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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

Stable oscillating motions in a model of a charged-particle accelerator


Author: L. D. Pustyl'nikov
Translated by: H. H. McFaden
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 65 (2004).
Journal: Trans. Moscow Math. Soc. 2004, 177-211
MSC (1991): Primary 34D20, 78A35; Secondary 37Kxx, 34C15, 70H08, 70H05, 70F10
Posted: October 1, 2004
MathSciNet review: 2193440
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to a qualitative study of oscillating motions in Hamiltonian systems with five-dimensional phase space that depend periodically on the independent variable. It is proved that in the phase space there exists an open set of initial data giving rise to oscillating motions that are Lyapunov stable. The main example of such systems is the classical model of an accelerator of charged particles moving in a variable periodic electric field and a constant magnetic field, and the resulting oscillating motions lead to unbounded growth of the energy of the particles. It is established that the property of stability of the solutions is preserved under a small change in the Hamiltonian function.


References [Enhancements On Off] (What's this?)

  • 1. V.I. Veksler, A new method for accelerating relativistic particles, Dokl. Akad. Nauk SSSR 43 (1944), 346-348. (Russian)
  • 2. S.P. Kapitsa and V.N. Melekhin, The microtron, ``Nauka'', Moscow, 1969. (Russian)
  • 3. L.A. Artsimovich and S.Yu. Luk'yanov, The motion of charged particles in electric and magnetic fields, 2nd ed., ``Nauka'', Moscow, 1978. (Russian)
  • 4. L. D. Pustyl′nikov, Oscillating motions in a dynamical system, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 5, 1010–1032, 1118 (Russian); English transl., Math. USSR-Izv. 31 (1988), no. 2, 325–347. MR 925092 (89a:78003)
  • 5. L. D. Pustyl′nikov, Stable and oscillating motions in nonautonomous dynamical systems. II, Trudy Moskov. Mat. Obšč. 34 (1977), 3–103 (Russian). MR 0477274 (57 #16815)
  • 6. L. D. Pustyl′nikov, Unbounded growth of the action variable in certain physical models, Trudy Moskov. Mat. Obshch. 46 (1983), 187–200 (Russian). MR 737907 (85b:70019)
  • 7. L. D. Pustyl′nikov, Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism, Uspekhi Mat. Nauk 50 (1995), no. 1(301), 143–186 (Russian); English transl., Russian Math. Surveys 50 (1995), no. 1, 145–189. MR 1331358 (96c:82035), http://dx.doi.org/10.1070/RM1995v050n01ABEH001663
  • 8. L. D. Pustyl′nikov, Some final motions in the 𝑛-body problem, Prikl. Mat. Mekh. 54 (1990), no. 2, 329–331 (Russian); English transl., J. Appl. Math. Mech. 54 (1990), no. 2, 272–274 (1991). MR 1065775 (91m:70012), http://dx.doi.org/10.1016/0021-8928(90)90045-C
  • 9. L. D. Pustyl′nikov, On the measure of one-way oscillating motions for the Kolmogorov model and its generalization in the 𝑛-body problem, Uspekhi Mat. Nauk 53 (1998), no. 5(323), 245–246 (Russian); English transl., Russian Math. Surveys 53 (1998), no. 5, 1102–1103. MR 1691200 (2001i:70015), http://dx.doi.org/10.1070/rm1998v053n05ABEH000084
  • 10. L.D. Pustyl'nikov, On the stability of solutions and absence of Arnol'd diffusion in a nonintegrable Hamiltonian system of a general form with three degrees of freedom, preprint no. 155, Weierstrass-Institut für Angewandte Analysis und Stochastik, Berlin, 1995.
  • 11. J. Chazy, Sur l'allure finale du mouvement dans le problème des trois corps quand le temps croit indéfiniment, Annales de l'École Norm. Sup. (3) 39 (1922), 29-130.
  • 12. K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl. 5 (1960), 647–650. MR 0127389 (23 #B435)
  • 13. V. M. Alekseev, Quasirandom dynamical systems. I. Quasirandom diffeomorphisms, Mat. Sb. (N.S.) 76 (118) (1968), 72–134 (Russian). MR 0276948 (43 #2687b)
    V. M. Alekseev, Quasirandom dynamical systems. II. One-dimensional nonlinear vibrations in a periodically perturbed field, Mat. Sb. (N.S.) 77 (119) (1968), 545–601 (Russian). MR 0276949 (43 #2688)
    V. M. Alekseev, Quasirandom dynamical systems. III. Quasirandom vibrations of one-dimensional oscillators, Mat. Sb. (N.S.) 78 (120) (1969), 3–50 (Russian). MR 0276950 (43 #2689)
  • 14. An. M. Leontovič, On the existence of unbounded oscillating trajectories in a billiard problem, Dokl. Akad. Nauk SSSR 145 (1962), 523–526 (Russian). MR 0138846 (25 #2287)
  • 15. V. I. Arnol′d, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR 156 (1964), 9–12 (Russian). MR 0163026 (29 #329)
  • 16. Carl Ludwig Siegel, Vorlesungen über Himmelsmechanik, Springer-Verlag, Berlin, 1956 (German). MR 0080009 (18,178c)

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Additional Information

L. D. Pustyl'nikov
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Pl. 4, Moscow 125047, Russia
Email: lpustyln@spp.keldysh.ru

DOI: http://dx.doi.org/10.1090/S0077-1554-04-00141-4
PII: S 0077-1554(04)00141-4
Posted: October 1, 2004
Additional Notes: This work was carried out with the financial support of the Russian Foundation for Basic Research, grant no. 02–01–01067.
Dedicated: Dedicated to the memory of V.M. Alekseev
Article copyright: © Copyright 2004 American Mathematical Society