynamical system belonging to this class displays remarkably neat properties and was therefore considered a “goldfish” (for a justification of this terminology, see [5]; subsequently this terminology has been often employed to identify N -body models belonging to this class, and this justifies its use also in the present paper, including its title). The behavior of the solutions of these N -body problems has been variously investigated and also used to arrive at related mathematical results, such as finite-dimensional representations of differential operators and Diophantine properties of the zeros of certain polynomials: see the two monographs [4] and [6] and the references quoted there (including the more recent ones added to the 2012 paperback version of [6]), and the more recent papers [1, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. The class of linear PDEs satisfied by the polynomial ψ(z, t) which subtended these investiga- tions was so far restricted to PDEs featuring derivatives up to second order. The restriction to mailto:obihun@cord.edu http://wwwp.cord.edu/faculty/obihun/Oksana%20Bihun.htm mailto:francesco.calogero@roma1.infn.it mailto:francesco.calogero@uniroma1.it http://dx.doi.org/10.3842/SIGMA.2013.059 2 O. Bihun and F. Calogero time-differentiations of second order was motivated by the interest in N -body problems featu- ring equations of motion of Newtonian type (i.e., “acceleration equals force”); the more general restriction to differentiations of at most second-order implied that the N -body problems under consideration only involved one-body and two-body (generally velocity-dependent) forces. In the present paper we extend our consideration to third-order z-derivatives – while maintaining the restriction to second-order time-differentiations so as to only treat N -body problems of New- tonian type. This entails that the corresponding solvable N -body problems thereby identified involve, additionally, three-body forces. In the following Section 2 we report our main results, namely we display the Newtonian equations of motion of the N -body problems identified in this paper and we indicate how they are solved by algebraic operations. In a terse Section 3 these findings are proven: these developments rely on identities reported and proven in Appendix A, constituting a substantial part of this paper. In Section 4 we highlight some special cases in which these N -body problems – or appropriate variants of them – are isochronous or asymptotically isochronous, i.e. all their solutions are periodic with a fixed period (independent of the initial data) or they feature this property up to contributions vanishing exponentially as t → ∞. A terse Section 5 entitled “Outlook” outlines possible future developments. 2 Main results In this section we report the main results of this paper, which are then proven in the following section. But firstly let us specify our notation. Notation 2.1. The N coordinates (“dependent variables”) of the point, unit-mass, moving particles which are the protagonists of the N -body problem treated in this paper are generally denoted as zn ≡ zn(t), with t (“time”) the “independent variable”. As usual differentiation with respect to time is denoted by a superimposed dot, hence żn ≡ dzn/dt, z̈n ≡ d2zn/dt2. As we just did here, often the indication of the t-dependence is not explicitly displayed. We generally assume that these coordinates zn are complex numbers, so that the points with coordinates zn(t) move in the complex z-plane; but special subcases in which the coordinates zn are real are of course possible; and it is of course also possible to reinterpret motions taking place in the complex z- plane as instead taking place in the real horizontal plane by identifying the real and imaginary parts of the complex numbers zn = xn + iyn as the Cartesian components of the real 2-vectors ~rn = (xn, yn) (as explained in Chapter 4 of [4], entitled “Solvable and/or integrable many-body problems in the plane, obtained by complexification”). Here and hereafter i is the imaginary unit, i2 = −1, N is an arbitrary positive integer (generally N ≥ 2), and it is understood that sub- scripts such as n (and also m, k, `, but not j; see below) run over the positive integers from 1 to N (unless otherwise specified); the reader is often (but not always) explicitly reminded of this fact. We also use occasionally the notation z to denote the (generally complex ) N -vector of compo- nents zn, z ≡ (z1, . . . , zN ); and likewise for other underlined letters (see below). A key role in our treatment is played by the time-dependent (monic) polynomial ψ(z, t) of degree N in the (scalar, generally complex ) variable z which features the N coordinates zn(t) as its N zeros, see (1.1). The solvable N -body problem treated in this paper is characterized by the following Newto- nian equations of motion: z̈n + Eżn = B0 +B1zn − (N − 1)[2A3 + 3(N − 2)F4]z 2 n + (N − 1)(N − 2)G3żnzn + N∑ `=1; ` 6=n { (zn − z`)−1 [ 2żnż` + 2 ( A0 +A1zn +A2z 2 n +A3z 3 n ) − (żn + ż`)(D0 +D1zn)−D2zn(żnz` + ż`zn) ]} Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 3 + N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ 3(F0 + F1zn + F2z 2 n + F3z 3 n + F4z 4 n) (zn − z`)(zn − zk) (2.1) − ( G0 +G1zn +G2z 2 n +G3z 3 n zn − z` )( żn + żk zn − zk + ż` + żk z` − zk )] , n = 1, . . . , N. Here and hereafter the 19 upper-case