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<p>We consider a pendulum attached to a satellite describing an elliptical orbit around a Newtonian center of attraction, where the rod of the pendulum varies proportionally to the radius vector of the orbit. The dynamics is described by a periodic one degree-of-freedom Hamiltonian system. There are two stable equilibrium positions, independent of the parameters of the system, namely, the length of the pendulum and the eccentricity of the orbit. The linear stability of these equilibria was studied in Burov (Adv Astronaut Sci 142: 3495–3507, 2012), Menezes Neto (Regul Chaotic Dyn 25: 323–329, 2020). Here we consider the question of their nonlinear stability. To this end, we use the Deprit–Hori method and Kolmogorov–Arnold–Moser Theory (KAM Theory), following the techniques developed by Markeev As reported by Markeev (Libration Points in Celestial Mechanics and Space Dynamics, Nauka, Moscow, 1978), Markeev (Linear Hamiltonian Systems and Some Problems on Stability of Motion of a Satellite About its Center of Mass, Institute of Computer Science, Izhevsk, 2009), Markeev (Mech Solids 39: 1–8, 2004), to normalize the Hamiltonian function and draw conclusions about the nonlinear stability.</p>

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Nonlinear stability of a pendulum with variable length in elliptic orbit

  • 期刊论文

  • 发表日期：

    2025年3月3日

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  • José Laudelino de Menezes Neto,
  • Hildeberto Eulálio Cabral

摘要

We consider a pendulum attached to a satellite describing an elliptical orbit around a Newtonian center of attraction, where the rod of the pendulum varies proportionally to the radius vector of the orbit. The dynamics is described by a periodic one degree-of-freedom Hamiltonian system. There are two stable equilibrium positions, independent of the parameters of the system, namely, the length of the pendulum and the eccentricity of the orbit. The linear stability of these equilibria was studied in Burov (Adv Astronaut Sci 142: 3495–3507, 2012), Menezes Neto (Regul Chaotic Dyn 25: 323–329, 2020). Here we consider the question of their nonlinear stability. To this end, we use the Deprit–Hori method and Kolmogorov–Arnold–Moser Theory (KAM Theory), following the techniques developed by Markeev As reported by Markeev (Libration Points in Celestial Mechanics and Space Dynamics, Nauka, Moscow, 1978), Markeev (Linear Hamiltonian Systems and Some Problems on Stability of Motion of a Satellite About its Center of Mass, Institute of Computer Science, Izhevsk, 2009), Markeev (Mech Solids 39: 1–8, 2004), to normalize the Hamiltonian function and draw conclusions about the nonlinear stability.