Kohn's theorem and Newton-Hooke symmetry for Hill's equations

TL;DR

This paper explores the symmetry properties of Hill's equations, revealing an extended Newton-Hooke symmetry that facilitates a generalized Kohn's theorem, using Duval's Bargmann framework to analyze center-of-mass separation.

Contribution

It demonstrates the presence of a two-parameter extended Newton-Hooke symmetry in Hill's equations and extends Kohn's theorem within this framework, employing a generalized chiral decomposition.

Findings

Extended Newton-Hooke symmetry without rotations

Generalized Kohn's theorem for Hill's equations

Center-of-mass separation via chiral decomposition

Abstract

Hill's equations, which first arose in the study of the Earth-Moon-Sun system, admit the two-parameter centrally extended Newton-Hooke symmetry without rotations. This symmetry allows for extending Kohn's theorem about the center-of-mass decomposition. Particular light is shed on the problem using Duval's "Bargmann" framework. The separation of the center-of-mass motion into that of a guiding center and relative motion is derived by a generalized chiral decomposition.