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224 5. Some worked examples arising from physical problems

so it is convenient to solve for u and first (and then t follows after
an integration or, at least, by quadrature). Thus, for the purposes of constructing a
solution, let us set with (and the condition on t
at is redundant at this stage). Thus our equations (with
become

and we seek a solution

which is periodic in T.
From equations (5.71) we obtain

and so on.
The exact solution of equations (5.72), which describe a Keplerian ellipse, is usually
written in the form

where is the eccentricity, denotes the position of the pericentre
(i.e. at and is the angular momentum. (We may write
where is the semi-major axis, if this is useful.) Equation (5.73b)
now becomes (with appropriate use of equations (5.74))

which may be integrated, at least formally, to give