letters Aj , Bj , Dj , E, Fj , Gj denote time-independent parameters (“coupling constants”). They are a priori arbitrary (possibly complex ) numbers; special assignments are occasionally considered below. This notation is chosen consistently with equation (2.3.3-2) of [4] (with C = 1), to which this system of N equations of motion reduces when the 9 “new” parameters vanish, i.e. F0 = F1 = F2 = F3 = F4 = G0 = G1 = G2 = G3 = 0. (2.2) Note that the terms associated with the 4 new parameters F0, F1, F2, F3, represent velocity- independent three-body forces, the terms associated with the 3 new parameters G0, G1, G2 represent velocity-dependent three-body forces, and the terms associated with the new parame- ters F4 respectively G3 represent velocity-independent respectively velocity-dependent one-body and three-body forces. (Of course terms representing one-body forces can always be absorbed in those representing many-body forces: for instance the one-body term −2(N − 1)A3z 2 n in the right-hand side of (2.1) could be eliminated by replacing z3n with z2nz` in the two-body term multiplying A3 in the first sum in the right-hand side of (2.1)). This Newtonian N -body system is solvable by algebraic operations because – as shown in the following Section 3 – the coordinates zn(t) evolving according to this system of N Ordinary Differential Equations (ODEs) coincide with the N zeros of the time-dependent polynomial (1.1) of degree N in z, itself evolving according to the following linear PDE: ψtt + {E − (N − 1)[D2 + (N − 2)G3]z}ψt + ( G0 +G1z +G2z 2 +G3z 3 ) ψzzt + ( D0 +D1z +D2z 2 ) ψzt + ( F0 + F1z + F2z 2 + F3z 3 + F4z 4 ) ψzzz + ( A0 +A1z +A2z 2 +A3z 3 ) ψzz + { B0 +B1z − (N − 1)[2A3 + 3(N − 2)F4]z 2 } ψz +N { −B1 − (N − 1)[A2 + (N − 2)F3] + (N − 1)[A3 + 2(N − 2)F4]z } ψ = 0. (2.3) Here the 19 upper-case letters are of course the same time-independent parameters featured by the Newtonian equations of motion (2.1). Again, this notation is consistent with that used in [4]: indeed this PDE is a natural generalization of equation (2.3.3-1) of [4] (with C = 1) to which it clearly reduces when the new parameters vanish, see (2.2). This PDE implies that the N coefficients cm ≡ cm(t), see (1.1), evolve according to the following system of N linear ODEs: c̈m + (N + 2−m)(N + 1−m)G0ċm−2 + (N + 1−m)[D0 + (N −m)G1]ċm−1 + {E + (N −m)[D1 + (N − 1−m)G2]}ċm −m[D2 + (2N − 3−m)G3]ċm+1 + (N + 3−m)(N + 2−m)(N + 1−m)F0cm−3 + (N + 2−m)(N + 1−m)[A0 + (N −m)F1]cm−2 + (N + 1−m) { (N −m)[A1 + (N − 1−m)F2] +B0 } cm−1 −m { (2N − 1−m)A2 +B1 + [ 3N2 − 6N + 2− 3(N − 1)m+m2 ] F3 } cm +m(m+ 1)[A3 + (3N − 5−m)F4]cm+1 = 0, m = 1, . . . , N, (2.4) of course with c0 = 1 and cm = 0 for m < 0 and for m > N . 4 O. Bihun and F. Calogero Remark 2.1. This system of N ODEs satisfied by the N coefficients cm(t) justifies the as- sertion made above that the linear PDE (2.3) admits a polynomial solution: specifically, the polynomial (1.1) of degree N in z. Of course this system of N ODEs reduces to equation (2.3.3-8) of [4] (with C = 1) when the new parameters vanish, see (2.2). Likewise, it reduces to equa- tion (4.54) of [6] (up to the correction of a trivial misprint in that equation, and to an obvious notational change). Note however that the analogous equation has been wrongly reported (as equation (3)) in [13]: due to a trivial misprint (a multiplicative factor cm−2 missing in the first term in the second line) and the mistake of inserting two terms in the right-hand side (which should instead just be zero); fortunately this mistake has no consequence on the remaining part of that paper – except for the mistaken Remark 1.1 which should of course be ignored. This system, (2.4), ofN autonomous linear ODEs is of course solvable by algebraic operations. Indeed its general solution reads c(t) = N∑ m=1 [ b(+) m v(+)(m) exp ( λ(+) m t ) + b(−)m v(−)(m) exp ( λ(−)m t )] , (2.5) where the N -vector c(t) has the N components cm(t), the 2N (time-independent) coefficients b (±) m are a priori arbitrary – to be fixed a posteriori in order to satisfy the 2N initial condi- tions cm(0) and ċm(0) – while the 2N (time-independent) N -vectors v(±)(m) respectively the 2N (time-independent) numbers λ (±) m are the 2N eigenvectors, respectively the 2N eigenvalues, of the following (time-independent) generalized matrix-vector eigenvalue problem:( λ2I + λU + V ) v = 0. (2.6a) Here of course I is the N × N unit matrix (Imn = δm,n where, here and below, δm,n is the Kronecker symbol) and the two N ×N matrices U and V are defined componentwise as follows: Umn = (N + 1−m)(N + 2−m)G0δn,m−2 + (N + 1−m)[D0 + (N −m)G1]δn,m−1 + {E + (N −m)[D1 + (N − 1−m)G2]}δn,m −m[D2 + (2N − 3−m)G3]δn,m+1, n,m = 1, . . . , N ; (2.6b) Vmn = (N + 3−m)(N + 2−m)(N + 1−m)F0δn,m−3 + (N + 2−m)(N + 1−m)[A0 + (N −m)F1]δn,m−2 + (N + 1−m) { (N −m)[A1 + (N − 1−m)F2] +B0 } δn,m−1 −m { (2N − 1−m)A2 +B1 + [ 3N2 − 6N + 2− 3(N − 1)m+m2 ] F3 } δn,m +m(m+ 1)[A3 + (3N − 5−m)F4]δn,m+1 n,m = 1, . . . , N. (2.6c) This implies of course that the 2N eigenvalues λ (±) m are the 2N roots of the following poly- nomial equation (of degree 2N in λ): det ( λ2I + λU + V ) = 0. (2.6d) These findings show that the solution of the system (2.4) is achieved by the algebraic opera- tion of determining the eigenvalues and eigenvectors of the matrix-vector generalized eigenvalue problem (2.6a). The algebraic equation (2.6d) can be explicitly