Method of Averaging

In many dynamical problems, the motion consists of a rapid oscillation superimposed on a slow secular drift. For such problems, the most efficient approach is to describe the evolution in terms of the average values of the dynamical variables. The method outlined below is adapted from a classic paper by Morozov and Solov'ev.

Consider the equation of motion

$$\frac{d{\bf z}}{dt} = {\bf f}({\bf z}, t, \tau),$$ (43)

where ${\bf f}$ is a periodic function of its last argument, with period $2\pi$, and

$$\tau = t/\epsilon.$$ (44)

Here, the small parameter $\epsilon$ characterizes the separation between the short oscillation period $\tau$ and the time-scale $t$ for the slow secular evolution of the ``position'' ${\bf z}$.

The basic idea of the averaging method is to treat $t$ and $\tau$ as distinct independent variables, and to look for solutions of the form ${\bf z}(t,\tau)$ which are periodic in $\tau$. Thus, we replace Eq. (43) by

$$\frac{\partial{\bf z}}{\partial t} +\frac{1}{\epsilon}\frac{\partial {\bf z}} {\partial \tau} = {\bf f}({\bf z}, t, \tau),$$ (45)

and reserve Eq. (44) for substitution in the final result. The indeterminacy introduced by increasing the number of variables is lifted by the requirement of periodicity in $\tau$. All of the secular drifts are thereby attributed to the $t$-variable, whilst the oscillations are described entirely by the $\tau$-variable.

Let us denote the $\tau$-average of ${\bf z}$ by ${\bf Z}$, and seek a change of variables of the form

$${\bf z}(t,\tau) = {\bf Z}(t) +\epsilon\,\mbox{\boldmath$\zeta$}({\bf Z}, t, \tau).$$ (46)

Here, $\mbox{\boldmath$\zeta$}$ is a periodic function of $\tau$ with vanishing mean. Thus,

$$\langle \mbox{\boldmath$\zeta$}({\bf Z}, t, \tau)\rangle\equ... ...pi}\oint \mbox{\boldmath$\zeta$}({\bf Z}, t, \tau)\,d\tau = 0,$$ (47)

where $\oint$ denotes the integral over a full period in $\tau$.

The evolution of ${\bf Z}$ is determined by substituting the expansions

$$\mbox{\boldmath$\zeta$} = \mbox{\boldmath$\zeta$}_0({\bf Z}, t, \tau) + \epsilon\, \mbox{\b... ...}, t, \tau) + \epsilon^2\,\mbox{\boldmath$\zeta$}_2({\bf Z}, t, \tau) + \cdots,$$ (48)

$$\frac{d{\bf Z}}{dt} = {\bf F}_0({\bf Z}, t) + \epsilon\, {\bf F}_1({\bf Z}, t) + \epsilon^2\,{\bf F}_2({\bf Z}, t) + \cdots,$$ (49)

into the equation of motion (45), and solving order by order in $\epsilon$.

To lowest order, we obtain

$${\bf F}_0({\bf Z}, t) +\frac{\partial\mbox{\boldmath$\zeta$}_0}{\partial\tau} = {\bf f} ({\bf Z}, t, \tau).$$ (50)

The solubility condition for this equation is

$${\bf F}_0({\bf Z}, t) =\langle {\bf f}({\bf Z}, t,\tau)\rangle.$$ (51)

Integrating the oscillating component of Eq. (50) yields

$$\mbox{\boldmath$\zeta$}_0 ({\bf Z}, t,\tau) = \int_0^\tau\left( {\bf f} - \langle{\bf f}\rangle\right)\,d\tau'.$$ (52)

To first order, we obtain

$${\bf F}_1 +\frac{\partial\mbox{\boldmath$\zeta$}_0}{\partial... ...artial\tau} = \mbox{\boldmath$\zeta$}_0\cdot \nabla {\bf f}.$$ (53)

The solubility condition for this equation yields

$${\bf F}_1 =\langle \mbox{\boldmath$\zeta$}_0\cdot\nabla{\bf f}\rangle.$$ (54)

The final result is obtained by combining Eqs. (51) and (54):

$$\frac{d{\bf Z}}{dt} = \langle {\bf f}\rangle + \epsilon\, \l... ...boldmath$\zeta$}_0\cdot \nabla {\bf f}\rangle + O(\epsilon^2).$$ (55)

Note that ${\bf f} = {\bf f}({\bf Z},t)$ in the above equation. Evidently, the secular motion of the ``guiding centre'' position ${\bf Z}$ is determined to lowest order by the average of the ``force'' ${\bf f}$, and to next order by the correlation between the oscillation in the ``position'' ${\bf z}$ and the oscillation in the spatial gradient of the ``force.''