Solving linearized equations of the N-body problem using the Lie-integration method

Several integration schemes exist 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 to solve the equations of motion of the N -body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work 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 per cent faster than other integration methods.