Symmetry, Integrability and Geometry: Methods and Applications Another New Solvable Many-Body Model of Goldfish Type ⋆

A new solvable many-body problem is identified. It is characterized by nonlinear Newtonian equations of motion ("acceleration equal force") featuring one-body and two-body velocity-dependent forces"of goldfish type"which determine the motion of an arbitrary number $N$ of unit-mass point-particles in a plane. The $N$ (generally complex) values $z_{n}(t)$ at time $t$ of the $N$ coordinates of these moving particles are given by the $N$ eigenvalues of a time-dependent $N\times N$ matrix $U(t)$ explicitly known in terms of the 2N initial data $z_{n}(0)$ and $\dot{z}_{n}(0)$. This model comes in two different variants, one featuring 3 arbitrary coupling constants, the other only 2; for special values of these parameters all solutions are completely periodic with the same period independent of the initial data ("isochrony"); for other special values of these parameters this property holds up to corrections vanishing exponentially as $t\rightarrow \infty$ ("asymptotic isochrony"). Other isochronous variants of these models are also reported. Alternative formulations, obtained by changing the dependent variables from the $N$ zeros of a monic polynomial of degree $N$ to its $N$ coefficients, are also exhibited. Some mathematical findings implied by some of these results - such as Diophantine properties of the zeros of certain polynomials - are outlined, but their analysis is postponed to a separate paper.


Introduction
The technique used in this paper to identify a new solvable many-body problem has become by now standard. Its more convenient version starts from the identification of a solvable matrix problem characterized by two first-order, generally autonomous, matrix ODEs defining the time evolution of two N × N matrices U ≡ U (t) and V ≡ V (t): (1.1a) where the two functions F (U, V ), G(U, V ) may depend on several scalar parameters but on no other matrix besides U and V . Here and hereafter superimposed dots denote of course differentiations with respect to the independent variable t ("time"). The solvable character of this system amounts to the possibility to obtain the solution of its initial-value problem,

1b)
This paper is a contribution to the Special Issue "Geometrical Methods in Mathematical Physics". The full collection is available at http://www.emis.de/journals/SIGMA/GMMP2012.html with the two matrix functions U (t; U 0 , V 0 ), V (t; U 0 , V 0 ) explicitly known. For instance, in this paper we shall focus (see (3.3) below) on a system (1.1a) which can be related to a linear (third-order, matrix) evolution equation (see (3.3) below), so that its initial-value problem can be explicitly solved.
One then introduces the eigenvalues z n (t) of one of these two matrices, by setting, say (1.2) Remark 1.1. The diagonalizing matrix R(t) is identified by this formula up to right-multiplication by an arbitrary diagonal N × N matrix D(t), R(t) ⇒R(t) ≡ R(t)D(t).
Then one introduces a new N × N matrix Y (t) by setting where of course the N × N matrix R(t) is that defined above, see (1.2). This matrix Y (t) is of course generally nondiagonal : Y nm (t) = δ nm y n (t) + (1 − δ nm )Y nm (t).
(1.3b) Notation 1.1. Indices such as n, m, run from 1 to N (unless otherwise explicitly indicated), and N is an arbitrary positive integer (indeed generally N ≥ 2). δ nm is the Kronecker symbol, δ nm = 1 if n = m, δ nm = 0 if n = m. Note that hereafter we use the notation Y nm to denote the N (N − 1) off-diagonal elements of the N × N matrix Y .
