 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4760 |
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| B.S. Bardin and E.A. Chekina, "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 (S2), 15-25 (2018) |
| Year |
2018 |
Volume |
53 |
Number |
S2 |
Pages |
15-25 |
| DOI |
10.3103/S0025654418050023 |
| Title |
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 |
| Author(s) |
B.S. Bardin (Moscow Aviation Institute (National Research University), Moscow, 125993 Russia; Mechanical Engineering Research Institute of the Russian Academy of Sciences, Moscow, 101990 Russia, bsbardin@yandex.ru)
E.A. Chekina (Moscow Aviation Institute (National Research University), Moscow, 125993 Russia, chekina_ev@mail.ru) |
| 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. |
| Keywords |
Hamiltonian system, stability, resonance of essential type, symplectic map, normalization, resonant rotation of a symmetric satellite |
| Received |
26 July 2017 |
| Link to Fulltext |
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