Instability of Equilibrium Solutions of Hamiltonian Systems with n-Degrees of Freedom Under the Existence of Multiple Resonances and an Application to the Spatial Satellite Problem
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In this paper, we prove the instability of one equilibrium point in a Hamiltonian system with n-degrees of freedom under two assumptions: the first is the existence of multiple resonance of odd order s (without resonance of lower order) but with the possible existence of resonance of higher order; and the second is the existence of an invariant ray solution of the truncated Hamiltonian system up to order s. It is shown that in the case of resonance without interaction, the necessary conditions for instability have important simplifications with respect to the general case. Examples in three, four and six degrees of freedom are given. An application of our main result to the spatial satellite problem is considered.
KeywordsHamiltonian system Equilibrium solution Stability Normal form Resonance Invariant ray solution Chetaev’s theorem
Mathematics Subject Classification37C75 34D20 34A25
The authors would like to thank the referee for valuable comments, which improved an earlier version of this paper. Claudio Vidal was partially supported by project Fondecyt 1180288. This paper is part of the Daniela Cárcamo-Díaz Ph.D. thesis in the Program Doctorado en Matemática Aplicada, Universidad del Bío-Bío. Daniela Cárcamo-Díaz acknowledges funding from CONICYT PhD/2016-21161143.
- 2.Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Encyclopaedia of Mathematical Sciences, 3rd edn. Springer, Berlin (2006)Google Scholar
- 10.Khazin, L., Shnol, E.: Stability of the equilibrium states (Translated from the Russian). Nonlinear Science: Theory and Applications. Manchester University Press, Manchester (1991)Google Scholar
- 14.Markeev, A.P.: Librations Points in Celestial Mechanics and Space Dynamics (in Russian). Nauka, Moscow (1978)Google Scholar
- 15.Markeev, A.P., Sokolskii, A.G.: On the Stability of Relative Equilibrium of a Satellite in a Circular Orbit. Nauka, Moscow (1975)Google Scholar
- 19.Moser, J.: Lectures on Hamiltonian system. Mem. Am. Math. 81, 10–13 (1963)Google Scholar