On the Periodic Solutions for the Perturbed Spatial Quantized Hill Problem

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.

Analysis of Equilibrium Points in Quantized Hill System

“…In System (1), the parameters α 1 , α 11 , and α 21 represent very small amounts with order of O(1/c 2 ), but α 22 is of order O(1/c 3 ) where c is the speed of light. Therefore, the value of α 1 − α 11 will tends to zero [27]. Hence α 1 − α 11 ∼ = 0.…”

“…Recently, in [26], the authors investigated the Hill's problem by assuming that the infinitesimal body varies its mass according to Jeans law, they investigated numerically the location of equilibrium points, regions of motion, and basins of attraction and also examined the stability status of these points by using Meshcherskii's space-time transformation. Furthermore, in [27] the authors investigated the differences and similarities among the classical perturbation theory, Poincaré-Lindstedt technique, multiple scales method, the KB averaging method, and averaging theory, while the latter is used to find periodic orbits in the framework of the spatial QHP. They stated that this model can be utilized to develop a lunar theory and families of periodic orbits.…”

Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications

“…Under the choice of T and hypothesis (a), it is clear that f j 1 (T /2, 0, 0, 0) = f j 2 (T /2, 0, 0, 0) = 0 for j = 1, 2. Moreover, by differentiating the system (2.13) with respect to (δL, δG) and evaluating at t = T /2, Y j,k = Y j,k 0 and = 0, we obtain that the Jacobian matrix satisfies (1) ∂δL ∂g (1) ∂δG…”

“…To find the solutions (in the variable G 0 ) of g (1) (T /2, Y 2,k 0 ) = 0, we define the normalized parameter B 1 = −B/γ. Moreover, we introduce the auxiliary functions…”

“…After that, we evaluate ∂ ∂δG ∂H 1 ∂G on the solution ϕ 0 (τ, Y 2,k 0 )). Next, using the change (2.25), after integration and simplification, we arrive at ∂g (1)…”