solved (for arbitrary N) in the two special cases in which the two N ×N matrices U and V are either both upper triangular or both lower triangular. The first of these two special cases obtains if, of the 19 parameters in (2.4), the following 4 vanish: A3 = D2 = F4 = G3 = 0. (2.7) Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 5 The second of these two special cases obtains if instead, of the 19 parameters in (2.4), the following 9 vanish: A0 = A1 = B0 = D0 = F0 = F1 = F2 = G0 = G1 = 0. (2.8) It is then easily seen that – in both these two cases – the 2N eigenvalues λ (±) m read λ(±)m = −1 2 { E + (N −m)[D1 + (N − 1−m)G2]±∆m } , (2.9a) ∆2 m = {E + (N −m)[D1 + (N − 1−m)G2]}2 + 4m { (2N − 1−m)A2 +B1 + [ 3N2 − 6N + 2− 3(N − 1)m+m2 ] F3 } . (2.9b) These findings imply that – at least in these cases – specific predictions on the actual behavior of the solutions of the Newtonian N -body problem (2.1) can be easily made. In Section 4 we identify in particular the cases in which this Newtonian N -body problem is isochronous or asymptotically isochronous, or an appropriate variant of it is isochronous; while the findings described in this Section 2 are proven in the following Section 3. 3 Derivation of the equations of motion of the Newtonian N -body problem Our task in this section is to obtain the equations of motion (2.1) characterizing the new N -body problem of goldfish type. The starting point is the linear third-order PDE (2.3) satisfied by the polynomial (1.1). We already saw in Section 2 the implication of this evolution PDE for the coefficients cm(t) of its polynomial solution (1.1), leading to the identification – via (1.1) and (2.5) – of its solution by algebraic operations. In this section we show that the fact that the polynomial (1.1) satisfies the PDE (2.3) implies that its zeros zn(t) indeed evolve according to the Newtonian equations of motion (2.1). This is in fact an immediate consequence – via trivial, if somewhat cumbersome, algebra – of (some of) the identities reported in Appendix A of [6] and of the additional identities (A.2) and (A.3) reported (and proven) in Appendix A of this paper, see below. Indeed these identities – valid for an arbitrary time-dependent polynomial ψ(z, t) of degree N in z, see (1.1)) – allow to transform the linear PDE (2.3) satisfied by the polynomial ψ(z, t) into the system of nonlinear ODEs (2.1) satisfied by its zeros zn(t). Note that the fact that this outcome obtains is both a consequence and a confirmation of the fact that the PDE (2.3) admits as its solution the polynomial ψ(z, t), see (1.1), of degree N in z, hence featuring N zeros zn(t). 4 Isochronous and asymptotically isochronous cases In this section we identify cases in which the Newtonian N -body problem (2.1) is isochronous or asymptotically isochronous, or variants of it are isochronous. First of all we note, see (2.5), that if the 2N eigenvalues λ (±) m are all imaginary and read λ(±)m = ir(±)m ω, r(±)m ≡ p (±) m q (±) m , m = 1, . . . , N, (4.1) – with ω a positive real number, the N numbers q (±) m all positive integers, the N numbers p (±) m all integers (positive, negative or vanishing, with p (±) m and q (±) m coprimes and the 2N real rational 6 O. Bihun and F. Calogero numbers r (±) m ≡ p(±)m /q (±) m all different among themselves) – then the N coefficients cm(t) evolve isochronously, i.e. cm(t+ T ) = cm(t), m = 1, . . . , N, (4.2a) with the period T = 2πq ω (4.2b) independent of the initial data. Here of course q is the minimum common multiple of the 2N denominators of the 2N rational numbers r (±) m ≡ p (±) m /q (±) m . And it is plain that the same property of isochrony is then shared by the N coordinates zn(t), namely the Newtonian N -body problem (2.1) is then isochronous as well; with the possibility that in some open regions of its phase space the periodicity only holds for a period which is a (generally small) integer multiple of T due to the fact that some of the zeros of the polynomial ψ(z, t) – itself evolving isochronously with period T , see (1.1) – might “exchange their roles” over the time evolution (for an analysis of this phenomenology, also explaining the meaning of the assertion made above that the integer multiple in question is generally small, see [18]). Let us then focus on the two cases – as identified at the end of Section 2, see (2.7) and (2.8) – in which the 2N eigenvalues λ (±) m can be explicitly obtained, see (2.9). It is then easy to identify the additional restrictions on the parameters which are necessary – and also sufficient, up to some minor additional restrictions to exclude the coincidence of eigenvalues – to guarantee that the Newtonian N -body problem (2.1) be isochronous with period T . They read: E = −2ir1ω, D1 = −2ir2ω, G2 = −2ir3ω, (4.3a) A2 = a2ω 2, B1 = b1ω 2, F3 = f3ω 2, (4.3b) with f3 = 2r3(s2r4 −R2), (4.3c) a2 = 2(s1s2 − 1)R1r3 −R2 2 + r24 − 3(N − 1)f3, (4.3d) b1 = 2R1(s1r4 −R2)− (2N − 1)a2 − ( 3N2 − 6N + 2 ) f3, (4.3e) R1 = r1 +N [r2 + (N − 1)r3], R2 = r2 + (2N − 1)r3. (4.3f) Here r1, r2, r3, r4 are 4 arbitrary rational numbers and s1 and s2 are two arbitrary signs (+ or −). Indeed with these assignments λ (±) m clearly satisfies the condition (4.1) with r(±)m = (1± s1)R1 − (R2 ± r4)m+ (1± 1)r3m 2. (4.4) It is moreover plain that, if the 2N eigenvalues λ (±) m satisfy, instead of the condition (4.1), the less restrictive condition λ(±)m = ( ir(±)m − ρ(±)m ) ω (4.5) – with ω again a positive real number, the 2N numbers r (±) m again all real and rational and the 2N numbers ρ (±) m all real and nonnegative, ρ (±) m ≥ 0, with at least one of them vanishing – then the N coefficients cm(t) are asymptotically isochronous, lim t→∞ [ cm(t)− c(asy)m (t) ] = 0, (4.6a) Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 7 with c (asy) m (t) periodic, c(asy)m (t+ T ) = c(asy)m (t). (4.6b) Here T is given again by (4.2b), but now with q being the minimum common multiple of the denominators q (±) m ’s of the rationals r (±) m ’s associated with vanishing ρ (±) m ’s, ρ (±) m = 0, see (4.5) and (4.1). And it is again plain that the same property is then shared by the N coordinates zn(t), namely that the Newtonian N -body problem (2.1) is then asymptotically isochronous as well (again, with a period which might be a, generally small, integer multiple of T [18]). Next, less us investigate the cases in which – via a well-known trick, see for instance Section 2.1 (entitled “The trick”) of [6] – isochronous variants can be manufactured of the Newtonian N - body problem (2.1). To this end, it is convenient to re-write the equations of motion (2.1) via the formal replacement of dependent and independent variables zn(t) ⇒ ζn(τ), so that the equations of motion read as follows: ζ ′′n + Eζ ′n = B0 +B1ζn − (N − 1)[2A3 + 3(N − 2)F4]ζ 2 n + (N − 1)(N − 2)G3ζ ′ nζn + N∑ `=1; 6̀=n { (ζn − ζ`)−1 [ 2ζ ′nζ ′ ` + 2 ( A0 +A1ζn +A2ζ 2 n +A3ζ 3 n ) − ( ζ ′n + ζ ′` ) (D0 +D1ζn)−D2ζn(ζ ′nζ` + ζ ′`ζn) ]} + N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ 3(F0 + F1ζn + F2ζ 2 n + F3ζ 3 n + F4ζ 4 n) (ζn − ζ`)(ζn − ζk) (4.7) − ( G0 +G1ζn +G2ζ 2 n +G3ζ 3 n ζn − ζ` )( ζ ′n + ζ ′k ζn − ζk + ζ ′` + ζ ′k ζ` − ζk )] , n = 1, . . . , N. Here and hereafter (in this section) appended primes denote differentiation with respect to the variable τ . We now perform the following change of dependent and independent variables (“the trick”): zn(t) = exp(iαωt)ζn(τ), τ ≡ τ(t) = exp(iωt)− 1 iω . (4.8a) Here and below ω is a positive constant, and α a nonvanishing number that we reserve to assign, see below. It is then plain that there hold the following formulas: żn − iαωzn = exp[i(α+ 1)ωt]ζ ′n, (4.8b) z̈n − (2α+ 1)iωżn − α(α+ 1)ω2zn = exp[i(α+ 2)ωt]ζ ′′n. (4.8c) And via these formulas one can easily obtain the equations of motion implied for the dependent variables zn(t) by the equations of motion (4.7) satisfied by the variables ζn(τ), and thereby ascertain for which assignments of the parameter α, and for which corresponding restrictions on the 19 coupling constants featured by these equations of motion, the resulting equations of motion satisfied by the N coordinates zn(t) are autonomous, i.e. they feature no explicit time- dependence. We list below all these cases, on the understanding that all the coupling constants which do not appear in these new equations of motion have been set to zero (while those that do appear are arbitrary). For α = −2, these Newtonian equations of motion read z̈n + 3iωżn − 2ω2zn = B0 + 2 N∑ `=1; `6=n (żn + 2iωzn)(ż` + 2iωz`) zn − z` 8 O. Bihun and F. Calogero + N∑ `,k=1; 6̀=n, k 6=n, ` 6=k 3F2z 2 n (zn − z`)(zn − zk) . (4.9) For α = −1, these Newtonian equations of motion read z̈n + iωżn = N∑ `=1; 6̀=n 2(żn + iωzn)(ż` + iωz`) + 2A0 −D0[żn + ż` + iω(zn + z`)] zn − z` + N∑ `,k=1; 6̀=n, k 6=n, ` 6=k [ 3F1zn (zn − z`)(zn − zk) − G1zn zn − z` ( żn + żk + iω(zn + zk) zn − zk + ż` + żk + iω(z` + zk) z` − zk )] . (4.10) For α = −2/3, these Newtonian equations of motion read z̈n + iω 3 żn + 2 9 ω2zn = 2 N∑ `=1; 6̀=n (żn + 2iωzn/3)(ż` + 2iωz`/3) zn − z` + N∑ `,k=1; ` 6=n, k 6=n, ` 6=k 3F0 (zn − z`)(zn − zk) . (4.11) For α = −1/2, these Newtonian equations of motion read z̈n + ω2 4 zn = 2 N∑ `=1; ` 6=n (żn + iωzn/2)(ż` + iωz`/2) zn − z` − N∑ `,k=1; 6̀=n, k 6=n, ` 6=k [ G0 zn − z` ( żn + żk + iω(zn + zk)/2 zn − zk + ż` + żk + iω(z` + zk)/2 z` − zk )] . (4.12) For α = 1, these Newtonian equations of motion read z̈n − 3iωżn − 2ω2zn = (N − 1)(N − 2)G3(żn − iωzn)zn + N∑ `=1; 6̀=n [ (zn − z`)−1 { 2(żn − iωzn)(ż` − iωz`) −D2zn[(żn − iωzn)z` + (ż` − iωz`)zn] }] − N∑ `,k=1; 6̀=n, k 6=n, ` 6=k [ G3z 3 n zn − z` ( żn + żk − iω(zn + zk) zn − zk + ż` + żk − iω(z` + zk) z` − zk )] . (4.13) For α = 2, these Newtonian equations of motion read z̈n − 5iωżn − 6ω2zn = −(N − 1)[2A3 + 3(N − 2)F4]z 2 n (4.14) + N∑ `=1; 6̀=n 2(żn − 2iωzn)(ż` − 2iωz`) + 2A3z 3 n zn − z` + N∑ `,k=1; `6=n, k 6=n, ` 6=k 3F4z 4 n (zn − z`)(zn − zk) . And it is plain that all these Newtonian models are isochronous: see (4.8a), and note that the solutions ζn(τ) of the solvable N -body model (4.7) have at most algebraic singularities as functions of τ . Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 9 5 Outlook The search for more general solvable models of goldfish type should continue: they might still yield interesting results. Another development likely to yield interesting findings is the investigation of the behavior, in the infinitesimal neighborhood of its equilibria, of the N -body model introduced above, especially in the isochronous cases. This investigation might yield new Diophantine findings for the zeros of interesting polynomials. We plan to pursue these results, which shall eventually be submitted to a journal devoted to special functions if they turn out to be sufficiently interesting to justify their publication. A Appendix: identities involving the zeros of a polynomial In this Appendix we report (and then prove) several identities for the time-dependent polynomial ψ ≡ ψ(z, t), see (1.1), of degree N in z. We of course use hereafter the notation introduced in Section 2, see Notation 2.1, and in addition the following convenient shorthand notation: as in [6] (see there equations (A.4) and (A.5)) Dψ ⇐⇒ Fn(z, ż) (A.1a) – with D a differential operator acting on the independent variables z and t of the polynomial ψ(z, t), see (1.1) – stands for the identity Dψ(z, t) = ψ(z, t) N∑ n=1 [z − zn(t)]−1Fn(z, ż). (A.1b) Below we often, for notational simplicity, omit to indicate explicitly the dependence on their arguments of ψ ≡ ψ(z, t), zn ≡ zn(t) and cm(t) (see (1.1)). We now list the following identities, which complement those reported in Appendix A of [6] (see in particular the 2012 paperback version, where the formulas denoted in [6] as (A.8k) and (A.8l) are corrected): zpψzzz ⇐⇒ 3zpn N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ] , p = 0, 1, 2, (A.2a) z3ψzzz −N(N − 1)(N − 2)ψ ⇐⇒ 3z3n N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ] , (A.2b) z4ψzzz + (N − 1)(N − 2)(3c1 −Nz)ψ ⇐⇒ 3z4n N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ] ; (A.2c) zpψzzt ⇐⇒ − N∑ k,`=1; k 6=n, ` 6=n, k 6=` [( zpn zn − zk )( żn + ż` zn − z` + żk + ż` zk − z` )] , p = 0, 1, 2, (A.3a) z3ψzzt − (N − 1)(N − 2)zψt ⇐⇒ (N − 1)(N − 2)żnzn − N∑ k,`=1; k 6=n, ` 6=n, k 6=` [( z3n zn − zk )( żn + ż` zn − z` + żk + ż` zk − z` )] . (A.3b) 10 O. Bihun and F. Calogero Of course in the last two, (A.3a) and (A.3b), superimposed dots indicate t-differentiations. Also note that in the third, (A.2c), of these identities, consistently with (1.1) c1 ≡ c1(t) = − N∑ n=1 [zn(t)]. (A.4) But remarkably – due to a neat cancellation, see (A.2c) and equation (A.6b) of [6] – this quantity does not appear in the equations of motion (2.1). Our task in this Appendix is to prove these identities. To perform these proofs it is convenient to introduce the following shorthand notation denoting sums over (dummy) indices restricted to take different values (among themselves): ∑′ nk ≡ N∑ n,k=1;n6=k ; ∑′ nk` ≡ N∑ n,k,`=1;n6=k, k 6=`, ` 6=n . (A.5) We moreover introduce the following shorthand notation for the “denominator” den(zn,zk,z`), den(zn, zk, z`) ≡ (zn − zk)(zn − z`)(zk − z`), (A.6a) which clearly has the property to be invariant under the cyclic exchange n→ k → ` of the three indices n, k, `, den(zn, zk, z`) = den(zk, z`, zn), (A.6b) and to be instead antisymmetric under the exchange of any two of its three arguments zn, zk, z`, den(zn, zk, z`) = −den(zk, zn, z`) = −den(z`, zk, zn) = −den(zn, z`, zk). (A.6c) Likewise, we denote corresponding “numerators” as num(n, k, `) (i.e., num(zn, zk, z`) ≡ num(n, k, `)) and, whenever one of them is the sum of an arbitrary number of terms numj(n, k, `), i.e. num(n, k, `) ≡ ∑ j=0 numj(n, k, `), (A.7) we take advantage (if need be) of the following Lemma A.1. If each of the addends numj(n, k, `) in the right-hand side of (A.7) is invariant under the exchange of any two of the three indices n, k, `, namely if each of the addends numj(n, k, `) satisfies at least one of the following three relations: numj(n, k, `) = numj(k, n, `), (A.8a) or numj(n, k, `) = numj(`, k, n), (A.8b) or numj(n, k, `) = numj(n, `, k), (A.8c) then the triple sum ∑′ nk`[num(n, k, `)/ den(zn, zk, z`)] vanishes:∑′ nk` [ num(n, k, `) den(zn, zk, z`) ] ≡ ∑′ nk` {∑ j=0 [ numj(n, k, `) den(zn, zk, z`) ]} = ∑ j=0 {∑′ nk` [ numj(n, k, `) den(zn, zk, z`) ]} = 0. (A.9) Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 11 The validity of this assertion is an obvious consequence of the antisymmetry – see (A.6c) and (A.8) – under the exchange of two appropriately chosen dummy indices in the triple sums∑′ nk`[numj(n, k, `)/ den(zn, zk, z`)], which therefore vanish (for all values of j). Next, let us report and prove the following identities, valid for any set of N arbitrary num- bers zn (for convenience, we always assume them to be all different among themselves).∑′ nk ( zn zn − zk ) = N(N − 1) 2 , (A.10a)∑′ nk` ( zk zk − z` ) = N(N − 1)(N − 2) 2 , (A.10b)∑′ nk` [ zpn (zn − zk)(zn − z`) ] = 0, p = 0, 1, (A.10c) ∑′ nk` [ z2n (zn − zk)(zn − z`) ] = N(N − 1)(N − 2) 3 , (A.10d)∑′ nk` ( znzk zk − z` ) = −1 2 (N − 1)(N − 2)c1, (A.10e) ∑′ nk` ( z2n zn − z` ) = −(N − 1)(N − 2)c1, (A.10f) ∑′ nk` [ z3n (zn − zk)(zn − z`) ] = −(N − 1)(N − 2)c1. (A.10g) In the last three formulas of course c1 is defined by (A.4). The proof of the (well-known) identity (A.10a) is trivial:∑′ nk ( zn zn − zk ) = 1 2 ∑′ nk ( zn − zk zn − zk ) = N(N − 1) 2 . (A.11) The first step is justified by adding to the left-hand side of (A.10a) what is obtained by the exchange of the dummy indices k and ` (which does not change the result) and dividing by 2 (this operation is often repeated below without describing it in as much detail as done here); the second step is immediately implied by the first definition (A.5). Then the proof of (A.10b) is no less trivial: it follows from (A.10a) and the second defini- tion (A.5). The proof of (A.10c) goes as follows. For p = 0 by adding to the right-hand side of this formula the sums obtained by performing sequentially two cyclic transformations of the three indices n, k, ` and by taking advantage of the invariance property (A.6a)) – and by