A SecondOrder Solution of the Ideal Resonance Problem by Lie Series
Abstract
A secondorder libration solution of theIdeal Resonance Problem is construeted using a Lieseries perturbation technique. The Ideal Resonance Problem is characterized by the equations <MediaObject> <ImageObject FileRef="10569_2005_Article_BF01227819_TeX2GIFE1.gif" Format="GIF" Color="BlackWhite" Type="Linedraw" Rendition="HTML"/> </MediaObject> begin{gathered}  F = B(x) + 2μ ^2 A(x)sin^2 y, \ dot x =  Fy,dot y = Fx, \ together with the property thatB _{ x } vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initialvalue problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished.
 Publication:

Celestial Mechanics
 Pub Date:
 January 1972
 DOI:
 10.1007/BF01227819
 Bibcode:
 1972CeMec...5....8J