Consider the equation of motion

where is a periodic function of its last argument, with period , and

Here, the small parameter characterizes the separation between the short oscillation period and the timescale for the slow secular evolution of the ``position'' .

The basic idea of the averaging method is to treat and as distinct independent variables, and to look for solutions of the form that are periodic in . Thus, we replace Equation (2.8) by

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 . All of the secular drifts are thereby attributed to the variable , while the oscillations are described entirely by the variable .

Let us denote the -average of by , and seek a change of variables of the form

(2.11) |

Here, is a periodic function of with vanishing mean. Thus,

(2.12) |

where denotes the integral over a full period in .

The evolution of is determined by substituting the expansions

(2.13) | ||

(2.14) |

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

To lowest order, we obtain

The solubility condition for this equation is

Integrating the oscillating component of Equation (2.15) yields

(2.17) |

To first order, Equation (2.10) gives,

(2.18) |

The solubility condition for this equation yields

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

(2.20) |

Evidently, the secular motion of the ``guiding center'' position is determined to lowest order by the average of the ``force'' , and to next order by the correlation between the oscillation in the ``position'' and the oscillation in the spatial gradient of the ``force.''