It often turns out that the time evolution of the eigenvalues z n (t) is then characterized by a system of N second-order ODEs which read as follows: z n = f (z n ,ż n ) + N =1, =n Y n Y n g (1) (z n ,ż n )g (2) (z ,ż ) z n − z , (1.4) where the three functions f (z,ż), g (1) (z,ż) and g (2) (z,ż) can be computed from the two matrix functions F (U, V ), G(U, V ) (see below). It is then natural to try and interpret this system of ODEs as the Newtonian equations of motion ("acceleration equal force") of a many-body problem characterizing the motion of N particles whose coordinates coincide with the N eigenvalues z n (t); an N -body problem generally featuring one-body and two-body velocity-dependent forces, with the N (N − 1) quantities Y n Y n playing the role of "coupling constants". But these quantities are not time-independent, nor can they be arbitrarily assigned: they are the off-diagonal elements of the N × N matrix Y , hence they should be themselves considered as dependent variables, the time evolution of which is characterized by the system of N (N − 1) ODEs implied for them by (1.1a) via (1.2) and (1.3). Two options are then open to provide nonetheless a "physical" interpretation for the equations of motion (1.4).
One option that we do not pursue here is to provide some kind of "physical" interpretation for these N (N − 1) quantities Y n Y n as additional (internal) degrees of freedom of the moving particles.
A second option -the one which we pursue below -is to find a (time-independent) ansatz expressing the N (N − 1) quantities Y n in terms of the N coordinates z m , or possibly of the 2N quantities z m ,ż m ; an ansatz consistent with the N (N − 1) equations of motion satisfied by the N (N − 1) quantities Y n , which satisfies these equations either identically (i.e., independently from the time evolution of the N coordinates z m (t)) or, as it were, self-consistently (i.e., thanks to the time evolution (1.4) of the N coordinates z m (t) with the N (N − 1) quantities Y n assigned according to the ansatz ). Given a matrix system of type (1.1a) no technique is known to assess a priori whether or not such an ansatz exist. However the experience accumulated over time suggests that, if such an ansatz does exist, it has one of the following two forms: of course with the functions appearing in the right-hand side of these formulas chosen appropriately. And as a rule the ansatz 1 should work identically, i.e. independently of the time evolution of the coordinates z m (t); while to ascertain the validity of ansatz 2 the equations of motion (1.4) should be used self-consistently.
For instance in the very simple case of the equations of motion (1.1a) with F (U, V ) = V and G(U, V ) = 0 implyingÜ = 0 and U (t) = U 0 + V 0 t, V (t) = V 0 , both ansätze exist: the ansatz 1 reads in this case Y n = ig/(z n − z ) with i the imaginary unit (introduced for convenience) and g an arbitrary constant, and it yields the prototypical "CM" model characterized by the equations of motion (1.6) while the ansatz 2 in this case reads Y n = (ż nż ) 1/2 , and it yields the prototypical "goldfish" model characterized by the equations of motion Nomenclature and historical remarks. The model characterized by the Newtonian equations of motion (1.6) -which obtain from the Hamiltonian -is usually associated with the names of those who first demonstrated the possibility to treat this many-body problem exactly, respectively in the quantal [1] and in the classical [2] contexts; accordingly, we usually call "many-body problems of CM-type" those featuring in the righthand ("forces") side of their Newtonian equations of motion a term such as that appearing in the right-hand side of (1.6) (in addition of course to other terms). Such models are typically produced by the ansatz 1.
The solvable character of the many-body problem characterized by the Newtonian equations of motion (1.7a) -which is also Hamiltonian, for instance with the Hamiltonian [3,4,5] -is demonstrated by the following neat Prescription [5,6,7]: the N values of the coordinates z n (t) providing the solution of the initial-value problem of the equations of motion (1.7a) are the N zeros of the following algebraic equation for the variable z: (1.7b) (Note that this formula amounts to a polynomial equation of degree N in z, as it is immediately seen by multiplying it by the polynomial N m=1 [z − z m (0)]). In [7] it was suggested that this model, in view of its neat character, be considered a "goldfish" (meaning, in Russian traditional lore, a very remarkable item, endowed with magical properties); accordingly, we usually call "many-body problems of goldfish-type" those featuring in the right-hand ("forces") side of their Newtonian equations of motion a term such as that appearing in the right-hand side of (1.7a) (in addition of course to other terms). Such models are typically produced by the ansatz 2.
