Francesco Calogero

• Sapienza University of Rome

It is shown that the solutions of certain systems of nonlinear \"Orst-order recursions with polynomial right-hand sides may be rather easily ascertained, and display interesting evolutions in their ticking time variable (taking integer values): for instance a remarkable kind of asymptotic periodicity.

The evolution, as functions of the "ticking time" $\ell =0,1,2,...$, of the solutions of the system of $N$ quadratic recursions \begin{eqnarray*} x_{n}\left( \ell +1\right) =c_{n}+\sum_{m=1}^{N}\left[ C_{nm}x_{m}\left( \ell \right) \right] +\sum_{m=1}^{N}\left\{ d_{nm}\left[ x_{m}\left( \ell \right) \right] ^{2}\right\} +\sum_{m_{1}>m_{2}=1}^{N}\le...

In this paper a procedure is described which allows to identify new systems of nonlinear recursions whose solutions are controllable and which may be asymptotically isochronous as functions of the independent variable (considered a ticking time).

In this paper we firstly review how to \textit{explicitly} solve a system of $3$ \textit{first-order linear recursions }and outline the main properties of these solutions. Next, via a change of variables, we identify a class of systems of $3$ \textit{first-order nonlinear recursions} which also are \textit{explicitly solvable}. These systems might...

In this paper a class of simple, but nonlinear, systems of recursions involving $2$ dependent variables $x_{j}\left( n\right) $ is identified, such that the solutions of their initial-values problems -- with arbitrary initial data $x_{j}\left( 0\right) $ -- may be explicitly obtained.

In this paper we introduce some conjectures analogous to the well-known Collatz conjecture.

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same...

A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we point out that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints.

The generic monic polynomial of degree N features N a priori arbitrary coefficients $c_m$ and N zeros $z_n$. In this paper we limit consideration to $N = 8$ and $N = 9$. We show that if the $N$ -- a priori arbitrary -- coefficients $c_m$ of these polynomials are appropriately defined -- as it were, a posteriori -- in terms of 6 arbitrary parameters...

The initial-values problem of the following nonlinear autonomous recursion of order p , z (s + p) = c product of [z (s + l)]^a_l ; with p an arbitrary positive integer, z (s) the dependent variable (possibly a complex number), s the independent variable (a non negative integer), c an arbitrarily assigned, possibly complex, number, and the p exponen...

In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit solutions. A novelty of these findings is to consider special systems characterized by constraints involving both th...

The asymmetric May-Leonard model is a prototypical system of 3 nonlinearly coupled first-order Ordinary Differential Equations with second-degree polynomial right-hand sides. In this short paper we identify a class of special solutions of this system which do not seem to have been previously advertised in spite of their rather elementary character.

We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides and the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. E...

We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides; the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equi...

Certain nonlinearly-coupled systems of N discrete-time evolution equations are identified, which can be solved by algebraic operations; and some remarkable Diophantine findings are thereby obtained. These results might be useful to test the accuracy of numerical routines yielding the N roots of polynomials of arbitrary degree N.

The chapter analyses the technical and political problems connected to the introduction of anti-ballistic missiles (ABMs), and to a lesser extent of multiple independently targetable re-entry vehicles and fractional orbital ballistic systems. These technological events are depicted as extremely dangerous as they, contrary to what their supporters h...

Recent findings concerning the zeros of generic polynomials are extended to entire functions featuring infinitely many distinct zeros, and related systems of infinitely many nonlinearly coupled evolution ODEs and PDEs are identified, the solutions of which display interesting properties.

We use previous results concerning the time evolution of the zeros xn(t) of time-dependent polynomials pN (z;t) or entire functions F(z;t) of the complex variable z, in order to identify lots of nonlinearly-coupled, finite or infinite, systems of Ordinary Differential Equations the solutions of which feature remarkable Diophantine properties.

Examples of many-body Hamiltonians are identified which yield time-evolutions, having the property that the projection of an arbitrary orbit in phase space onto configuration space is periodic with a period independent of initial data (“isochrony”), while the evolution in phase space is not periodic. These include a variation on the harmonic oscill...

We evaluate the number of monic polynomials (of arbitrary degree $N$) the zeros of which equal their coefficients when these are allowed to take arbitrary complex values. In the following, we call polynomials with this property {\em peculiar\/} polynomials. We further show that the problem of determining the peculiar polynomials of degree $N$ simpl...

The notion of generations of monic polynomials such that the coefficients of the polynomials of the next generation coincide with the zeros of the polynomials of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems of "goldfish type" is demonstrated.

