Integrability of the generalised Hill problem

Thierry Combot,Maria Przybylska,Andrzej J Maciejewski

doi:10.1007/s11071-021-07040-8

Thierry Combot, Maria Przybylska + Show 1 more

Open Access

https://doi.org/10.1007/s11071-021-07040-8

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Abstract

We consider a certain two-parameter generalisation of the planar Hill lunar problem. We prove that for nonzero values of these parameters the system is not integrable in the Liouville sense. For special choices of parameters the system coincides with the classical Hill system, the integrable synodical Kepler problem or the integrable parametric Hénon system. We prove that the synodical Kepler problem is not super-integrable, and that the parametric Hénon problem is super-integrable for infinitely many values of the parameter.

Highlights

We consider a certain version of the planar circular Hill problem
The Hill problem was developed by George Hill, see [2] in order to construct the theory of motion of the Moon in the Sun–Earth–Moon system
A pseudoNewtonian Hill problem was investigated in [4,5,6]. In this model the Newtonian potential is replaced by the Paczynski-Wiita potential [7]. It can be considered as the zeroth order of the general relativistic Hill problem

Summary

Introduction

We consider a certain version of the planar circular Hill problem. In the classical formulation the Hill problem is a limiting case of the restricted three. A quite general method of Hill’s type approximation of the equations of motion is described in [8] Another generalisation of the Hill problem was considered in [9]. We consider a generalised Hill problem proposed by [11,12] and described by the following Hamiltonian function. We analyse super-integrability of the parametric Hénon system and the synodical Kepler problem. We show that the parametric Hénon system for infinitely many rational values of parameter ε appears to be super-integrable and this problem is discussed in Sect. 5, the problem of existence of one more functionally independent first integral for the integrable parametric Hénon system and the synodical Kepler problem is analysed

General properties of the dynamics of the system

Regularised equations of motion

Integrability analysis

The parametric Hénon case

The synodical Kepler case