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<p>We investigate the spatial motion of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the mutual distances between the primaries are small compared to their distance from the particle. By applying a Lie-Deprit normalization, we simplify the Hamiltonian, relegating both the mean anomaly and the argument of periapsis to third-order terms or higher. After reducing the symmetries associated with the Kepler flow and the central action of the angular momentum, we examine the relative equilibria in the first and second reduced spaces. We are able to identify the conditions for the existence of circular periodic orbits and KAM tori, thus providing insight into the system’s long-term stability and dynamical structure.</p>

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Stability, Periodic Orbits and KAM Tori in the Dynamics of the Three Fixed Centers Problem

  • 期刊论文

  • 发表日期：

    2025年12月22日

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  • Edward A. Turner,
  • Francisco Crespo,
  • Jhon Vidarte,
  • Jersson Villafañe,
  • Jorge L. Zapata

摘要

We investigate the spatial motion of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the mutual distances between the primaries are small compared to their distance from the particle. By applying a Lie-Deprit normalization, we simplify the Hamiltonian, relegating both the mean anomaly and the argument of periapsis to third-order terms or higher. After reducing the symmetries associated with the Kepler flow and the central action of the angular momentum, we examine the relative equilibria in the first and second reduced spaces. We are able to identify the conditions for the existence of circular periodic orbits and KAM tori, thus providing insight into the system’s long-term stability and dynamical structure.