Fiberwise convexity of Hill’s lunar problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

A new solution of Hill's lunar problem

The two second order differential equations of Hill's lunar problem for rectangular co-ordinates in the rotating system are transformed into a single equation of the fourth order for the geocentric radius vector r, which can be reduced, using Jacobi's integral, to a differential equation of the third order. A discussion of this equation leads to a new exhibition of Hill's “variation orbits”. The first terms of Hill's power series, which represent these periodic solutions of the problem, are exactly confirmed by using a very simple approximative assumption about the mathematical character of the solution.