-7-

The equations (3.8, 3.9) are sufficient to generate a power series
expansion for the transformation i and the averaged driving terms I. The
result is conveniently written down in terms of the integrating and averaging
operators (ti,-) defined as follows: for any function f, periodic in v,
f =     (f - )dV'                                    (3.10)
f =     l   ofdv'
Notice that f contains a constant of integration or initial condition and
so f = 0 but f } 0. Without further ado, we write the transformation to
0(c3
I - I - Ei + e    G          + 0(c3)                              (3.11)
Q . Y + cG + c2(1Z1-z        + 0(c3)
Tha average coordinates 2 have the equation of motion
Zt -   + c              -  Z - g.;)  + 0(3)                      (3.12)
where a = (1,0,0,...,0). Notice that the phase 0 does not appear on the right
of this equation, as desired. There are many Iescriptions of the :ecnod
of averaging in the text books but equations (3.11, 3.12) are the answer for
the plasma physicist. In celestial mechanics one is usually interested
in a hirh order of accuracy and so requires many orders of the expansion.
The best method is due to Ceprit (1969) and is well described in Nayfeh's
book (1973). The method uses a generating function or Lie transform,
i(Y), which allows the manipulations to be computerized on an algebraic
manipulator. This generating function approach also allows any function
of theold variables to be expanded directly in the new variables.
We observe that the original problem has merely been transformed to
a simpler one which still must be solved, equations (3,12). As a final answer,