dividing the sum of the three equal sums thereby obtained by 3 – we clearly get∑′ nk` [ 1 (zn − zk)(zn − z`) ] = 1 3 ∑′ nk` [ (zk − z`) + (z` − zn) + (zn − zk) den(zn, zk, z`) ] , (A.12) and it is then plain that this quantity vanishes. For p = 1 via (A.6a)∑′ nk` [ zn (zn − zk)(zn − z`) ] = ∑′ nk` [ znzk − znz` den(zn, zk, z`) ] (A.13) and it is plain that this quantity vanishes thanks to Lemma A.1, since in the right-hand side the first term in the numerator is invariant under the exchange of dummy indices n↔ k and in the second under the exchange n↔ `. The proof of (A.10d) goes through the following steps:∑′ nk` [ z2n (zn − zk)(zn − z`) ] = ∑′ nk` [ z2n (zn − zk)(zn − z`) − zk zk − z` ] + ∑′ nk` zk zk − z` 12 O. Bihun and F. Calogero = − ∑′ nk` [ z2kz` − (z2nzk + z2kzn − znzkz`) den(zn, zk, z`) ] + N(N − 1)(N − 2) 2 = − ∑′ nk` [ z2kz` den(zn, zk, z`) ] + N(N − 1)(N − 2) 2 = ∑′ nk` [ z2nz` den(zn, zk, z`) ] + N(N − 1)(N − 2) 2 = −1 2 ∑′ nk` [ z2n (zn − zk)(zn − z`) ] + N(N − 1)(N − 2) 2 , (A.14) the first of which is quite trivial, the second is given by standard algebra together with the definition (A.6a) and the formula (A.10b) just proven, the third of which is implied by the formula (A.9) (since z2nzk + z2kzn − znzkz` is symmetrical under the exchange of dummy indices n ↔ k), the fourth of which obtains via the exchange n ↔ k (see (A.6c)), and the last of which – which clearly implies (A.10d) – is clearly implied by the property (A.6c) and by the definition (A.6a) of den(zn, zk, z`). The proof of (A.10e) is again quite trivial:∑′ nk` ( znzk zk − z` ) = 1 2 ∑′ nk` (zn) = −(N − 1)(N − 2) 2 c1, (A.15) the first step being implied by the antisymmetry of the summand under the exchange of the dummy indices k ↔ ` and the second by the definitions of the symbol ∑′ nk` (see (A.5)) and of c1 (see (A.4)). The proof of (A.10f) is analogous:∑′ nk` ( z2n zn − z` ) = 1 2 ∑′ nk` (zn + z`) = ∑′ nk` (zn) = −(N − 1)(N − 2)c1, (A.16) with the first step justified by the replacement z2n → (z2n − z2` )/2 due to the antisymmetry of the summand under the exchange of dummy indices n ↔ `, the second step justified by the symmetry of the summand, and the third step justified as just above. Finally, the proof of (A.10g) is analogous:∑′ nk` [ z3n (zn − zk)(zn − z`) ] = ∑′ nk` [ z3n (zn − zk)(zn − z`) − z2n zk − z` ] = − ∑′ nk` [ z2n(zn − zk) (zn − z`)(zk − z`) ] = −1 2 ∑′ nk` [ (z2n − z2k)(zn − zk) (zn − z`)(zk − z`) ] = 1 2 ∑′ nk` ( z2n − z2k zn − z` − z2n − z2k zk − z` ) = 1 2 ∑′ nk` ( z2n zn − z` + z2k zk − z` ) = ∑′ nk` ( z2n zn − z` ) = −(N − 1)(N − 2)c1. (A.17) Here the first step is justified by the vanishing of (the sum over) the added term (due to the antisymmetry of the summand under the exchange of dummy indices k ↔ `), the second step follows by trivial algebra, the third step is justified by the symmetry of the summand under the exchange of dummy indices n↔ k, the fourth step is justified by the identity − zn − zk (zn − z`)(zk − z`) = 1 zn − z` − 1 zk − z` , (A.18) the fifth step by the elimination of two addends antisymmetric under the exchanges of dummy indices n ↔ ` respectively k ↔ `, the sixth step by the symmetry of the summand under the exchange of dummy indices n↔ `, and the last step by (A.10f). Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 13 Now we can finally proceed and prove the formulas (A.2) and (A.3). We start from reporting equations (A.1) (which coincides with (1.1)), (A.2), (A.3), (A.8a) and (A.9a) of [6]: ψ = N∏ n=1 [z − zn] = zN + N∑ m=1 cmz N−m = N∑ m=0 cmz N−m, c0 = 1, (A.19) ψz = ψ N∑ n=1 (z − zn)−1, (A.20) ψt = −ψ N∑ n=1 (z − zn)−1żn, (A.21) ψzz = 2ψ N∑ n=1 [ (z − zn)−1 N∑ `=1; 6̀=n (zn − z`)−1 ] , (A.22) ψzt = −ψ N∑ n=1 { (z − zn)−1 N∑ `=1; 6̀=n [ (żn + ż`)(zn − z`)−1 ]} . (A.23) The last two equations correspond of course to (A8.a) and (A.9a) via the convention defining the symbol ⇐⇒, see (A.1). Partial differentiation of (A.22) with respect to z gives ψzzz = 2ψz N∑ n=1 (z − zn)−1 N∑ `=1; ` 6=n (zn − z`)−1 − 2ψ N∑ n=1 (z − zn)−2 N∑ `=1; `6=n (zn − z`)−1. (A.24a) Via (A.20) this becomes (after a convenient cancellation and change of dummy index from n to k) ψzzz = 2ψ N∑ n=1 [ (z − zn)−1 N∑ k=1; k 6=n (z − zk)−1 N∑ `=1; `6=n (zn − z`)(zn − z`) ] . (A.24b) We then use the identity (z − zn)−1(z − zk)−1 = (zn − zk)−1 [ (z − zn)−1 − (z − zk)−1 ] , (A.25) getting thereby ψzzz = 2ψ N∑ n,k=1; k 6=n {[ (z − zn)−1 − (z − zk)−1 ] (zn − zk)−1 N∑ `=1; `6=n (zn − z`)−1 } . (A.26a) We then exchange the dummy indices n and k in the second of the two sums over these indices, getting thereby ψzzz = 2ψ N∑ n,k=1; k 6=n { 1 (z − zn)(zn − zk) [ N∑ `=1; ` 6=n ( 1 zn − z` ) + N∑ `=1; ` 6=k ( 1 zk − z` )]} , (A.26b) which can also be written as follows: ψzzz = 2ψ N∑ n=1 {( 1 z − zn ) N∑ k,`=1; k 6=n, ` 6=n, k 6=` [( 1 zn − zk )( 1 zn − z` + 1 zk − z` )]} . (A.26c) 14 O. Bihun and F. Calogero This implies (using the definition of the symbols ⇔ and ∑′ nk`, see above) ψzzz ⇐⇒ 2 N∑ k,`=1; k 6=n, ` 6=n, k 6=` { (zn − zk)−1 [ (zn − z`)−1 + (zk − z`)−1 ]} , (A.27a) ψzzz ⇐⇒ 2 ∑′ nk` [ (zn + zk − 2z`)(zk − z`)−1(zn − zk)−1(zn − z`)−1 ] ; (A.27b) and finally, taking advantage of the fact that the part of the summand antisymmetric under the exchange of the two dummy indices k and ` can be eliminated – so that (zn + zk− 2z`)/(zk− z`) ≡ 3/2+(2zn−zk−z`)/[2(zk−z`)] can be replaced by 3/2 – one gets (A.2a) with p = 0, which is