Higher order approximation to the Hill problem dynamics about the libration points

Abstract

An analytical solution to the Hill problem Hamiltonian expanded about the libration points has been obtained by means of perturbation techniques. In order to compute the higher orders of the perturbation solution that are needed to capture all the relevant periodic orbits originated from the libration points within a reasonable accuracy, the normalization is approached in complex variables. The validity of the solution extends to energy values considerably far away from that of the libration points and, therefore, can be used in the computation of Halo orbits as an alternative to the classical Lindstedt-Poincaré approach. Furthermore, the theory correctly predicts the existence of the twolane bridge of periodic orbits linking the families of planar and vertical Lyapunov orbits. * Commun Nonlinear Sci Numer Simulat (2018) https://doi.

Complex variables approach to the short-axis-mode rotation of a rigid body

“…For perturbed harmonic oscillators, it is well known that the use of complex variables makes the whole perturbation approach very efficient, cf. [31][32][33]. Therefore, the perturbation Hamiltonian given by Eqs.…”

“…In these variables, the free rigid body Hamiltonian takes the form of the Hamiltonian of the simple harmonic oscillator disturbed by additional quartic polynomial terms. Hence, since it is well known that the use of complex variables renders very efficient the construction of higher order analytical solutions to perturbed harmonic oscillators (see, for instance, [31][32][33]), an additional transformation to complex variables is carried out in Section 3 that converts the integration of the free rigid body Hamiltonian by perturbations into a simple exercise of polynomial algebra.…”

A Quantized Hill’s Dynamical System

“…A considerable study on the circular Hill's problem has been established by Hénon [16,17], where he determined the main families of periodic orbits, revealed the phase space by means of surface of sections of the Poincaré map, and found the stability regions in the parameters' plane. Recently, for the same problem, Lara et al in [18] have employed a normalization approach in complex variables to compute a single perturbation solution with enough accuracy. e solution captures the main four families of periodic orbits for the Hill problem originated from the libration points.…”

Dynamics Near the Three-Body Libration Points via Koopman Operator Theory

“…to Lindstedt-Poincaré method, Lie-Deprit methods have been successfully applied to the central gravity and RTBP problems [3], with extensive applications, from reduction center manifold [4] to long term propagation [5]. Particularly, the evaluation of periodic orbits around the libration points is quite efficient when approached in complex variables [6].…”