Relation between various formulations of perturbation equations of celestial mechanics

TL;DR

This paper compares two formulations of perturbation equations in celestial mechanics, demonstrating that one can be derived from the other through simple mathematical operations, thus clarifying their relationship.

Contribution

It shows that the second set of perturbation equations can be derived from the first set using straightforward mathematical steps, linking different formulations.

Findings

The second formulation can be obtained from the first using simple operations.

Clarifies the relationship between different perturbation equations.

Provides a unified understanding of perturbation equations in celestial mechanics.

Abstract

Orbital motion of a body can be found from Newtonian equation of motion. However, it is useful to express the motion through time derivatives of Keplerian orbital elements, mainly if the motion is perturbed by small perturbing force. The first set of equations for the time derivatives of the orbital elements can be derived from the equation of motion using Lagrange brackets. The second one by using equation of motion and perturbation acceleration decomposed to radial, transversal and normal components. This paper shows that the second type of the perturbation equations can be derived from the first type using simple mathematical operations.