SOLVABLE (NONRELATIVISTIC, CLASSICAL) N-BODY PROBLEMS IN MULTIDIMENSIONS .1.

Several solvable n-body problems are exhibited. They are characterized by equations of motion of Newtonian type, m(j) r(j) over arrow pointi ng right = F(j) over arrow pointing right, j = 1,...,n, with the ''for ces'' F(j) over arrow pointing right given as explicit functions of th e ''particle coordinates'' r(k) over arrow pointing right and their ve locities r(k) over arrow pointing right, k = 1,...,n; the forces F(j) over arrow pointing right generally also depend on some free parameter s, so that in each case there is actually a class of solvable models. The particle coordinates r(j) over arrow pointing right, and the force s F(j) over arrow pointing right, are vectors in N-dimensional space, with N = 1,2,3. In this article we focus mainly on few-body problems ( n = 1,2,3,4) in two- and three-dimensional space (N = 2,3); all these models are rotation-invariant, and some are also translation-invariant , but they are generally not Galilei-invariant. The solution of the in itial-value problem is given in each case. We also exhibit one-dimensi onal problems (N = 1), including cases with n>4; these models are gene rally translation-invariant, and some are also Galilei-invariant.