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

Abstract

Hill's equations, which first arose in the study of the Earth-Moon-Sun system, admit the twoparameter centrally extended Newton-Hooke symmetry without rotations. This symmetry allows us to extend 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. PACS numbers: 11.30.-j, 02.40.Yy, 02.20.Sv, 96.12.De, Phys. Rev. D85 (2012) 045031

Newton–Hooke-type symmetry of anisotropic oscillators

“…Then, for approximately circular orbits, the first-order approximation to Newton's gravitational equations provides us with Hill's equations [6,12,29],…”

“…In the Hill case, rotation-less "Newton-Hooke type" symmetry, with one (or, in the "exotic" case, with two) central extensions could be established [6,7].…”

“…Yet another example was found recently, however, namely for Hill's equations [6,7]. Remember that Hill originally devised his method for finding approximate solutions of the three-body problem, and in particular for the Moon-Earth-Sun system [8,9].…”

“…Recently, the method was extended to the Hill problem [6,7] which is in fact a "maximally anisotropic oscillator"; here we further extend it to arbitrary anisotropy.…”

“…From the technical point of view, we will find it convenient to use chiral decomposition [6,7,19,[23][24][25]. The motion in the (ordinary) Hall effect can in fact be decomposed into two uncoupled chiral oscillators with opposite chirality [23].…”