Title:  Algebraic proof of the non-integrability of Hill's Problem

Authors: Juan J. Morales-Ruiz (1), Carles Sim\'o (2) and Sergi Simon (2)

(1) Departament de Matem\`atica Aplicada II, Universitat Polit\`ecnica
    de Catalunya, Pau Gargallo, 5, 08028 Barcelona, Spain 
    juan.morales-ruiz@upc.es

(2) Departament de Matem\`atica Aplicada i An\`alisi, Universitat de
    Barcelona, Gran Via, 585, 08007 Barcelona, Spain
    carles@maia.ub.es, sergi@mat.ub.es

Abstract

Hill's lunar problem appears in Celestial Mechanics as a limit of the
Restricted Three-Body Problem. Besides, information on the former shows
light on several other three-body problems. It contains no parameters
and is globally far from any simple well--known problem. Strong
numerical evidences of its lack of integrability have been given in the
past. Here an algebraic proof of non--integrability is presented. Beyond
the result in itself, the paper can also be considered as an example of
the application of differential Galois theory to a significant problem.