A technique is introduced which allows to generate -- starting from any solvable discrete-time dynamical system involving N time-dependent variables -- new, generally nonlinear, generations of discrete-time dynamical systems, also involving N time-dependent variables and being as well solvable by algebraic operations (essentially by finding the N z...

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1...

A new solvable many-body problem of goldfish type is introduced and the behavior of its solutions is tersely discussed.

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the ordering assignment of the N zeros of the Hermite polynomial of order N. We construct two NxN matrices M_1 and M_2 defined in terms of the N zeros of th...

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers zn , and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f(z) as follows: [f(z + a) − f(z)]/a respectively [f(qz) − f(z)]/[(q − 1) z]. These rep...

We define the generalized hypergeometric polynomial of degree N in terms of the generalized hypergeometric function that depends on p parameters a_1, ..., a_p and q parameters b_1, ..., b_q. The parameters are "generic", possibly complex, numbers. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros z_n of this...

In this paper we provide properties -- which are, to the best of our knowledge, new -- of the zeros of the polynomials belonging to the q-Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.

A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points z n ≡z n (ℓ), where ℓ=0,±1,±2,⋯ is the discrete-time independent variable. Both these equations of motion, and the solution of their initial-value problem, only in...

Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants of motion in terms of the dependent variables and their time-derivatives. Some of these models are moreover s...

A class of macroscopic systems is described which have the remarkable feature that they can sustain undamped compressional radial oscillations. They consist of an arbitrary number of particles confined by a harmonic potential and interacting among themselves through conservative forces scaling as the inverse cube of distances. The radial oscillatio...

In the context of an ambient space with an arbitrary number d of dimensions, the many-body problem consisting of an arbitrary number N of particles confined by a common, external harmonic potential (realizing a container with soft walls) and interacting among themselves and with the environment with arbitrary conservative repulsive forces scaling a...

A new discrete-time N-body problem is introduced. Its equations of motion—which become Newtonian equations of motion (accelerations equal forces) in the continuous-time limit—are nonautonomous, featuring an arbitrary function f(ℓ) of the discrete-time variable ℓ = 0, 1, 2, 3… . They are nevertheless solvable by algebraic operations: the solution of...

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 system...

The class of solvable N-body problems of “goldfish” type has been recently extended by including (the additional presence of) three-body forces. In this paper we show that the equilibria of some of these systems are simply related to the N roots xn of the polynomial equation (x)= w, where (x) is the Jacobi polynomial of order N, the parameters α an...

A linear second-order ODE featuring an arbitrary number of free parameters is identified, all solutions of which are polynomials.

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.

By investigating the behavior of two solvable isochronous N-body problems in the immediate vicinity of their equilibria, functional equations satisfied by the para-Jacobi polynomial ${p_{N} \left(0, 1; \gamma; x \right)}$ and by the Jacobi polynomial ${P_{N}^{\left(-N-1,-N-1 \right)} \left(x \right)}$ (or, equivalently, by the Gegenbauer polyno...

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather explicitly known) polynomials satisfying a (recen...

Recently a solvable N-body problem featuring several free parameters has been investigated, and conditions on these parameters have been identified which guarantee that this system is isochronous (all its solutions are periodic with a fixed period) and that it possesses equilibria. The N coordinates characterizing the equilibrium configurations ar...

Some properties of a solvable N-body problem featuring several free parameters (“coupling constants”) are investigated. Restrictions on its parameters are reported which guarantee that all its solutions are completely periodic with a fixed period independent of the initial data (isochrony). The restrictions on its parameters which guarantee the exi...

A new class of solvable $N$-body problems is identified. They describe $N$ unit-mass point particles whose time-evolution, generally taking place in the complex plane, is characterized by Newtonian equations of motion "of goldfish type" (acceleration equal force, with specific velocity-dependent one-body and two-body forces) featuring several arbit...

A new solvable discrete-time many-body problem is identified. It extends a model treated in a previous paper by introducing in its equations of motion an additional free parameter. Hence, it features 6 parameters, 2 of which can be eliminated (say, replaced by unity) by appropriate rescalings. Assignments of these parameters are identified which en...

The starting point is an N×N matrix, U ≡ U(ℓ), evolving in the discrete-time independent variable ℓ = 0, 1, 2, ... according to a solvable matrix evolution equation. One then focuses on the evolution of its N eigenvalues zn(ℓ). This evolution generally also involves N(N−1) additional variables. In some cases via a compatible ansatz these additional...

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_...

