Solving Linearized Equations of the $N$-body Problem Using the Lie-integration Method

TL;DR

This paper develops recurrence formulas for linearized equations in the N-body problem using Lie-integration, demonstrating a 30-40% speed improvement over existing methods.

Contribution

It introduces simplified recurrence formulas for linearized N-body equations and compares the Lie-integrator's efficiency with other methods.

Findings

Lie-integrator can be 30%-40% faster than other methods

Recurrence formulas simplify linearized equations derivation

Optimal step size and order are determined

Abstract

Several integration schemes exits to solve the equations of motion of the $N$-body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s this method was applied for the $N$-body problem by giving the recurrence formula for the calculation of the Lie-terms. The aim of this works is to present the recurrence formulae for the linearized equations of motion of $N$-body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30%-40% faster than other integration methods.