Contribution to the study of periodic solutions of the Hill equation.

Summary In this work, we are interested in the study of periodic solutions of the Hill equation: q+k q/q#32j q3#2aq=0. We show that for a fixed period t>0, there are at least two periodic anti-t/2 solutions of the Hill equation in the neighborhood of a variety of circular solutions of the Kepler equation, and this for small values of . Then, we relate these anti-t/2 periodic solutions to the circular solutions of the Kepler equation by applying the implicit function theorem. This leads us to a Taylor development of the anti-t/2 periodic solutions of the Hill equation as a function of the parameter and the period t. We use this development to make a numerical experiment whose goal is to verify the theoretical results. We end with a study of the Floquet multipliers of the anti-t/2 periodic solutions of the Hill equation, in particular we show that the Floquet multipliers of the circular solutions of the Kepler equation are all equal to 1.