(24) where x2 is given at Eq. (23) and ( )' = d( )/d/x. This equation is valid to 0(e) as long as the flow stays in region 2a (Fig. 1), and it can be shown that if jii(0)>0 and e>0, then trajectories originating in region 2a remain in that region.9 Numerical comparisons show that solutions to Eq. (24) agree quite well with the exact solution to Eqs. (2) and (3). In this example there is only one region of phase space where the unperturbed solution is periodic. In case there is more than one such region [e.g., when V(x; ju) is quartic], then the form of the e = 0 solution and, hence, the form of the right-hand side of Eq. (14) are different in different regions. When the slow flow passes from one region to another, the averaged equation may lose validity. This is because the transition may involve crossing an instantaneous separatrix of the unperturbed system. At a separatrix, the period of the e = 0 solution becomes infinite, so that the average computed in Eq. (13) is over an infinite time interval, violating the conditions of the averaging theorem (see Ref. 9 for further discussion of separatrix crossing). Conclusions We have presented a general formulation for application of the method of averaging to a specific class of nonlinear equations. The method exploits the existence of an energy integral (the Hamiltonian) for the unperturbed system and leads to a single first-order equation for the slow evolution of the Hamiltonian. By using the canonical coordinate x as the fast variable, the need to identify the rapidly varying phase angle (as in Kruskal's method) is eliminated. As shown in the example, application is relatively straightforward when the form of the potential leads to an explicit solution to the unperturbed problem. References
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