Abstract
We consider a non-autonomous Hamiltonian system with two degrees of freedom, whose Hamiltonian function is a 2π-periodic function of time and is analytic in the neighborhood of an equilibrium point. It is assumed that the system exhibits a first-order resonance, i.e., the linearized system in the neighborhood of the equilibrium point has a unit multiplier of multiplicity two. The case of the general position is considered when the monodromy matrix is not reduced to the diagonal form, and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system. In this paper, a constructive algorithm for the rigorous-stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed by Markeev. The sufficient conditions for the instability of the equilibrium position, as well as the conditions for its formal stability and stability in the third approximation, are expressed in terms of the coefficients of the normalized map. Explicit formulas are obtained that allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of the symplectic map. The developed algorithm is used to solve the problem of the stability of the resonant rotation of a symmetric satellite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Markeyev, A.P., A constructive algorithm for the normalization of a periodic hamiltonian, J. Appl. Math. Mech. (Engl. Transl.), 2005, vol. 69, no. 3, pp. 323–337.
Lyapunov, A.M., General problem on motion stability, in Sobranie sochinenii (Collection of Scientific Works), Moscow, Leningrad: USSR Acad. Sci., 1956, vol. 2, pp. 7–263.
Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966.
Markeev, A.P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike (Libration Points in Celestial Mechanics and Cosmo-Dynamics), Moscow: Nauka, 1978.
Birkhoff, G.D., Dynamical Systems, New York: American Mathematical Society, 1927.
Giacaglia, G.E.O., Perturbation Methods in Non-linear Systems, New York: Springer, 1972.
Bardin, B.S. and Chekina, E.A., On the constructive algorithm for stability analysis of an equilibrium point of a periodic Hamiltonian system with two degrees of freedom in the second-order resonance case, Regular Chaotic Dyn., 2017, vol. 22, no. 7, pp. 808–824.
Bardin, B.S. and Chekina, E.A., On the stability of planar oscillations of a satellite-plate in the case of essential type resonance, Rus. J. Nonlinear Dyn., 2017, vol. 13, no. 4, pp. 465–476.
Ivanov, A.P. and Sokol’skii, A.G., On the stability of a nonautonomous hamiltonian system under a parametric resonance of essential type, J. Appl. Math. Mech. (Engl. Transl.), 1980, vol. 44, no. 6, pp. 687–691.
Markeev, A.P., Lineinye gamil’tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel’no tsentra mass (Linear Hamiltonian Systems and Some Problems on Stability of Satellite Motion Relatively Center of Mass), Moscow, Izhevsk: Regulyarnaya i Haoticheskaya Dinamika, 2009.
Bardin, B.S and Chekina, E.A., On stability of resonant rotation of dynamically symmetrical satellite in the plane of elliptic orbit, Tr. Inst.–Mosk. Aviats. Inst. im. Sergo Ordzhonikidze, 2016, no. 89.
Beletskii, V.V. and Shlyakhtin, A.N., Resonance satellite rotations under interaction between magnetic and gravitational fields, Preprint of Keldysh Institute of Applied Mathematics, USSR Acad. Sci., Moscow, 1980, no. 46.
Khentov, A.A., On one rotating motion of satellite, Kosm. Issled., 1984, vol. 22, no. 1, pp. 130–131.
Markeev, A.P. and Bardin, B.S., Planar rotating motions of satellite in elliptic orbit, Kosm. Issled., 1994, vol. 32, no. 6, pp. 43–49.
Bardin, B.S., Chekina, E.A., and Chekin, A.M., On stability of a planar rotation of a satellite in an elliptic orbit, Regular Chaotic Dyn., 2015, vol. 20, no. 1, pp. 63–73.
Bardin, B.C. and Chekina, E.A., On the stability of a resonant rotation of a satellite in an elliptic orbit, Rus. J. Nonlinear Dyn., 2016, vol. 12, no. 4, pp. 619–632.
Bardin, B.S. and Chekina, E.A., On stability of a resonant rotation of a symmetric satellite in an elliptic orbit, Regular Chaotic Dyn., 2016, vol. 21, no. 4, pp. 377–389.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © B.S. Bardin, E.A. Chekina, 2018, published in Prikladnaya Matematika i Mekhanika, 2018, Vol. 82, No. 4, pp. 414–426.
About this article
Cite this article
Bardin, B.S., Chekina, E.A. On the Constructive Algorithm for Stability Investigation of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in First-Order Resonance Case. Mech. Solids 53 (Suppl 2), 15–25 (2018). https://doi.org/10.3103/S0025654418050023
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0025654418050023