Abstract
An approximate renormalization procedure is derived for the HamiltonianH(v,x,t)=v2/2−M cosx−P cosk(x−t). It gives an estimate of the large scale stochastic instability threshold which agrees within 5–10% with the results obtained from direct numerical integration of the canonical equations. It shows that this instability is related to the destruction of KAM tori between the two resonances and makes the connection with KAM theory. Possible improvements of the method are proposed. The results obtained forH allow us to estimate the threshold for a large class of Hamiltonian systems with two degrees of freedom.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Casati and J. Ford, eds.,Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Springer, Berlin, 1979).
S. Jorna, ed.,Topics in Nonlinear Dynamics (American Institute of Physics, New York, 1978).
M. Month and J. C. Herrera, eds.,Nonlinear Dynamics and the Beam-Beam Interaction (American Institute of Physics, New York, 1978).
G. Laval and D. Gresillon, eds.Intrinsic stochasticity in plasmas (Editions de Physique, Orsay, 1979).
B. V. Chirikov,Phys. Rep. 52:263–379 (1979).
K. C. Mo,Physica (Utrecht) 57:445–454 (1972).
D. F. Escande, Primary resonances do not overlap, inIntrinsic stochasticity in plasmas, G. Laval and D. Gresillon, eds. (Editions de Physique, Orsay, 1979).
D. F. Escande and F. Doveil, Charged particle trajectories in the field of two electrostatic waves, inProceedings of International Conference on Plasma Physics 1980, Vol. 1, (Fusion Research Association of Japan, Nagoya, 1980), p. 387.
V. I. Arnold and A. Avez,Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
J. M. Greene,J. Math. Phys. 20:1183–1201 (1979).
J. M. Greene, KAM Surfaces computed from the Henon-Heiles Hamiltonian, inNonlinear Dynamics and the Beam-Beam Interaction, M. Month and J. C. Herrera, eds. (American Institute of Physics, New York, 1978).
K. Wilson,Rev. Mod. Phys. 47:773–840 (1975).
L. M. Milne-Thomson, Elliptic integrals, in M. Abramowitz and I. A. Stegun, eds.,Handbook of Mathematical Functions (Dover, New York, 1972).
G. R. Smith and N. R. Pereira,Phys. Fluids 21:2253–2262 (1978).
J. B. Taylor and E. W. Laing,Phys. Rev. Lett. 35:1306–1307 (1975).
E. T. Whittaker and G. N. Watson,A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1969).
L. D. Landau and E. M. Lifshitz,Mechanics (Pergamon Press, Oxford, 1960).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Escande, D.F., Doveil, F. Renormalization method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems. J Stat Phys 26, 257–284 (1981). https://doi.org/10.1007/BF01013171
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01013171