The original idea of the approach described above is due to Olshanetsky and Perelomov, who introduced it to solve, in the classical context, the many-body model characterized by the Newtonian equations of motion (1.6) [8]. For a more detailed description of their work see the review paper [9], the book [10], Section 2.1.3.2 (entitled "The technique of solution of Olshanetsky and Perelomov") in [11], and other references cited in these books. In Section 4.2.2 (entitled "Goldfishing") of [5] several many-body problems, mainly "of goldfish type", are reviewed, the solvable character of which has been ascertained by this approach; and several other such models are discussed in more recent papers [12,13,14,15].
The present paper provides two further additions to the list of solvable many-body problems "of goldfish type"; and we expect that other items will be added to this list in the future, possibly by a continuation of the case-by-case search of solvable matrix evolution equations allowing -via the route outlined above and described in more detail, in a specific case, below (see Section 3)the identification of working ansätze leading to new systems of Newtonian equations (thereby shown to be themselves solvable, inasmuch as their solution is reduced to the algebraic task of computing the N eigenvalues of an explicitly known N × N matrix). The identification of a new model of this kind constitutes -in our opinion -an interesting finding (even if several analogous models have been previously discovered): we view these many-body problems as gems embedded in the magma of the generic many-body problems which are not amenable to exact treatments (although the latter include of course more examples of applicative interest and are also mathematically interesting to investigate the emergence and phenomenology of chaotic behaviors).
In the following Section 2 the main findings of this paper are reported, including in particular a description of two new solvable many-body problems (one featuring 3, the other only 2, a priori arbitrary parameters), of the algebraic solution of their initial-value problems, and of the variety of behaviors (including isochrony and asymptotic isochrony) featured by them for certain assignments of their parameters; two additional isochronous systems are moreover exhibited in Section 2.1. In Section 3 these results are proven. Section 4 provides the alternative formulations of these models, obtained by changing the dependent variables from the N zeros of a monic polynomial of degree N to its N coefficients. A final Section 5 entitled "Outlook" outlines further developments, including in particular the identification of Diophantine properties of the zeros of certain polynomials; but their detailed discussion is postponed to a separate paper.

Main f indings
The two models treated in this paper are characterized by the following two sets of Newtonian equations of motion "of goldfish type".
Model (i): where a, b and γ are 3 a priori arbitrary parameters.
Model (ii): where a and b are 2 a priori arbitrary parameters.
Notation 2.1. These models describe the motion of an arbitrary number N (generally N ≥ 2) of points moving in the complex z-plane (but see below Remark 2.1). Their positions are identified by the complex dependent variables z n ≡ z n (t). The independent variable t ("time") is instead real. Superimposed dots denote of course time-differentiations. The parameters featured by these models are generally arbitrary complex numbers; unless otherwise specified when discussing special cases.
Remark 2.1. It is possible to reformulate these models so that they describe the motion of pointlike "physical particles" moving in a real -say, horizontal -plane: see for instance, in [11], Section 4.1 entitled "How to obtain by complexification rotation-invariant many-body models in the plane from certain many-body models on the line". This task is left to the interested reader. But hereafter we feel free to refer to the models identified by the Newtonian equations of motion (2.1) as many-body problems characterizing the motion of N particles in a plane.
Remark 2.2. Additional parameters could be inserted in these models by shifting or rescaling the dependent variables z n or the independent variable t. We will not indulge in such trivial exercises (see also Remark 3.1 below).
The solvable character of these two models is demonstrated by the following Proposition 2.1. The solution of the initial-value problems of the two many-body models characterized by the Newtonian equations of motion "of goldfish type" (2.1) are given by the eigenvalues of the N × N matrix U (t), the explicit expression of which in terms of the 2N initial data z n (0),ż n (0) and of the time t is given by the following formulas (see (3.6a) with (3.7)): with the two constant N × N matrices A and B defined as follows:

2c)
Here and hereafter I is the N × N unit matrix, the N × N matrix U 0 ≡ U (0) is diagonal and is given in terms of the initial particle positions z n (0) as follows, is the sum of a diagonal and a dyadic matrix, being given componentwise by the following formulas in terms of the initial particle positions z n (0) and velocitiesż n (0): with c = 1 for model (i) and c = 1/2 for model (ii) (see (2.6)). As for the 3 constants ω j appearing in (2.2), they are defined by the following formula (see (3.5b)): with α and β respectively α, β and γ given by the following expressions for model (i) respectively for model (ii) (see (3.21a) respectively (3.21b)): Note that the 3 a priori arbitrary parameters ω j have the dimension of an inverse time; above and hereafter we assume for simplicity that they are different among each other (except in the following Subsection 2.1, where they are all assumed to vanish).
