Introduction

In the preceding chapter, we have considered several methods for dealing with the perturbed two-body problem. Formally all these methods work or quite large perturbations, provided the regularized time-step is chosen sufficiently small. However, the selection of the dominant pair in a triple resonance interaction frequently calls for new initializations where the intruder is combined with one of the components. Likewise, one may have situations in which two hard binaries approach each other with small impact parameter. Hence a description in terms of one dominant two-body motion tends to break down during strong interactions, precisely at times when interesting outcomes are likely to occur. Since the switching of dominant components reduces the efficiency and also degrades the quality of the results, it it highly desirable to seek alternative methods for improved treatment.

In this chapter, we discuss several multiple regularization methods that have turned out to be very beneficial in practical simulations. By multiple regularization it is understood that at least two separations in a compact subsystem are subject to special treatment where the two-body singularities are removed. We begin by describing a three-body formulation that may be considered the Rosetta Stone for later developments. The generalization to more members with just one reference body is also included for completeness. A subsequent section outlines the elegant global formulation and is followed by a detailed discussion of the powerful chain regularization.