Qualitative Aspects of theN-Body Problem for Approximately Relativistic Equations of Motion

Abstract

Following the pioneering work of Jacobi, numerous qualitative results were obtained for dynamical systems, especially on the asymptotic time development of such systems. All these studies were based on Newtonian concepts and treated the basic equations as mathematically exact, whereas even within classical theory they only represent approximations to relativistic equations. In this paper the question of generalizing known Newtonian theorems is investigated for the $N$-body problem on the basis of the Einstein-Infeld-Hoffmann (EIH), Coulomb, and Darwin approximate equations of motion. Proofs of two theorems are given in a manner pointing the way to further generalization in higher order approximations. Two additional theorems are considered for the EIH case utilizing the method of continuous induction; the reason for the failure to prove one of them is discussed, as is the question whether some Newtonian results are invalidated if the basic equations cannot be considered as exact. A number of theorems previously proved for general dynamical systems are shown to hold for the three cases considered before as well as for the canonical formalism of special-relativistic particle interactions, and some limitations of the applicability of such results are discussed.