Geodesic
Adiabatic approximation for a Model of Cyclotron Motion
Matematičeskie zametki, Tome 101 (2017) no. 5, pp. 733-749.

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A specific problem is used to illustrate the limits of the approach resulting in an adiabatic approximation. The system of differential equations modeling the cyclotron motion of a charged relativistic particle in the field of an electromagnetic wave is considered. The problem of resonance capture of a particle with significantly varying energy is studied. The main result is the description of the capture area, i.e., the set of initial points on the phase plane from which the resonance trajectories issue. Such a description is obtained by the method of asymptotic approximation in a small parameter which corresponds to the rate of variation in the magnetic field. It is discovered that such an approximation is inapplicable in the case of small amplitudes of the electromagnetic wave.
Keywords: nonlinear oscillations, small parameter, asymptotics, resonance capture. 
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L. A. Kalyakin. Adiabatic approximation for a Model of Cyclotron Motion. Matematičeskie zametki, Tome 101 (2017) no. 5, pp. 733-749. http://geodesic.mathdoc.fr/item/MZM_2017_101_5_a8/

[1] L. D. Landau, Teriya polya, Nauka, M., 1973

[2] C. S. Roberts, S. J. Buchsbaum, “Motion of a charged particle in a constant magnetic field and a transverse electromagnetic wave propagating along the field”, Phys. Rev. (2), 135 (1964), A381–A389 | DOI | MR

[3] V. P. Milantev, “Tsiklotronnyi avtorezonans (k 50-letiyu otkrytiya yavleniya)”, UFN, 183:8 (2013), 875–884 | DOI

[4] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt, Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, VINITI, M., 1985

[5] A. P. Itin, A. I. Neishtadt, A. A. Vasiliev, “Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave”, Phys. D, 141:3-4 (2000), 281–296 | DOI | MR | Zbl

[6] L. A. Kalyakin, “Asimptoticheskii analiz modelei avtorezonansa”, UMN, 63:5 (383) (2008), 3–72 | DOI | MR | Zbl

[7] I. Bryuning, S. Yu. Dobrokhotov, M. A. Poteryakhin, “Ob usrednenii dlya gamiltonovykh sistem s odnoi bystroi fazoi i malymi amplitudami”, Matem. zametki, 70:5 (2001), 660–669 | DOI | MR | Zbl

[8] K. S. Golovanivsky, “Autoresonant acceleration of electrons at nonlinear ECR in a magnetic field which is smoothly growing in time”, Phys. Scripta, 22:2 (1980), 126–133 | DOI

[9] K. S. Golovanivsky, “The gyromagnetic autoresonance”, IEEE Transactions on Plasma Sci., 11:1 (1983), 28–35 | DOI

[10] A. I. Neishtadt, “Prokhozhdenie cherez separatrisu v rezonansnoi zadache s medlenno menyayuschimsya parametrom”, Prikl. matem. mekh., 39:4-6 (1975), 1331–1334 | MR

[11] O. M. Kiselev, S. G. Glebov, “An asymptotic solution slowly crossing the separatrix near a saddle-centre bifurcation point”, Nonlinearity, 16:1 (2003), 327–362 | DOI | MR | Zbl

[12] V. D. Azhotkin, V. M. Babich, “O primenenii metoda dvukhmasshtabnykh razlozhenii k odnochastotnoi zadache teorii nelineinykh kolebanii”, Prikl. matem. mekh., 49:3 (1985), 377–383 | Zbl

[13] J. A. Murdock, Perturbations. Theory and Methods, John Wiley Sons, New York, 1991 | MR | Zbl

[14] O. M. Kiselev, “Ostsillyatsii okolo separatrisy v uravnenii Dyuffinga”, Tr. IMM UrO RAN, 18, no. 2, 2012, 141–153

[15] A. I. Neishtadt, “O razdelenii dvizhenii v sistemakh s bystro vraschayuscheisya fazoi”, Prikl. matem. mekh., 48:2 (1984), 197–204 | MR

[16] L. A. Kalyakin, “Asimptoticheskii analiz modeli tsiklotronnogo giromagnitnogo avtorezonansa”, Vestn. Chelyabinskogo gos. un-ta. Fizika, 21 (2015), 68–74

[17] L. A. Kalyakin, “Asimptoticheskii analiz modeli giromagnitnogo avtorezonansa”, Zh. vychisl. matem. i matem. fiz., 57:2 (2017), 285–301