D. Minenkov
prosp.Vernadskogo 101/1, 119526, Moscow, Russian Federation
Institute for Problems in Mechanics, Russian Academy of Sciences
Publications:
Dobrokhotov S. Y., Minenkov D. S.
On various averaging methods for a nonlinear oscillator with slow timedependent potential and a nonconservative perturbation
2010, vol. 15, no. 23, pp. 285299
Abstract
The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the onedimensional anharmonic oscillator with potential $V(x, \tau)$ depending on the slow time $\tau=\varepsilon t$ and with a small nonconservative term $\varepsilon g(\dot x, x, \tau)$, $\varepsilon \ll 1$. This problem was discussed in numerous papers, and in some sense the present paper looks like a "methodological" one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $X(\frac{S(\tau)+\varepsilon \phi(\tau))}{\varepsilon}, I(\tau), \tau)$, where the phase $S$, the "slow" parameter $I$, and the socalled phase shift $\phi$ are found from the system of "averaged" equations. The pragmatic result is that one can take into account the phase shift $\phi$ by considering it as a part of $S$ and by simultaneously changing the initial data for the equation for $I$ in an appropriate way.
