# On the integrability of Hill's equation of the motion of the moon

## TL;DR

This paper investigates the integrability of Hill's equation of the moon's motion using complex analytic foliations, establishing conditions for the existence of various types of first integrals and proposing a new classification framework.

## Contribution

It introduces the concept of Hill foliation, constructs a Hill fundamental form, and proves the existence of Bessel and Laurent-Fourier type first integrals for complex Hill equations.

## Key findings

- Existence of rational or Liouvillian first integrals in simple cases
- Construction of a Hill fundamental form for the foliation
- Proof of Laurent-Fourier formal first integrals in general cases

## Abstract

We study under the standpoint of integrable complex analytic 1-forms (complex analytic foliations), a class of second order ordinary differential equations with periodic coefficients. More precisely, we study Hill's equations of motion of the moon, which are related to the dynamics of the system Sun-Earth-Moon. We associate to the {\em complex Hill equation} an integrable complex analytic one-form in dimension three. This defines a {\it Hill foliation}. The existence of first integral for a Hill foliation is then studied. The simple cases correspond to the existence of rational or Liouvillian first integrals. We then prove the existence of a {\it Bessel type} first integral in a more general case. We construct a standard two dimensional model for the foliation which we call {\it Hill fundamental form}. This plane foliation is then studied also under the standpoint of reduction of singularities and existence of first integral. For the more general case of the Hill equation, we prove for the corresponding Hill foliation, the existence of a Laurent-Fourier type formal first integral. Our approach suggests that there may be a class of plane foliations admitting Bessel type first integrals, in connection with the classification of (holonomy) groups of germs of complex diffeomorphisms associate to a certain class of second order ODEs.