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 (Morozov and Solev'ev 1966; Hazeltine and Waelbroeck 2004).

Consider the equation of motion

$$\frac{d{\bf z}}{dt} = {\bf f}({\bf z}, t, \tau),$$ (2.8)

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

$$\tau = t/\epsilon.$$ (2.9)

Here, the small parameter $\epsilon$ characterizes the separation between the short oscillation period and the timescale 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 ${\bf z}(t,\tau)$ that are periodic in $\tau$. Thus, we replace Equation (2.8) by

$$\frac{\partial{\bf z}}{\partial t} +\frac{1}{\epsilon}\frac{\partial {\bf z}} {\partial \tau} = {\bf f}({\bf z}, t, \tau),$$ (2.10)

and reserve Equation (2.9) for substitution into 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 variable $t$, while the oscillations are described entirely by the variable $\tau$.

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

$${\bf z}(t,\tau) = {\bf Z}(t) +\epsilon\, \mbox{\boldmath$\zeta$} ({\bf Z}, t, \tau).$$ (2.11)

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

$$\langle \mbox{\boldmath$\zeta$} ({\bf Z}, t, \tau)\rangle\equiv \frac{1}{2\pi}\oint \mbox{\boldmath$\zeta$} ({\bf Z}, t, \tau)\,d\tau = 0,$$ (2.12)

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

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

$$\mbox{\boldmath$\zeta$} = \mbox{\boldmath$\zeta$} _0({\bf Z}, t, \tau) + \epsilon\, \mbox{\boldmath$\zeta$} _1({\bf Z}, t, \tau) + \epsilon^2\, \mbox{\boldmath$\zeta$} _2({\bf Z}, t, \tau) + \cdots,$$ (2.13)

$$\frac{d{\bf Z}}{dt} ={\bf F}_0({\bf Z}, t) + \epsilon\, {\bf F}_1({\bf Z}, t) + \epsilon^2\,{\bf F}_2({\bf Z}, t) + \cdots,$$ (2.14)

into the equation of motion, Equation (2.10), and solving order by order in $\epsilon$.

To lowest order, we obtain

$${\bf F}_0({\bf Z}, t) +\frac{\partial\mbox{\boldmath$\zeta$}_0}{\partial\tau} = {\bf f} ({\bf Z}, t, \tau).$$ (2.15)

The solubility condition for this equation is

$${\bf F}_0({\bf Z}, t) =\langle {\bf f}({\bf Z}, t,\tau)\rangle\equiv \langle {\bf f}\rangle({\bf Z},t).$$ (2.16)

Integrating the oscillating component of Equation (2.15) yields

$$\mbox{\boldmath$\zeta$} _0 ({\bf Z}, t,\tau) = \int_0^\tau\left[ {\bf f}({\bf Z},t,\tau') - \langle{\bf f}\rangle({\bf Z},t)\right]d\tau'.$$ (2.17)

To first order, Equation (2.10) gives,

$${\bf F}_1 +\frac{\partial\mbox{\boldmath$\zeta$}_0}{\partial t} +... ...{\partial\tau} = \mbox{\boldmath$\zeta$}_0\cdot \nabla {\bf f}({\bf Z},t,\tau).$$ (2.18)

The solubility condition for this equation yields

$${\bf F}_1({\bf Z},t) =\langle \mbox{\boldmath$\zeta$} _0({\bf Z},t,\tau)\cdot\nabla{\bf f}({\bf Z},t,\tau)\rangle\equiv \langle \mbox{\boldmath$\zeta$} _0\cdot\nabla{\bf f}\rangle({\bf Z},t).$$ (2.19)

The final result is obtained by combining Equations (2.14), (2.16), and (2.19):

$$\frac{d{\bf Z}}{dt} = \langle {\bf f}\rangle({\bf Z},t) + \epsilon\, \langle \mbox{\boldmath$\zeta$} _0\cdot \nabla {\bf f}\rangle({\bf Z},t) + {\cal O}(\epsilon^2).$$ (2.20)

Evidently, the secular motion of the “guiding center” 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.”