It is plain (see (2.2)) that, if the 3 constants ω j are integer multiples of a single real constant ω, then the matrix U (t) is periodic, Here and throughout the 3 parameters k j are integer numbers (positive, negative or vanishing, but different among themselves); their definition, as well as that of the positive parameter ω, is made unequivocal (up to permutations; once the 3 parameters α, β, γ are assigned, compatibly via (2.5b) with (2.7a)) by the requirement that this positive parameter ω be assigned the largest value for which (2.7a) holds. More generally, if the real parts of the 3 constants ω j are integer multiples of a single real constant ω and the imaginary parts of 2 of them coincide while the imaginary part of the third is larger, say, Re(ω j ) = k j ω, j = 1, 2, 3; Im(ω 1 ) = Im(ω 2 ) < Im(ω 3 ), (2.9a) then the matrix U (t) is asymptotically periodic with period T , namely it becomes periodic with period T in the remote future up to exponentially vanishing corrections, so that While for generic values of the parameters, implying via (2.5) that the imaginary parts of the 3 quantities ω j are different among themselves, Im(ω 1 ) = Im(ω 2 ), Im(ω 2 ) = Im(ω 3 ), Im(ω 3 ) = Im(ω 1 ), (2.10a) then clearly U (t) tends to a time-independent matrix as t → ±∞, Let us emphasize that these outcomes, (2.7b) or (2.9b) or (2.10b), obtain -provided the 3 parameters α, β, γ satisfy (2.5b) with (2.7a) or (2.9a) or (2.10a) -for arbitrary initial data, and that the period T is as well independent of the initial data. Let us also note that these properties hold unless U (t) is singular ; clearly a nongeneric circumstance, in which case U (t) would in fact still feature the properties indicated above, but only in the sense in which, for instance, the function tan[ω(t−t 0 )/2] (with ω and t 0 two real numbers) is periodic with period T .
These properties of the N × N matrix U (t) carry of course over to its eigenvalues z n (t), hence to the generic solutions of the many-body problems "of goldfish type" (2.1); these models are therefore isochronous respectively asymptotically isochronous if their parameters satisfy the relevant conditions, see (2.5b) with (2.7a) respectively (2.9a).

Remark 2.3.
Let us however recall that the periods of the time evolution of individual eigenvalues of a periodic matrix may be a (generally small) positive integer multiple of the period of the matrix, due to the possibility that different eigenvalues exchange their roles over the time evolution (for a discussion of this possibility see [16], where a justification is also provided of the statement made above that the relevant positive integer multiple is "generally small").

Two additional isochronous many-body models
Additional isochronous models obtain by applying, to the special cases of the two many-body models (2.1) with all parameters vanishing, with c = 1 respectively c = 1/2, the standard "isochronizing" trick, see for instance Section 2.1 (entitled "The trick") of [5]. It amounts in these cases to the following change of dependent and independent variables, Here and below ω is again an arbitrary real constant to which we associate the period T , see (2.8), and of course appended primes denote differentiations with respect to the argument of the function they are appended to (hence z n (τ ) = dz n (τ )/dτ ). Hence clearly the Newtonian equations of motion of these two models read as follows: again with c = 1 respectively c = 1/2. The solutions of the corresponding initial-value problems are clearly given by the following variant (of the case with ω j = 0) of Proposition 2.1: The solution of the initial-value problems of the two many-body models characterized by the Newtonian equations of motion "of goldfish type" (2.13) are given by the eigenvalues of the N × N matrixŨ (t) given by the following formulas in terms of the 2N initial dataz n (0), · z n (0) and of the time t: with the two constant N × N matricesÃ andB defined as follows: of course always with c = 1 respectively c = 1/2 and τ ≡ τ (t) defined in terms of the time t by (2.12a). Note that the N × N matrixÃ is diagonal and the N × N matrixB is now dyadic.
