Abstract
This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.
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References
Arnold V.: Proof of A. N. Kolmogorov’s theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian. Russian Math. Surveys 18(5), 9–36 (1963)
Arnold V.: Small denominators and the problems of stability of motion in classical and celestial mechanics. Usp. Matem. Nauka 18, 6 (1963)
Alfriend K.T., Richardson D.L.: Third and fourth order resonances in Hamiltonian systems. Celestial Mechanics 7, 408–420 (1973)
Cabral H.E., Meyer K.R.: Stability of equilibria and fixed points of conservative systems. Nonlinearity 12, 1351–1362 (1999)
Chetaev N.: The Stability of Motion. Pergamom Press, New York (1961)
Elipe A., Lanchares V., López-Moratalla T., Riaguas A.: Nonlinear stability in resonant cases: a geometrical approach. J. Nonlinear Sci 11(3), 211–222 (2001)
Khazin L.G.: On the stability of Hamiltonian systems in the presence of resonances. App. Math. and Mech 35, 384–391 (1971)
Markeev A.P.: On the stability of a non-autonomous Hamiltonian system with two degrees of freedom. Prikl. Mat. Mekh 34(6), 563–569 (1969)
Markeev A.P.: Librations points in Celestial Mechanics and space dynamics. Nauka, Moscow (1978) (in Russian)
Markeev A.P.: The problem of the stability of the equilibrium position of a Hamiltonian system at 3:1 resonance. J. Appl. Maths. and Mechs 65, 639–645 (2001)
Meyer, K., Hall, G.R.: Introduction to Hamiltonian dinamical systems and the n-body problem. Vol. 90, New York. Springer-Verlag. Applied mathematical science (1992)
Meyer K., Schmidt D.: The stability of the Lagrange triangular point and a theorem of Arnold. J. of Diff. Eqs 62(2), 222–236 (1986)
Moser J.: New aspects in the theory of stability of Hamiltonian systems. Comm. Appl. Math 11(1), 81–114 (1958)
Moser J.: Lectures on Hamiltonian system. Mem. Amer. Math 81, 10–13 (1963)
Siegel C., Moser J.: Lectures on Celestial Mechanics. Springer, New York (1971)
Vidal C., dos Santos F.: Stability of equilibrium positions of periodic Hamiltonian systems umder third and fourth order resonances. Regular and Chaotic Dynamics, Moscow 10(01), 95–110 (2005)
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C. Vidal was partially supported by Fondecyt 1080112.
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dos Santos, F., Mansilla, J.E. & Vidal, C. Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with n-Degrees of Freedom in the Case of Single Resonance. J Dyn Diff Equat 22, 805–821 (2010). https://doi.org/10.1007/s10884-010-9176-z
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DOI: https://doi.org/10.1007/s10884-010-9176-z
Keywords
- Hamiltonian system
- Autonomous case
- Periodic case
- Equilibrium solution
- Type of stability
- Normal form
- Resonances
- Lyapunov’s theorem
- Chetaev’s theorem