Approximation Solution of the Nonlinear Circular Sitnikov Restricted Four–Body Problem

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.

The Existence of Odd Symmetric Periodic Solutions in the Generalized Elliptic Sitnikov (N+1)-Body Problem

“…Since then, many other authors have studied this problem. We refer the reader to [2][3][4][5][6][7][8][9][10][11][12][13] and the references therein for a more detailed introduction.…”