In the context of classical mechanics, Newton-equivalent Hamiltonians are those that yield the same Newtonian equations of motion (‘acceleration equal force’). Various such Hamiltonians are discussed, all of them characterizing a rotator in the plane. Some of them are well known, while others are perhaps new. Their quantization is also discussed. O...

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. Its Newtonian equations of motion read as follows: $$\ddot z_n = \frac{{\dot z_n^2 }} {{z_n }} + c_1 \frac{{\dot z_n }} {{z_n }} + c_2 \dot z_n + c_2 c_3 z_n + c_1 c_2 + \sum\limits_{m = 1,m \ne n}^N {\frac{{\left( {\dot z_n + c_3 z...

A new solvableN-body model of goldfish type is identified. Its Newtonian equations of motion read as follows: where zn ≡ zn(t) are the N dependent variables (with N an arbitrary positive integer), t is the independent variable ("time") and the dots indicate time-differentiations. Its isochronous variant is also obtained and discussed. Other new sol...

This paper is a terse highlight of the remarkable similarities and differences among two matrix ODEs — one solvable and one integrable — and the many-body problems related to them.

We first provide a terse review of the status and prospects of nuclear-weapon proliferation, and we then discuss the present perspective of a future transition to a Nuclear-Weapon-Free World

Matrices and of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case, this property states that the 2N zeros of the polynomial are all integers; indeed we conjecture that The results in the para-Jacobi case are somewhat...

The answer to the question posed in the title is affirmative. Indeed, while the second-order ODE (1 − x2)p'' + [β − α − (α + β + 2)x]p' + N(N + α + β + 1)p = 0 for generic values of the two parameters α and β (and N a non-negative integer) identifies uniquely (up to an overall multiplicative constant characterizing its normalization) the Jacobi pol...

We indicate how one can extend any dynamical system (namely, any system of nonlinearly coupled autonomous ordinary differential equations) so that the extended dynamical system thereby obtained is either isochronous or asymptotically isochronous or multi-periodic, namely its generic solutions are either completely periodic with a fixed period or te...

We identify a solvable dynamical system — interpretable to some extent as a many-body problem — and point out that — for an appropriate assignment of its parameters — it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the init...

Recently we highlighted the remarkable nature of an explicitly invertible transformation, we reported some generalizations of it and examples of its expediency in several mathematical contexts: algebraic and Diophantine equations, dynamical systems (with continuous and discrete time), nonlinear PDEs, analytical geometry, functional equations. In th...

Some years ago, Mikhailov and Sokolov identified as integrable the neat system of two evolution equations = V2, = U2, where U ≡ U(t) and V ≡ V(t) are two N × N matrices, N is an arbitrary positive integer, t (“time”) is the independent variable, and superimposed dots indicate the time derivatives. This entails, rather trivially, that the generic s...

The original continuous-time "goldfish" dynamical system is characterized by two neat formulas, the first of which provides the $N$ Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initial-value problem. Several other, more general, solvable dynamical systems "of goldfish type" have...

It is shown how, given an arbitrary dynamical system, other systems can be manufactured which are isochronous (periodic in all their degrees of freedom with an arbitrarily assigned fixed period T) and whose dynamics coincides exactly with that of the original system over a fraction 1 − σ of that period where σ is a number that can be arbitrarily as...

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. A version of its Newtonian equations of motion reads as follows: $$\ddot z_n = i\omega \dot z_n + \sum\limits_{m = 1,m \ne n}^N {\frac{{2\dot z_n \dot z_m }} {{z_n - z_m }}} .$$ (1) where zn ≡ zn(t) are the N dependent variables, t is...

In this paper (as in previous ones) we identify and investigate polynomials p (ν) n (x) featuring at least one additional parameter ν besides their argument x and the integer n identifying their degree. They are orthogonal (provided the parameters they generally feature fit into appropriate ranges) inasmuch as they are defined via standard three-te...

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained is revealed. Various contexts are considered: a...

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a sim...

A simple mathematical model involving two first-order Ordinary Differential Equations (ODEs) with fourth-degree polynomial nonlinearities is introduced. The initial-value problem for this system of two ODEs is solved in terms of elementary functions: for an open set of initial data, this solution is isochronous, i.e., completely periodic with a fix...

This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic i...

We call partially superintegrable (indeed isochronous) those dynamical systems all solutions of which are completely periodic with a fixed period (“isochronous”) in a part of their phase space, and we review a recently introduced trick that allows to manufacture many such systems. Several examples are discussed.