It is plain that the matrixŨ (t) is isochronous with period T (see (2.14a), (2.12a) and (2.8)), and the same property of isochrony holds therefore for the generic solutions of the two manybody models (2.13), up to the observation made above (see Remark 2.3).

Proofs
The starting point of our treatment is the following system of two coupled matrix ODEs satisfied by the two N × N matrices U ≡ U (t) and V ≡ V (t): Notation 3.1. The 3 scalars α, β, γ are 3, a priori arbitrary, constant parameters; I is the unit N × N matrix; and we trust the rest of the notation to be self-evident (see also Sections 1 and 2).
Remark 3.1. Additional parameters could of course be introduced by scalar shifts or rescalings of the dependent variables U , V or of the independent variable t ("time"). We forsake any discussion of such trivial transformations (see Remark 2.2).
To solve this matrix system we introduce the N × N matrix W ≡ W (t) by setting It is then easily seen that the system (3.1) entails that the matrix W satisfy the following linear third-order matrix ordinary differential equation (ODE): ... W = αW + βẆ + γẄ ; (3.3) and the converse is as well true (in fact, perhaps easier to verify), namely if W satisfies this linear third-order matrix ODE, then the two matrices U and V defined by (3.2) satisfy the system (3.1).
Clearly the general solution of this linear third-order matrix ODE reads where the 3 constant matrices W (j) are arbitrary, i is the imaginary unit (i 2 = −1; introduced here for notational convenience), and the 3 scalars ω j are the 3 roots of the following cubic equation in ω: It is then easily seen that the general solution of the system (3.1) can be written as follows: where A and B are two, a priori arbitrary, constant N × N matrices. And a trivial if tedious computation shows that these formulas provide the solution of the initial-value problem for the system (3.1) if the two matrices A and B are expressed in terms of the initial values U 0 ≡ U (0), V 0 ≡ V (0) as follows: To derive, from the solvable matrix system (3.1), the solvable many-body problem reported in the preceding section, we follow the procedure outlined in Section 1. This requires that we introduce -in addition to the diagonal N × N matrix Z respectively the nondiagonal N × N matrix Y associated to U respectively V via (1.2) respectively (1.3a) -the auxiliary N × N matrix M ≡ M (t) defined as follows in terms of the diagonalizing matrix R(t), see (1.2): (3.8a) In the following we indicate as µ n ≡ µ n (t) respectively M nm ≡ M nm (t) the diagonal respectively off-diagonal elements of this matrix: Let us now look separately at the diagonal and off-diagonal parts of these two matrix equations, (3.9).
The diagonal part of the first of these two equations reads (see (1.2) and (1.3b)) Likewise, the off-diagonal part of the first of these two equations reads The diagonal part of the second of these two equations reads (see (1.2) and (1.3b)) y n = −z n y n + α + βz n + γy n + N =1, =n implying, via (3.10b) and (3.11b), y n = −ż n z n + γż n − z 3 n + γz 2 n + βz n + α + 2 We now note that, via this equation, time-differentiation of (3.10a) yields the following set of Newtonian-like equations of motion: confirming the treatment outlined in Section 1, see in particular (1.4).
Finally we consider the off-diagonal elements of the second of the matrix equations (3.9). The relevant equations read, componentwise, as follows: And via (3.10b) and (3.11b) (and a tiny bit of algebra) this becomeṡ The next step is to try out the ansätze (1.5), to see if one can thereby get rid of the quantities Y nm .
