N-Body Systems*

Abstract

We shall now model systems of N bodies interacting through mechanical forces that result from springs and dashpots, see Fig. 48.1, or from gravitational or electrostatic forces. We shall use two different modes of description. In the first formulation, we describe the system through the coordinates of (the centers of gravity of) the bodies. In the second, we use the displacements of the bodies measured from an initial reference configuration. In the latter case, we also linearize under an assumption of small displacements to obtain a linear system of equations. In the first formulation, the initial configuration is only used to initialize the system and is “forgotten” at a later time in the sense that the description of the system only contains the present position of the masses. In the second formulation, the reference configuration is retrievable through the evolution since the unknown is the displacement from the reference position. The different formulations have different advantages and ranges of application.

The reader will find no figures in this work. The methods which I set forth do not require either geometrical or mechanical reasonings, but only algebraic operations, subject to a regular and uniform rule of procedure. (Lagrange in Méchanique Analytique)