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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 their initial-value problem are the zeros of a polynomial of degree N in z, PN(z; ℓ), explicitly known for all time—via an appropriate discrete-time quadrature—in terms of the initial data. This model generalizes a previously known model, in which the arbitrary function f(ℓ) is an arbitrary constant η. [ABSTRACT FROM AUTHOR]

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: where c, c, and c are arbitrary constants, z ≡ z(t) are the N dependent variables, N is an arbitrary positive number (N > 1), t is the independent variable ('time') and the dots indicate time-differentiations. [ABSTRACT FROM AUTHOR]

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: where z ≡ z(t) are the N dependent variables, t is the independent variable ('time'), the dots indicate time differentiations, and β, η are two arbitrary constants. [ABSTRACT FROM AUTHOR]

A recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many-body problems, being characterized by equations of motion of Newtonian type (generally in two-dimensional space). In this paper we tersely review results previously obtained in this manner and present novel findings of this kind: mainly solvable variants of the goldfish many-body model, including models that feature isochronous classes of completely periodic solutions. Different formulations of these models are presented. The behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated, and in this manner some remarkable Diophantine findings are obtained. [ABSTRACT FROM AUTHOR]

A recent technique to identify solvable many-body problems in two-dimensional space yields, via a new twist, new many-body problems of “goldfish” type. Some of these models are isochronous, namely their generic solutions are completely periodic with a fixed period (independent of the initial data). The investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations. [ABSTRACT FROM AUTHOR]

An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, a (quite large) class of such systems is identified, with equations of motion ‘of Newtonian type.’ Subcases are exhibited in which the equations of motion are Hamiltonian, and/or are interpretable as the equations of motion of a many-body problem, possibly with one- and two-body forces only, possibly invariant under rotations and/or translations, possibly constituting a one-parameter deformation of a classical physical problem (such as the many-body gravitational problem in ordinary three-dimensional space). [ABSTRACT FROM AUTHOR]

A novel solvable extension of the goldfish N-body problem is presented. Its Newtonian equations of motion read ζn=2aζnζn+2∑m=1,m≠nN(ζn-aζn2)(ζm-aζm2)/(ζn-ζm), n=1,...,N, where a is an arbitrary (nonvanishing) constant and the rest of the notation is self-evident. The isochronous version of this model is characterized by the Newtonian equations of motion zn-3iωżn-2ω2zn=2a(żn-iωzn)zn+2∑m=1,m≠nN(żn-iωzn-azn2)(żm-iωzm-azm2)/(zn-zm), n=1,...,N, where ω is an arbitrary positive constant and the points zn(t) move now necessarily in the complex z-plane. The generic solution of this second model is completely periodic with a period Tk=kT which is an integer multiple k (not larger than N!, indeed generally much smaller) of the basic period T=2π/ω and which is independent of the initial data (for sufficiently small, but otherwise arbitrary, changes of such data). These many-body models have an intriguing variety of equilibrium configurations (genuine: with no two particles sitting at the same place), but only for small values of N (N=2,3,4 for the first model, N=2,3,4,5 for the second). Other versions of these models are also discussed. The study of the behavior of the second, isochronous model around its equilibrium configurations yields some amusing diophantine results. [ABSTRACT FROM AUTHOR]

Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion (“acceleration equal force”), with linear and cubic forces, in N-dimensional space (N being an arbitrary positive integer, with special attention to N=2, namely, motions in a plane, and N=3, namely, motions in ordinary three-dimensional space). All the equations of motion are written in covariant form (“N-vector equal N-vector”), entailing their rotational invariance. The corresponding Hamiltonians are of normal type, with the kinetic energy quadratic in the canonical momenta, and the potential energy quadratic and quartic in the canonical coordinates. [Copyright &y& Elsevier]

Presents a solvable model for the motion of strings in the plane. Description of the motion in the plane of a string; Equations of motion which characterize the models; Analysis of the phenomenology of the string motions.

Discusses an n-body problem in the plane, characterized by equations of motion of Newtonian type. Invariance properties of the equations of motion; Behavior as a vector under rotations in the plane; N-body problem solution for arbitrary n and for arbitrary values of the eight coupling constants.

Several solvable n-body problems on the line are exhibited. They are characterized by equations of motion of Newtonian type, mjxj=Fj, j=1,...,n, with the ‘‘forces’’ Fj given as explicit functions of the ‘‘particle coordinates’’ xk and their velocities xk, k=1,...,n; the forces Fj generally also depend on some free parameters, so that in each case there is actually a class of solvable models. In this paper the emphasis is on translation-invariant models, and on values of n sufficiently small (‘‘few-body problems’’) to allow the display of the solution of the initial-value problem in completely explicit form. [ABSTRACT FROM AUTHOR]

In classical mechanics, Newton-equivalent Hamiltonians yield the same equations of motion and hence identical dynamics. In the quantum context, they instead generally yield different discrete energy spectra in the confined case and different scattering amplitudes in the unconfined case. [ABSTRACT FROM AUTHOR]

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 entail that the many-body model is asymptotically isochronous, namely, that its generic solution-when the discrete-time variable ℓ diverges, ℓ → ∞-becomes completely periodic up to exponentially vanishing corrections, with a fixed period independent of the initial data. [ABSTRACT FROM AUTHOR]