According to Equation (G.6), we have

(G.13) 
If this expression, and the analogous expressions for and ,
were differentiated with respect to time, and the results substituted into Equations (G.3)–(G.5), then we would obtain three time evolution
equations for the six variables
. In order to make
the problem definite, three additional conditions must be introduced into the problem.
It is convenient to choose

(G.14) 
for . Hence, it follows from Equations (G.12) and (G.13) that
Differentiation of these equations with respect to time yields
Substitution into Equations (G.3)–(G.5) gives
where
. Because , , and are the respective solutions to Equation (G.3)–(G.5) when the righthand sides are
zero, and the orbital elements are thus constants, it follows that the
first two terms in each of the preceding three equations cancel one another. Hence,
writing as , and as , and so on, Equations (G.14)
and (G.21)–(G.23) yield
These six equations are equivalent to the three original equations of motion
[(G.3)–(G.5)].