Solvable (nonrelativistic, classical) n-body problems on the line. I

Abstract

Several solvable n-body problems on the line are exhibited. They are characterized by equations of motion of Newtonian type, mjẍj=Fj, j=1,...,n, with the ‘‘forces’’ Fj given as explicit functions of the ‘‘particle coordinates’’ xk and their velocities ẋk, 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.