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Abstract <p>This paper extends the classical Sitnikov problem by considering three primary bodies of equal mass positioned at the vertices of a triangle, each moving in circular orbit around their common center of mass. The equations of motion are derived for an infinitesimal mass constrained to move along the <i>z</i>-axis, with the plane of motion rotating at a constant unit angular velocity. The analysis includes the determination of zero-velocity regions on the <i>xy</i>, <i>yz</i>, and <i>zx</i> planes. The coordinates of the equilibrium points are calculated, and their stability is examined in detail. The study also explores the motion of the fourth body using Poincaré surfaces of section to reveal the system’s dynamic behavior. Over time, the emergence of regular orbital patterns is observed, highlighting the interplay between symmetry and stability in the system’s evolution.</p>

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Mathematical Insights into Triangular Configurations of the Sitnikov Four-Body Dynamics

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • M. Shahbaz Ullah,
  • M. Javed Idrisi

摘要

Abstract

This paper extends the classical Sitnikov problem by considering three primary bodies of equal mass positioned at the vertices of a triangle, each moving in circular orbit around their common center of mass. The equations of motion are derived for an infinitesimal mass constrained to move along the z-axis, with the plane of motion rotating at a constant unit angular velocity. The analysis includes the determination of zero-velocity regions on the xy, yz, and zx planes. The coordinates of the equilibrium points are calculated, and their stability is examined in detail. The study also explores the motion of the fourth body using Poincaré surfaces of section to reveal the system’s dynamic behavior. Over time, the emergence of regular orbital patterns is observed, highlighting the interplay between symmetry and stability in the system’s evolution.