thereby proven. This is the first of the new formulas analogous to those reported in Appendix A of [6]. To proceed it is convenient to write in longhand the formula we just proved: ψzzz = ψ N∑ n=1 { 3(z − zn)−1 N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ]} . (A.28) Multiplication by zp then yields zpψzzz = ψ N∑ n=1 { 3zp(z − zn)−1 N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ]} . (A.29) By replacing the zp in the numerator with zpn + (zp − zpn) we get zpψzzz = ψ N∑ n=1 { 3zpn z − zn N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ]} + 3ψ ∑′ nk` [ zp − zpn z − zn (zn − zk)−1(zn − z`)−1 ] (A.30) (see (A.5)). It is then plain that, for p = 1 and p = 2, the formula (A.10c) implies that the last sum in this equation vanishes. Hence (A.2a) is now proven also for p = 1 and p = 2. To prove (A.2b) we set p = 3 in (A.30), which then reads z3ψzzz = ψ N∑ n=1 { 3z3n z − zn N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ]} + 3ψ ∑′ nk` [ (z2 + zzn + z2n)(zn − zk)−1(zn − z`)−1 ] (A.31) (via the identity z3 − z3n = (z − zn)(z2 + zzn + z2n)). And it is then plain that this formula, via (A.10c) and (A.10d), yields (A.2b), which is thereby proven. Likewise, to prove (A.2c) we set p = 4 in (A.30), which then reads z4ψzzz = ψ N∑ n=1 { 3(z − zn)−1z4n N∑ k,`=1; k 6=n, ` 6=n, k 6=` [ (zn − zk)−1(zn − z`)−1 ]} + 3ψ ∑′ nk` [( z3 + z2zn + zz2n + z3n ) (zn − zk)−1(zn − z`)−1 ] . (A.32) Then it is easily seen that, via (A.10c), (A.10d) and (A.10g), this equation yields (A.2c), which is thereby proven. Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 15 Next, to prove (A.3a) (to begin with, with p = 0), we z-differentiate (A.23) getting thereby ψzzt = −ψz N∑ n=1 [ (z − zn)−1 N∑ `=1; `6=n ( żn + ż` zn − z` )] + ψ N∑ n=1 [ (z − zn)−2 N∑ k=1; k 6=n ( żn + żk zn − zk )] = −ψ N∑ k=1 { (z − zk)−1 N∑ n=1;n6=k [ (z − zn)−1 N∑ `=1; ` 6=n ( żn + ż` zn − z` )]} = −ψ ∑′ nk` {[ (z − zk)−1 − (z − zn)−1 ] (zk − zn)−1 ( żn + ż` zn − z` )} = −ψ N∑ n=1 1 z − zn { k∑ k,`=1; k 6=n, ` 6=n, ` 6=k [ 1 zn − zk ( żn + ż` zn − z` + żk + ż` zk − z` )]} . (A.33) The first equality corresponds clearly to the differentiation of (A.23); the second obtains via (A.20) (also taking account of the cancellation occurring for n = k); the third equality obtains via the identity (z − zk)−1(z − zn)−1 = [ (z − zk)−1 − (z − zn)−1 ] (zk − zn)−1; (A.34) and the fourth equality obtains via the exchange of dummy indices k ↔ n in the sum containing the term (z−zk)−1 (and note the cancellations of the term with ` = n in the resulting sum, with the term with ` = k in the second sum, justifying the exclusion of the 3 addends with k = n, ` = n and ` = k in the sum in the last formula). The final result corresponds – of course, via the notation (A.1) – to (A.3a) with p = 0, which is thereby proven. To prove (A.3a) with p = 1 and p = 2 we start from the longhand version of the equation we just proved, multiplied by zp ≡ zpn + (zp − zpn): zpψzzt = −ψ ∑′ nk` { zpn z − zn [ (zn − zk)−1 ( żn + ż` zn − z` + żk + ż` zk − z` )]} − ψ ∑′ nk` {( zp − zpn z − zn )[ (zn − zk)−1 ( żn + ż` zn − z` + żk + ż` zk − z` )]} . (A.35) From this formula it is plain that (A.3a) is proven also for p = 1 and p = 2 if, for these two values of p, there holds the identity σ(p; z, ż) ≡ ∑′ nk` [( zp − zpn z − zn )( 1 zn − zk )( żn + ż` zn − z` + żk + ż` zk − z` )] = 0. (A.36) Note that we have now employed the convenient notation (A.5). Indeed, for p = 1, σ(1; z, ż) = ∑′ nk` [( 1 zn − zk )( żn + ż` zn − z` + żk + ż` zk − z` )] = ∑′ nk` [ żnzk + żkzn + ż`(zn + zk)− (żn + żk + 2ż`)z` den(zn, zk, z`) ] . (A.37) The second equality obtains via trivial algebra and the definition (A.6a) of den(zn, zk, z`); and it clearly implies that σ(1; z, ż) vanishes via Lemma A.1, since the numerator of the summand in the sum in the right-hand side of the second equality is clearly invariant under the exchange of dummy indices n↔ k. Likewise, for p = 2, σ(2; z, ż) = ∑′ nk` [( z + zn zn − zk )( żn + ż` zn − z` + żk + ż` zk − z` )] = zσ(1; z, ż) 16 O. Bihun and F. Calogero + ∑′ nk` {[ zn[żnzk + żkzn + ż`(zn + zk)− (żn + żk + 2ż`)z`] den(zn, zk, z`) ]} = ∑′ nk` {[ zn[żnzk + żkzn + ż`(zn + zk)− (żn + żk + 2ż`)z`] den(zn, zk, z`) ]} = ∑′ nk` {[ den(zn, zk, z`)] −1× × [ (żk + ż`)z 2 n − (żn + ż`)znz` + żnznzk + ż`zkzn − (żk + ż`)znz` ]} = ∑′ nk` {[ żnznzk + ż`zkzn − żkznz` − ż`znz` den(zn, zk, z`) ]} = ∑′ nk` {[ żnzn(zk + z`) + 2ż`zkzn den(zn, zk, z`) ]} = 0. (A.38) The first equality is implied by the definition (A.36) with p = 2 and the identity z2 − z2n = (z + zn)(z − zn); the second equality is a consequence of the vanishing of σ(1; z, ż) proven above; the third equality is obtained via trivial algebra; the fourth equality obtains – thanks to Lemma A.1 – because the first two addends in the numerator in the right-hand side are invariant under the exchanges of dummy indices k ↔ ` respectively n ↔ `; the fifth equality obtains by performing on the fourth term in the numerator the exchange of dummy indices n↔ ` (entailing a change of sign, since the denominator changes sign under this exchange of indices, see (A.6c)) and on the third term the exchange of dummy indices k ↔ ` (entailing likewise a change of sign, see (A.6c)). Finally the last equality is implied by Lemma A.1, since the first term in the numerator is now invariant under the exchange of dummy indices k ↔ ` and the second under the exchange n↔ k. Finally, let us prove (A.3b). Now we start from the identity z2ψzzt − (N − 1)(N − 2)ψt = (N − 1)(N − 2)ψ N∑ n=1 ( żn z − zn ) − ψ ∑′ nk` { z2n z − zn [ (zn − zk)−1 ( żn + ż` zn − z` + żk + ż` zk − z` )]} = −ψ ∑′ nk` { 1 z − zn [ z2n zn − zk ( żn + ż` zn − z` + żk + ż` zk − z` ) − żn ]} . (A.39) This identity is justified by (A.21) and by (the longhand version of) the identity we just proved, (A.3a). To write the second equality we also used the trivial observation that the very definition of the symbol ∑′ nk`, see (A.5), implies the identity ∑′ nk` f(n) = (N −1)(N −2) N∑ n=1 f(n) for any function f(n) (and we will feel free to use this identity again below). We now multiply this identity by z ≡ (z − zn) + zn, and we thereby obtain z2ψzzt − (N − 1)(N − 2)ψt = −σ̃(3; z, ż)ψ + ψ N∑ n=1 [( 1 z − zn ){ (N − 1)(N − 2)żnzn − ∑ k,`=1; k 6=n, ` 6=n, ` 6=k [ z3n z − zn ( żn + ż` zn − z` + żk + ż` zk − z` )]}] , (A.40a) with σ̃(3; z, ż) ≡ ∑′ nk` [ z2n zn − zk ( żn + ż` zn − z` + żk + ż` zk − z` ) − żn ] . (A.40b) Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces 17 It is now plain that (A.3b) is proven if we show that σ̃(3; z, ż) vanishes, σ̃(3; z, ż) = 0. To prove this we make the following steps: σ̃(3; z, ż) = ∑′ nk` ( [den(zn, zk, z`)] −1{z2n[(żn + ż`)(zk − z`) + (żk + ż`)(zn − z`)] − żn(zn − zk)(zn − z`)(zk − z`) }) = ∑′ nk` ( [den(zn, zk, z`)] −1{żn(zk − z`) [ z2n − (zn − zk)(zn − z`) ] + żkz 2 n(zn − z`) + ż`z 2 n(zn + zk − 2z`) }) = ∑′ nk` ( żn den(zn, zk, z`) × { (zk − z`) [ z2n − (zn − zk)(zn − z`)− z2k ] − z2` (z` + zk − 2zn) }) = ∑′ nk` { żn −(z3k + z3` ) + zn(z2k + z2` ) den(zn, zk, z`) } = 0. (A.41) Here the first equality is justified by the definition (A.6a) and a bit of trivial algebra, the second equality obtains by trivial algebra, the third equality obtains by the exchange of the dummy indices n↔ k in the term multiplying żk and likewise the exchange n↔ ` in the term multiplying ż` (under these exchanges the denominator changes sign, see (A.6c)), the fourth equality obtains by trivial algebra, and the final equality to zero is yielded by Lemma A.1 since the numerator is invariant under the exchange k ↔ `. Acknowledgements One of us (OB) would like to acknowledge with thanks the hospitality of the Physics Department of the University of Rome “La Sapienza” on the occasion of two two-week visits in June 2012 and May 2013, and the financial support for these trips provided by the NSF-AWM Travel Grant. References [1] Bihun O., Calogero F., Solvable and/or integrable many-body models on a circle, J. Geom. Symmetry Phys. 30 (2013), 1–18. [2] Bihun O., Calogero F., Yi G., Diophantine properties associated to the equilibrium configurations of an isochronous N -body problem, J. Nonlinear Math. Phys. 20 (2013), 158–178. [3] Calogero F., Motion of poles and zeros of special solutions of nonlinear and linear partial differential equa- tions and related “solvable” many-body problems, Nuovo Cimento B 43 (1978), 177–241. [4] Calogero F., Classical many-body problems amenable to exact treatments, Lecture Notes in Physics. New Series m: Monographs, Vol. 66, Springer-Verlag, Berlin, 2001. [5] Calogero F., The neatest many-body problem amenable to exact treatments (a “goldfish”?), Phys. D 152– 153 (2001), 78–84. [6] Calogero F., Isochronous systems, Oxford University Press, Oxford, 2008. [7] Calogero F., An integrable many-body problem, J. Math. Phys. 52 (2011), 102702, 5 pages. [8] Calogero F., Another new goldfish model, Theoret. and Math. Phys. 171 (2012), 629–640. [9] Calogero F., New solvable many-body model of goldfish type, J. Nonlinear Math. Phys. 19 (2012), 1250006, 19 pages. [10] Calogero F., Two quite similar matrix ODEs and the many-body problems related to them, Int. J. Geom. Methods Mod. Phys. 9 (2012), 1260002, 6 pages. [11] Calogero F., A linear second-order ODE with only polynomial solutions, J. 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[14] Calogero F., Yi G., A new class of solvable many-body problems, SIGMA 8 (2012), 066, 29 pages, arXiv:1210.0651. [15] Calogero F., Yi G., Can the general solution of the second-order ODE characterizing Jacobi polynomials be polynomial?, J. Phys. A: Math. Theor. 45 (2012), 095206, 4 pages. [16] Calogero F., Yi G., Diophantine properties of the zeros of certain Laguerre and para-Jacobi polynomials, J. Phys. A: Math. Theor. 45 (2012), 095207, 9 pages. [17] Calogero F., Yi G., Polynomials satisfying functional and differential equations and Diophantine properties of their zeros, Lett. Math. Phys. 103 (2013), 629–651. [18] Gomez-Ullate D., Sommacal M., Periods of the goldfish many-body problem, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 351–362. http://dx.doi.org/10.1080/14029251.2013.805565 http://dx.doi.org/10.1063/1.4773571 http://dx.doi.org/10.3842/SIGMA.2012.066 http://arxiv.org/abs/1210.0651 http://dx.doi.org/10.1088/1751-8113/45/9/095206 http://dx.doi.org/10.1088/1751-8113/45/9/095207 http://dx.doi.org/10.1007/s11005-013-0612-y http://dx.doi.org/10.2991/jnmp.2005.12.s1.28 1 Introduction 2 Main results 3 Derivation of the equations of motion of the Newtonian N-body problem 4 Isochronous and asymptotically isochronous cases 5 Outlook A Appendix: identities involving the zeros of a polynomial References

Solvable Many-Body Models of Goldfish Type with One-, Two- and Three-Body Forces

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