We consider systems of ordinary differential equations in the plane featuring at most quadratic nonlinearities. It is known that, up to linear transformations of the variables, there are only four systems for which the origin is an isochronous center, that is, for which all orbits in the vicinity of the origin are periodic with the same, fixed peri...

Two new solvable dynamical systems of goldfish type are identified, as well as their isochronous variants. The equilibrium configurations of these isochronous variants are simply related to the zeros of appropriate Laguerre and Jacobi polynomials.

A simple technique is identified to manufacture solvable nonlinear dynamical systems, and in particular three classes whose generic solutions are, respectively, isochronous, multi-periodic, or asymptotically isochronous.

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian where N is an arbitrary positive integer, Ω, b and ωn2 are N + 2 arbitrary real constants, q̂m, q̂m with m = 0,1,⋯,N are the 2N + 2 canonical coordinates and p̂m, p̌m the corresponding 2N + 2 canonical momenta. In the classical...

The isochronous variant is exhibited of the dynamical system corresponding to the Mth ordinary differential equation of the stationary Korteweg-de Vries KdV hi-erarchy. New Diophantine relations are thereby obtained, in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbi-trary degree having int...

An isochronous system is introduced by modifying the Nth ODE of the stationary Burgers hierarchy, and then, by investigating its behaviour near its equilibria, neat Diophantine relations are identified, involving (well-known) polynomials of arbitrary degree having integer zeros, or equivalently matrices the determinants of which yield such polynomi...

An isochronous variant of the Ruijsenaars–Toda integrable many-body problem is introduced, an equilibrium configuration of this dynamical system is identified and by investigating the motions in its neighborhood Diophantine relations are obtained.

We study a simple mathematical model that can be interpreted as a description of the kinetics of the following four reactions involving the two chemicals U and W: (i) U + U U with rate α, (ii) U + W U with rate β, (iii) W + W U with rate γ and (iv) W + W W + W + W with rate δ + 2γ. The model can be generally solved by quadratures, and in the specia...

Certain nonlinear evolution PDEs in 1+1 variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period T=2π/ω (or perhaps T˜=pT with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but th...

In this paper, we apply to (almost) all the “named” polynomials of the Askey scheme, as defined by their standard three-term recursion relations, the machinery developed in previous papers. For each of these polynomials we identify at least one additional recursion relation involving a shift in some of the parameters they feature, and for several o...

Given an arbitrary (autonomous) Hamiltonian , where the N components pn of the N-vector are the canonical momenta, the N components qn of the N-vector are the corresponding canonical coordinates and N is an arbitrary positive integer, we show how to manufacture (autonomous) Hamiltonians , featuring the N + 1 canonical momenta and the corresponding...

We identify a solvable dynamical system-interpretable to some extent as a many-body problem- and point out that-for an appropriate assignment of its parameters-it is entirely isochronous, namely all its nonsingular solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial dat...

The claim made in [Eur. Phys. J. D 53, 123 (2009)] is invalid.

In earlier works we have shown how it is possible to modify a largely arbitrary Hamiltonian H to a different Hamiltonian depending on an arbitrarily assigned (positive) parameter Ω, so that all orbits of have period but the dynamics of resembles closely that yielded by H over times much shorter than the period . In this paper, we apply our approach...

A new class of isochronous dynamical systems is introduced and briefly discussed. These systems feature in their phase space a fully dimensional region (part of which can be explicitly identified) where all their solutions are completely periodic (periodic in all their degrees of freedom) with the same period. But in other regions of their phase sp...

We investigate the dynamics defined by the following set of three coupled first-order ODEs: It is shown that the system can be reduced to quadratures which can be expressed in terms of elementary functions. Despite the integrable character of the model, the general solution is a multiple-valued function of time (considered as a complex variable),...

A new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an i...

Recently, a general technique has been introduced to Ω-modify a Hamiltonian so that the Hamiltonian thereby produced is, in the classical context, isochronous. In this paper, we introduce and discuss simple examples of isochronous Hamiltonians manufactured in this manner. We also outline their quantal treatment, yielding equispaced spectra.

A (classical) dynamical system is called isochronous if it features an open (hence fully dimensional) region in its phase space in which all its solutions are completely periodic (i. e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data, provided they are inside the isochrony region). When the isochrony...

Recently a technique has been introduced to Ω-modify a Hamiltonian so that the Ω-modifiedHamiltonian thereby produced is isochronous: all its solutions are periodic in all degrees of freedom with the same period . In this paper—after briefly reviewing this approach—we focus in particular on the Ω-modified version of the most general realistic many-...