Gaussian variational equations for osculating elements of an arbitrary separable reference orbit

If a satellite orbit is described by means of osculating Jacobi α's and β's of a separable problem, the paper shows that a perturbing forceF makes them vary according to $$dot alpha _kappa = {text{F}} cdot partial {text{r/}}partial beta _k {text{ }}dot beta _k = {text{ - F}} cdot partial {text{r/}}partial alpha _k ,{text{ (}}k = 1,2,3).{text{ (A1)}}$$ Herer is the position vector of the satellite andF is any perturbing force, conservative or non-conservative. There are two special cases of (A1) that have been previously derived rigorously. If the reference orbit is Keplerian, equations equivalent to (A1), withF arbitrary, were derived by Brouwer and Clemence (1961), by Danby (1962), and by Battin (1964). IfF=?gradV ₁(t), whereV ₁ may or may not depend explicitly on the time, Equations (A1) reduce to the well known forms (e.g. Garfinkel, 1966) $$dot alpha _kappa = {text{ - }}partial V_1 {text{/}}partial beta _k {text{ }}dot beta _k = partial V_1 {text{/}}partial alpha _k ,{text{ (}}k = 1,2,3).{text{ (A2)}}$$ holding for all separable reference orbits. Equations (A1) can of course be guessed from Equations (A2), if one assumes that (dot alpha _k (t)) and (dot beta _k (t)) depend only onF(t) and thatF(t) can always be modeled instantaneously as a potential gradient. The main point of the present paper is the rigorous derivation of (A1), without resort to any such modeling procedure. Applications to the Keplerian and spheroidal reference orbits are indicated.