Abstract
In this work, we investigated the differences and similarities among some perturbation approaches, such as the classical perturbation theory, Poincaré–Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory. The necessary conditions to construct the periodic solutions for the spatial quantized Hill problem—in this context, the periodic solutions emerging from the equilibrium points for the spatial Hill problem—were evaluated by using the averaging theory, under the perturbation effect of quantum corrections. This model can be used to develop a Lunar theory and the families of periodic orbits in the frame work for the spatial quantized Hill problem. Thereby, these applications serve to reinforce the obtained results on these periodic solutions and gain its own significance.
Highlights
Accepted: 14 February 2022A three-body problem plays a vital role in space science; in particular, the related fields of solar system motions, stars, planets, and their moons
Periodic solutions of a dynamical system are solutions that characterize some repeated phenomena identically at regular intervals. These solutions play a vital role in many branches of science, such as physics and engineering, but appear in celestial mechanics, to study the dynamical structures of the two-body problem [6,7]; an analysis the infinitesimal body motion within the frame of the three-body problem [8,9] or N-body problem [10,11,12]
We aimed to find the periodic solutions of the dynamical system of the spatial quantized Hill problem
Summary
A three-body problem plays a vital role in space science; in particular, the related fields of solar system motions, stars, planets, and their moons. Periodic solutions of a dynamical system are solutions that characterize some repeated phenomena identically at regular intervals These solutions play a vital role in many branches of science, such as physics and engineering, but appear in celestial mechanics, to study the dynamical structures of the two-body problem [6,7]; an analysis the infinitesimal body motion within the frame of the three-body problem [8,9] or N-body problem [10,11,12]. We evaluated the equilibria points of the linear system, and the necessary conditions were used to calculate the periodic solutions arising from the equilibria points for the spatial quantized Hill problem, by using the averaging theory This system was constructed for the first time by Abouelmagd et al (2020) [5]; this motivated us to study the dynamical structures of this system, through finding its own periodic solutions
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