By B. V. Chirikov (auth.), Acad B. B. Kadomtsev (eds.)
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Extra resources for Reviews of Plasma Physics: Volume 13
Example text
Since the function k( £) has a singularity at £ = 0, the new parameter k cannot be obtained, as we have verified also directly, from the asymptotic expansion in £. , [12]). Thus, for sufficiently small k-+ o and nonresonant p 1:- 211r we can expect the variation of p to be quasiperiodic and bounded. In exactly the same manner, for k -+ 0 and resonant p (for example, p,. 0) we can neglect all the resonances except the given one (r = 0). 2) which turn out to be canonical with a Hamiltonian HrC8, p) = L2 + K cos 8.
In this case, the quasiperiodic variations cancel out at both reflection points. 1 was first obtained in [16] from the solution of the corresponding quantum-mechanical problem. The relatively simple classicalmechanics technique described above was proposed in [20] and then developed in a number of papers (see, e. , [5, 15, 21]). The most extensive calculations were carried out in [5]. 8. 1. 1. This problem can be answered in two ways. 10). The second is to describe the particle motion by mapping or transforms, i.
Clearly, when the resonances overlap, the trajectory of the system can move over from the region of one resonance to that of a neighboring and the motion becomes infinite, at least for certain initial conditions. The character of this motion will be considered below (Section 11). We note here merely that the overestimate of the critical value by the simple resonance-overlap criterion is due mainly to the fact that in this form the criterion takes into account only resonance of first-order approximation in the adiabaticity parameter k.