In the nonlinear terms we can simply use
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Expanding and organizing Equation (287) with the linear terms given explicitly and the quadratic terms
collected in the expression for
, we obtain the long expression with all terms functions of either
or
:
Using Equations (269) and (271
), the
and
in Equation (288
) are replaced by
. On the
right-hand side all the variables, e.g.,
., are functions of ‘
’; their functional forms are the
same as when they were functions of
and
the linear terms are again explicitly given and the
quadratic terms are collected in the
To proceed, we use the complex center-of-mass condition, namely, , and solve for
. This is accomplished by first reversing the calculation via
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and then, before extracting the harmonic component, replacing the
by
, via the inverse of
Equation (271
),
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using Equation 280
The final expression for , given in terms of the complex world line
expressed as a function of
, then becomes our basic equation:
We emphasize that prior to this discussion/derivation, the and the
were independent
quantities but in the final expression the
is now a function of the
.
Note that the linear term
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coincides with the earlier results in the stationary case, Equation (259). From
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we have
We will see shortly that there is a great deal of physical content to be found in the nonlinear terms in Equation (292http://www.livingreviews.org/lrr-2009-6 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |