Acta Applicandae Mathematicae 2, 185-207. 0167-8019/84/0022-0185503.45 185 © 1984 by D. Reidel Publishing Company. Discrete Mechanics and its Application to the Solution of the n-Body Problem ANDRZEJ MARCINIAK Department of Numerical Methods, Institute of Mathematics, A. Mickiewicz University, Pozna~, Poland (Received: 2 June 1982; revised: 14 December 1983) Abstract. A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented. AMS (MOS) subject classifications (1980). 70.65, 70FI0, 65L05, 65M10. Key words. Discrete mechanics, n-body problem, numerical solutions of n-body problem, stability and convergence of discrete mechanics methods. 1. Introduction Discrete mechanics is a form of mechanics adapted to calculate the possibilities of modern digital computers. The symmetry with respect to translation and rotation and under the uniform motion of the frame of references and a law of energy conservation for the difference dynamical equations of this mechanics are its fundamental properties. The basic concepts were given by Greenspan [6-13 ], who defined the fundamental equations of the mechanics and the discrete form of the n-body force law. Moreover, Greenspan proved the symmetry mentioned above [ 10] and the law of energy conser- vation [7, 13]. In this paper it is shown how Greenspan's discrete n-body force law can be obtained from an adequate continuous formula. In view of the practical applications, e.g., to the study of a planetary system's motion an orbit calculation near to the equilibrium points [2], and in order to obtain the discrete Hill equations [15], from Greenspan equations, the adequate formulas for related motion of n-bodies and for the motion in the rotating frame are worked out. In Section 4, based Stetter's [ 18] theory, the problems of consistency, stability, and convergence of the discrete mechanics equations are studied. Finally, some numerical results are presented and compared with the results obtained from conventional me- thods of a similar order.