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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

(43) 
where is a periodic function of its last argument, with
period , and

(44) 
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
which are periodic in . Thus, we replace Eq. (43) by

(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 . All of the secular drifts
are thereby attributed to the variable, whilst the oscillations are
described entirely by the variable.
Let us denote the average of by , and seek a
change of variables of the form

(46) 
Here,
is a periodic function of with vanishing mean.
Thus,

(47) 
where denotes the integral over a full period in .
The evolution of is determined by substituting the
expansions
into the equation of motion (45), and solving order by order in .
To lowest order, we obtain

(50) 
The solubility condition for this equation is

(51) 
Integrating the oscillating component of Eq. (50) yields

(52) 
To first order, we obtain

(53) 
The solubility condition for this equation yields

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

(55) 
Note that
in the above equation.
Evidently, the secular motion of the ``guiding centre'' 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.''
Next: Guiding Centre Motion
Up: Charged Particle Motion
Previous: Motion in Uniform Fields
Richard Fitzpatrick
20110331