Study of errors in the integration of the two-body problem using generalized Sundman's anomalies

Abstract

[EN] As is well known, the numerical integration of the two body problem with constant step presents problems depending on the type of coordinates chosen. It is usual that errors in Runge-Lenz's vector cause an artificial and secular precession of the periaster although the form remains symplectic, theoretically, even when using symplectic methods. Provided that it is impossible to preserve the exact form and all the constants of the problem using a numerical method, a possible option is to make a change in the variable of integration, enabling the errors in the position of the periaster and in the speed in the apoaster to be minimized for any eccentricity value between 0 and 1.

The present work considers this casuistry. We provide the errors in norm infinite, of different quantities such as the Energy, the module of the Angular Moment vector and the components of Runge-Lenz's vector, for a large enough number of orbital revolutions.