Mathieu equation as a result of Laplace perturbation theory in the restricted three body problem

TL;DR

This paper explores the application of Laplace perturbation theory to the restricted three-body problem, deriving a linear equation with periodic coefficients and proposing a modification for longer-term analysis.

Contribution

It introduces a modified Laplace method that extends the analysis of the RTBP over longer time intervals, enhancing classical perturbation approaches.

Findings

Derivation of a linear equation with periodic coefficients for RTBP

Development of a modified Laplace method for extended analysis

Improved understanding of long-term planetary motion dynamics

Abstract

Linear equations with periodic coefficients describe the behavior of various dynamical systems. This studying is devoted to their applications to the planetary restricted three-body problem (RTBP). Here we consider the Laplace method for determining perturbation in coordinates. We show that classical theory of perturbation leads to a linear equation with periodic coefficients. Than we present a modification of Laplace method. This modification allows us to study motion over a longer time interval.