We leave the (rather easy but unfortunately unproductive) task to verify that the ansatz (1.5a) does not work, i.e. that it does not allow to eliminate the quantities Y nm by finding an assignment of the functions g (1) (z) and g (2) (z) which, when inserted in (1.5a), yield N (N − 1) quantities Y nm that satisfy the N (N − 1) ODEs (3.14c) (even by taking into account the possibility to assign freely -see Remark 3.2 -the N quantities µ n ).
We show that instead the ansatz (1.5b) allows the elimination of the quantities Y nm and leads to the Newtonian equations of motion "of goldfish type" (2.1). Indeed the insertion in (3.14c) of (1.5b) with and the assignment (see Remark 3.2) entails that the N (N − 1) equations (3.14c) can be re-formulated as follows: Here and below primes indicate differentiations with respect to the argument of the functions they are appended to; and the convenient shorthand notation +((n ⇒ m)) denotes addition of whatever comes before it, with the index n replaced by the index m.
It is easily seen that these equations can be re-written as follows: This suggests the assignments with the 3 parameters a, b, c a priori arbitrary, since thereby the N (N − 1) equations (3.16b) get reduced to the N equations z n +ż n (b + 2cz n ) z n + a + bz n + cz 2 n = (2c − 3)z n + γ + b + 2 N =1, =n ż + a + bz + cz 2 z n − z , (3.18a) or equivalently (in Newtonian form) Consistency requires now that this set of N Newtonian equations of motions coincide with the N analogous equations (3.13), which, via (1.5b) with (3.15a) and (3.17), now read z n = −3ż n z n + γż n − z 3 n + γz 2 n + βz n + α + N =1, =n 2 ż n + a + bz n + cz 2 n ż + a + bz + cz 2 z n − z . (3.19) This clearly requires that the following 4 constraints on the 6 parameters α, β, γ, a, b, c be satisfied (note that, somewhat miraculously, the two velocity-dependent one-body terms in the right-hand sides of the last two equations match automatically, as well as the two-body terms): And it is easily seen that this entails two alternative possibilities: model (i): a, b, γ arbitrary and Clearly these two possibilities correspond to the two solvable many-body models "of goldfish type" (2.1).
Next, we must justify the assertions made in the preceding section (see Proposition 2.1) concerning the solution of the initial-value problems for the many-body models characterized by the Newtonian equations of motion (2.1). The treatment given above (in this section) entails that these solutions are provided by the eigenvalues of the N ×N matrix U (t) evolving according to the explicit formula (3.6a) with (3.7); the missing detail is to express the two initial N × N matrices U 0 ≡ U (0) and V 0 ≡ V (0) appearing in the right-hand side of (3.7) in terms of the 2N initial data, z n (0) andż n (0), of the many-body problems (2.1).
To simplify the derivation of these formulas it is convenient to make the assumption (allowed by the treatment given above) that the matrix U (t) be initially diagonal, namely (see (1.2)) that R(0) = I. (3.22) This entails (see (1.2) and (1.3a)) that The formula (2.3) for U 0 is thereby immediately implied. The formula (2.4) for V 0 ≡ V (0) = Y (0) is also easily obtained, since the expression (3.10b) of the diagonal elements of the matrix Y and the ansatz (1.5b) with (3.15a) and (3.17) for the off-diagonal elements of Y entail that, componentwise, and this clearly yields (2.4). This completes the proof of the results of the preceding Section 2. As for the findings reported in Section 2.1, we consider their derivation sufficiently obvious -for instance via the treatment detailed in Section 2.1 of [5]; and by repeating the relevant treatment as given above, especially in the last part of this section -to justify us to dispense here from any further elaboration.

