Abstract
Small perturbations imposed on an integrable system produce an evolution of the motion. In the course of this evolution the system can pass through a state of resonance. Quasi-random phenomena of capture into the resonance and scattering on the resonance are discussed in this paper for the case of a two-frequency system. Major topics are calculation of probability of capture, description of the motion of the captured phase points, calculation of a probabilistic distribution of the scattering amplitude.
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References
Arnold V.I. (1978) Mathematical methods of classical mechanics. Springer, New York
Arnold V.I. (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York
Arnold V.I. (1963) Small denominators and problems of stability of motion in classical and celestial mechanics, Usp. Mat. Nauk, 18, pp. 91–192 (Russian) [English transl.: (1963) Russ. Math. Surv., 18, pp. 85–192]
Arnold V.I. (1965) Conditions of the applicability and estimates of the error of averaging method for systems which pass through states of resonance in the course of their evolution, Dokl. Akad. Nauk SSSR, 161, pp. 9–12 (Russian) [English transl.: (1965) Soviet Math. Dokl., 6, pp. 331–334 ]
Arnold V.I., Kozlov V.V., Neishtadt A.I. (1988) Mathematical aspects of classical and celestial mechanics. Encyclopaedia of mathematical sciences, vol. 3. Springer, Berlin
Beaugé C., Ferraz-Mello S. (1994) Capture in exterior mean-motion resonances due to Poynting-Robertson drag, Icarus, 110, pp. 239–260
Bogolyubov N.N., Mitropol'skii Yu.A. (1961) Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York
Chirikov B.V. (1959) Passage of nonlinear oscillatory system through resonance, Dokl. Akad. Nauk SSSR, 125, pp. 1015–1018 (Russian) [English transl.: (1959) Soviet Phys. Dokl., 4, pp. 390–394]
Dirac P.A.M. (1925) The adiabatic invariance of the quantum integrals. Proc. Roy. Soc., A107, pp. 725–734
Goldreich P., Peale S. (1966) Spin-orbit coupling in the Solar System, Astron. J., 71, pp. 425–438
Greenberg R. (1978) Evolution of satellite resonances by tidal dissipation, Astron. J., 78, pp. 338–346
Henrard J., Lemaitre A. (1983) A second fundamental model for resonance, Celest. Mech. Dynam. Astron., 27, pp. 3–22
Kevorkian J. (1974) On a model for reentry roll resonance, SIAM J. Appl.Math., 26, pp. 638–669
Lifshits I.M., Slutskin A.A., Nabutovskii V.M. (1961) On phenomenon of scattering of charged quasi-particles at singular points in p-space, Dokl. Akad. Nauk SSSR, 137, pp. 553–556 (Russian)
Lochak P., Meunier C. (1988) Multiphase averaging for classical systems. Springer, New York
Melnikov V.K. (1963) On the stability of the center for time-periodic perturbations, Tr. Mosk. Mt. O.-va, 12, pp. 3–52 (Russian) [English transl.: (1965) Trans. Mosc. Math. Soc. 1963, pp. 1–56
Moltchanov A.M. (1968) The resonant structure of the Solar System, Icarus, 8, pp. 203–215
Morozov A.D. (1976) A complete qualitative investigation of Duffing's equation, Differ. Uravn., 12, pp. 241–255 (Russian) [English transl.: (1976) Differ. Equations, 12, pp. 164–174]
Neishtadt A.I. (1975) Passage through resonance in the two-frequency problem, Dokl. Akad. Nauk SSSR, 221, pp. 301–374 (Russian) [English transl.: (1975) Soviet Phys. Dokl., 20, pp. 189–191]
Neishtadt A.I. (1975) Passage through a separatrix in a resonance problem with slowly varying parameter, Prikl.Mat. Mekh., 39, pp. 621–632 (Russian) [English transl.: (1975) JAppl.Math.Mech., 39, pp. 594–605]
Neishtadt A.I. (1975) Some resonant problems in nonlinear systems. Ph.D.Thesis. Moscow State Univ, Moscow (Russian)
Neishtadt A.I. (1990) Averaging, capture into resonances and chaos in nonlinear systems. In: Campbell D. (ed.)Chaos/Xaoc AIP, New York, pp. 261–275.
Neishtadt A.I. (1991) Probability phenomena due to separatrix crossing, Chaos, 1, pp. 42–48
Neishtadt A.I. (1993) Probability phenomenain perturbed systems, Selecta Matematica Sovetica, 12, pp. 195–209
Neishtadt A.I. (1993) On destruction of adiabatic invariants in multi-frequency systems, In: Perelló C., Simó C., Solà-Morales 7. (Eds.) Equadiff 91, International conference on differential equations, Vol. 1, pp.195–207.
Neishtadt A.I. (1996) On adiabatic invariance in two-frequency systems. In: A.Delshams, R.de la Llave, C., Simó (Eds.) Hamiltonian systems with 3 or more degrees of freedom. Proceedings of NATO ASI, in press
Sidorenko V.V. (1994) Capture and escape from the resonance in the dynamics of the rigid body in viscous medium., Journal ofNonlinear Science, 4, pp. 35–57
Sinclair A.T. (1972, 1974) On the origin of commensurabilities among the satellites of Saturn, I, II, Monthly Notices Roy. Astron. Soc., 160, pp. 169–187, 166, pp. 165–179.
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Neishtadt, A. Scattering by resonances. Celestial Mech Dyn Astr 65, 1–20 (1996). https://doi.org/10.1007/BF00048435
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DOI: https://doi.org/10.1007/BF00048435