Abstract
A general method is presented for constructing a nonlinear canonical transformation, which makes it possible to introduce local variables in a neighborhood of periodic motions of an autonomous Hamiltonian system with two degrees of freedom. This method can be used for investigating the behavior of the Hamiltonian system in the vicinity of its periodic trajectories. In particular, it can be applied to solve the problem of orbital stability of periodic motions.
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Notes
The solution (2.2) can be stable in the sense of Lyapunov in the special case (the so-called isochronic case), when the period of the periodic solution does not depend on initial conditions. For instance, such a special case takes place for periodic motions of the harmonic oscillator.
References
Poincaré, H., Les méthodes nouvelles de la mécanique céleste: T. 2. Méthodes de Newcomb, Glyden, Lindstedt et Bohlin, Paris: Gauthier-Villars, 1892.
Birkhoff, G. D., Dynamical Systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Providence, R.I.: AMS, 1966.
Markeev, A. P., Stability of Plane Oscillations and Rotations of a Satellite in a Circular Orbit, Cosmic Research, 1975, vol. 13, no. 3, pp. 285–298; see also: Kosmicheskie Issledovaniya, 1975, vol. 13, no. 3, pp. 322-336.
Markeev, A. P. and Bardin, B. S., On the Stability of Planar Oscillations and Rotations of a Satellite in a Circular Orbit, Celest. Mech. Dynam. Astronom., 2003, vol. 85, no. 1, pp. 51–66.
Markeyev, A. P., The Stability of the Plane Motions of a Rigid Body in the Kovalevskaya Case, J. Appl. Math. Mech., 2001, vol. 65, no. 1, pp. 47–54; see also: Prikl. Mat. Mekh., 2001, vol. 65, no. 1, pp. 51-58.
Bardin, B. S., On the Orbital Stability of Pendulum-Like Motions of a Rigid Body in the Bobylev – Steklov Case, Regul. Chaotic Dyn., 2010, vol. 15, no. 6, pp. 702–714.
Bardin, B. S., Rudenko, T. V., and Savin, A. A., On the Orbital Stability of Planar Periodic Motions of a Rigid Body in the Bobylev – Steklov Case, Regul. Cahotic Dyn., 2012, vol. 17, no. 6, pp. 533–546.
Bardin, B. S. and Savin, A. A., On the Orbital Stability of Pendulum-Like Oscillations and Rotations of a Symmetric Rigid Body with a Fixed Point, Regul. Chaotic Dyn., 2012, vol. 17, no. 3–4, pp. 243–257.
Bardin, B. S. and Savin, A. A., The Stability of the Plane Periodic Motions of a Symmetrical Rigid Body with a Fixed Point, J. Appl. Math. Mech., 2013, vol. 77, no. 6, pp. 578–587; see also: Prikl. Mat. Mekh., 2013, vol. 77, no. 6, pp. 806-821.
Markeyev, A. P., An Algorithm for Normalizing Hamiltonian Systems in the Problem of the Orbital Stability of Periodic Motions, J. Appl. Math. Mech., 2002, vol. 66, no. 6, pp. 889–896; see also: Prikl. Mat. Mekh., 2002, vol. 66, no. 6, pp. 929-938.
Bardin, B. S., On a Method of Introducing Local Coordinates in the Problem of the Orbital Stability of Planar Periodic Motions of a Rigid Body, Russian J. Nonlinear Dyn., 2020, vol. 16, no. 4, pp. 581–594.
Bardin, B. S. and Chekina, E. A., On the Orbital Stability of Pendulum-Like Oscillations of a Heavy Rigid Body with a Fixed Point in the Bobylev – Steklov Case, Russian J. Nonlinear Dyn., 2021, vol. 17, no. 4, pp. 453–464.
Bardin, B. S., Chekina, E. A., and Chekin, A. M., On the Orbital Stability of Pendulum Oscillations of a Dynamically Symmetric Satellite, Russian J. Nonlinear Dyn., 2022, vol. 18, no. 4, pp. 589–607.
Malkin, I. G., Theory of Stability of Motion, Washington, D.C.: U.S. Atomic Energy Commission, 1952.
Arnol’d, V. I., Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Russian Math. Surveys, 1963, vol. 18, no. 6, pp. 85–191; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 6(114), pp. 91-192.
Siegel, C. L. and Moser, J. K., Lectures on Celestial Mechanics, Grundlehren Math. Wiss., vol. 187, New York: Springer, 1971.
Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., vol. 8, New York: Springer, 1972.
Markeev, A. P., Linear Hamiltonian Systems and Some Problems of Stability of the Satellite Center of Mass, Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).
Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
Markeyev, A. P., A Constructive Algorithm for the Normalization of a Periodic Hamiltonian, J. Appl. Math. Mech., 2005, vol. 69, no. 3, pp. 323–337; see also: Prikl. Mat. Mekh., 2005, vol. 69, no. 3, pp. 355-371.
Moser, J. K., Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., vol. 81, Providence, R.I.: AMS, 1968.
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., 1980, vol. 44, no. 6, pp. 687–691; see also: Prikl. Mat. Mekh., 1980, vol. 44, no. 6, pp. 963-970.
Ivanov, A. P. and Sokol’skii, A. G., On the Stability of a Nonautonomous Hamiltonian System under Second-Order Resonance, J. Appl. Math. Mech., 1980, vol. 44, no. 5, pp. 574–581; see also: Prikl. Mat. Mekh., 1980, vol. 44, no. 5, pp. 811-822.
Funding
This research was supported by the grant of the Russian Science Foundation (project 22-21-00729) and was carried out at the Moscow Aviation Institute (National Research University).
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MSC2010
34D20, 37J40, 70K30, 70K45, 37N05
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Bardin, B.S. On the Method of Introduction of Local Variables in a Neighborhood of Periodic Solution of a Hamiltonian System with Two Degrees of Freedom. Regul. Chaot. Dyn. 28, 878–887 (2023). https://doi.org/10.1134/S1560354723060059
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DOI: https://doi.org/10.1134/S1560354723060059