A nonautonomous yet solvable discrete-time N-body problem

Abstract

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.