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

for . Hence, it follows from Equations (G.12) and (G.13) that

(G.15) | ||||||

(G.16) | ||||||

and | (G.17) |

Differentiation of these equations with respect to time yields

(G.18) | ||||||

(G.19) | ||||||

and | (G.20) |

Substitution into Equations (G.3)-(G.5) gives

where . Because , , and are the respective solutions to Equation (G.3)-(G.5) when the right-hand 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)].