Additional f indings
In this section we introduce the time-dependent (monic) polynomial ψ(z, t) whose zeros are the N eigenvalues z n (t) of the N × N matrix U (t): The last of these formulas introduces the N coefficients c m ≡ c m (t) of the monic polynomial ψ(z, t); of course it implies that these coefficients are related to the zeros z n (t) as follows: and so on. The fact that the initial-value problem associated with the time evolution (2.1) of the N coordinates z n can be solved by algebraic operations implies that the same solvable character holds for the time evolution of the monic polynomial ψ(z, t) and of the N coefficients c m (t). In this section we display explicitly the equations that characterize these time evolutions. The procedure to obtain these equations from the equations of motion (2.1) is quite tedious but standard; a key role in this development are the identities reported, for instance, in Appendix A of [5] (but there are 2 misprints in these formulas: in equation (A.8k) the term (N + 1) inside the square bracket should instead read (N − 3); in equation (A.8l) the term N 2 inside the square bracket should instead read N (N − 2)). Here we limit our presentation to reporting the final result.
This of course entails that the conditions on the 3 a priori free parameters a, b and γ of the version (4.4) of model (i), or on the 2 a priori free parameters a and b of the version (4.5) of model (ii), which are necessary and sufficient to imply that these models be isochronous respectively asymptotically isochronous, are those indicated in Section 1, namely are those implied by (2.6) with (2.5b) and (2.7a) respectively (2.9a); while in the generic case, see (2.10), the coefficients c m (t) tend asymptotically to time-independent values, lim t→±∞ [c m (t)] = c m (±∞). (4.6c)

Two additional isochronous models
Here we report formulas analogous to those reported above, but related to the many-body models discussed in Section 2.1 rather than to those reported in Section 2. We consider the derivation of these results sufficiently straightforward not to require any detailed elaboration here beyond the terse hints provided below. The starting point are the two special cases of the two systems (4.4a) and (4.5a) which obtain by setting all the free parameters to vanish. We write them in compact form as follows: with c = 1 respectively c = 1/2. It is remarkable that the number N does not appear in these equations; although we always assume that these equations hold for m = 1, 2, . . . , N and that the dependent variables c n vanish identically for n > N , with N an arbitrary positive integer.
The diligent reader may also check that this equation also holds identically for m = 0 with c 0 = 1 (see (4.1)). We then make the following change of dependent and independent variables: c m (t) = exp(imωt)c m (τ ), τ = exp(iωt) − 1 iω , (4.8) with ω again a real arbitrary constant to which we associate the period T (see (2.8)). One thereby easily sees that the new dependent variablesc m (t) satisfy the following system of autonomous Newtonian equations of motion:  It is plain from the way these two models (with c = 1 or c = 1/2) have been derived that they are isochronous, namely the generic solutions of these nonlinear Newtonian equations of motion satisfy the conditioñ c m (t + T ) =c m (t). (4.10)

Outlook
It is clearly far from trivial that the Newtonian many-body models introduced in this papersee (2.1), (2.13), (4.4), (4.5), and (4.9) -can be solved, for arbitrary initial data, by algebraic operations. Also remarkable is that, for special assignments of their parameters, the systems of autonomous nonlinear ODEs (2.1), (4.4) and (4.5) are isochronous or asymptotically isochronous, and the systems of autonomous nonlinear ODEs (2.13) and (4.9) are isochronous. Let us end this paper by pointing out that Diophantine findings can be obtained from a nonlinear autonomous isochronous dynamical system by investigating its behavior in the infinitesimal vicinity of its equilibria. The relevant equations of motion become then generally linear, but they of course retain the properties to be autonomous and isochronous. For a system of linear autonomous ODEs, the property of isochrony implies that all the eigenvalues of the matrix of its coefficients are integer numbers (up to a common rescaling factor). When the linear system describes the behavior of a nonlinear autonomous system in the infinitesimal vicinity of its equilibria, these matrices can generally be explicitly computed in terms of the values at equilibrium of the dependent variables of the original, nonlinear model. In this manner nontrivial Diophantine findings and conjectures have been discovered and proposed: see for instance the review of such developments in Appendix C (entitled "Diophantine findings and conjectures") of [5]. Analogous results obtained by applying this approach to the isochronous systems of autonomous nonlinear ODEs introduced above will be reported in a separate paper if they turn out to be novel and interesting.