On the periodic orbits and the integrability of the regularized Hill lunar problem

Abstract

The classical Hill's problem is a simplified version of the restricted three-body problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill's variables. The Levi-Civita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C 1 integrability or non-integrability of the regularized Hill lunar problem. C

Fiberwise convexity of Hill’s lunar problem

“…Morales-Ruiz, Simó and Simon gave an algebraic proof of meromorphic non-integrability in [15]. Recently, Llibre and Roberto in [11] discussed the C 1 integrability. With these results about non-integrability, Simó and Stuchi in their numerical research [17] pointed out the importance of 'numerical methods guided by the geometry of dynamical systems' by observing the chaotic behavior in the Levi-Civita regularization of Hill's lunar problem.…”

Doubly-symmetric periodic orbits in the spatial Hill's lunar problem with oblate secondary primary

“…The Hill's lunar problem is studied by both analytical and numerical methods, see Michalodimitrakis (1980) [2], Howision & Meyer (2000I, 2000II) [3,4], Maciejewski & Rybicki (2001) [5], Llibre & Roberto (2011) [6], Belbruno et.al. (2019) [7].…”

Spectrum estimates of Hill's lunar problem

“…Morales-Ruiz, Simó and Simon gave an algebraic proof of meromorphic non-integrability in [25]. Recently, Llibre and Roberto in [21] discussed the C 1 integrability based on the existence of two periodic orbits on every positive energy level. One can see the chaotic feature of Hill's lunar problem in the numerical research of Simó and Stuchi in [30].…”