Abstract
In this paper, the approximated periodic solutions of the circular Sitnikov restricted four–body problem (RFBP) were constructed using the Lindstedt–Poincaré method, by removing the secular terms, and compared with numerical solution. It can be observed that, in the numerical as well as approximated solutions patterns, the initial conditions are important. In the sense of a numerical solution, the motion is periodic in a certain interval, but beyond this interval, the motion is not periodic. But, the Lindstedt–Poincaré method constantly gives regular and periodic motion all time. Finally, we observed that the solution obtained by the Lindstedt–Poincaré method gives the true motion of the circular Sitnikov RFBP and the fourth approximate solution has more accuracy than the first, second, and third approximate solutions.
Highlights
The dynamical system of the restricted three-body problem (RTBP) has a significant role in celestial mechanics
The RTBP is a special case of three-body problem whereas the restricted four-body (RFBP) problem is a generalization of the RTBP
The investigation includes the numerical solution of Equation (5) and the first, second, third and fourth-order approximated solutions of Equation (10) obtained using the Lindstedt–Poincaré method which are given in Equations (45) and (46), respectively
Summary
The dynamical system of the restricted three-body problem (RTBP) has a significant role in celestial mechanics. It has many applications in the astrodynamics and stellar dynamics fields [1,2,3,4]. In the RTBP, both primaries circumambulate around their common center of mass while the infinitesimal body does not have gravitational influence on the primaries bodies. Several studies have been carried out on the RTBP to analyze infinitesimal body motion [5,6,7,8]. Further considerable work in the frame of perturbed RTBP was undertaken in [9,10] to explore the equilibrium points, linear stability and feature of motion about these points
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