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Session S03 - Geometric and Variational Methods in Celestial Mechanics

Monday, July 12, 13:15 ~ 13:40 UTC-3

Nonlinear stability of equilibria: The satellite problem

Patricia Yanguas

Universidad Pública de Navarra, Spain   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We study the nonlinear stability of elliptic equilibria in Hamiltonian systems. Specifically, a kind of formal stability, the so-called Lie stability, is described. The concept of Lie stability appears as a strong alternative in cases where a sort of nonlinear stability is required but Liapunov stability cannot be ensured. Moreover, Lie stability is obtained even for some Hamiltonian systems that do not satisfy Nekhoroshev's theory necessary conditions [1].

The determination of Lie stability passes through the obtention of the required convexity of the Hamiltonian function restricted to a certain subspace that is contained in the orthogonal space related to the frequency vector.

In the Lie stable cases, the error estimates of the solutions over exponentially long times are obtained through a result based in the determination of error bounds for adiabatic invariants in Hamiltonian systems.

As an application we analyse the attitude nonlinear stability of the spatial satellite problem and enlarge previous results by Markeev and Sokol'skii [2]. Furthermore, KAM tori related to Lie stable, as well as unstable equilibria, are also calculated [3].

[1] Nonlinear stability of elliptic equilibria in Hamiltonian systems with exponential time estimates, D. Cárcamo-Díaz,  J.F. Palacián, C. Vidal, P. Yanguas, accepted in Discrete & Continuous Dynamical Systems (2021).

[2] On the stability of relative equilibrium of a satellite in a circular orbit, A.P. Markeev, A.G. Sokol'skii, Kosmicheskie Issledovaniya, 13(2), 139-146 (1975); Cosm. Res., 13(2), 119-125 (1975).

[3] Nonlinear stability in the spatial attitude motion of a satellite in a circular orbit, D. Cárcamo-Díaz,  J.F. Palacián, C. Vidal, P. Yanguas, preprint (2021).

Joint work with Daniela Cárcamo-Díaz (Universidad del Bío-Bío, Chile), Jesús F. Palacián (Universidad Pública de Navarra, Spain) and Claudio Vidal (Universidad del Bío-Bío, Chile).

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