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Planetary, stellar and galactic physics often rely on the general restrictedgravitational N-body problem to model the motion of a small-mass object underthe influence of much more massive objects. Here, I formulate the generalrestricted problem entirely and specifically in terms of the commonly-usedorbital elements of semimajor axis, eccentricity, inclination, longitude ofascending node, argument of pericentre, and true anomaly, without anyassumptions about their magnitudes. I derive the equations of motion in thegeneral, unaveraged case, as well as specific cases, with respect to both abodycentric and barycentric origin. I then reduce the equations to three-bodysystems, and present compact singly- and doubly-averaged expressions which canbe readily applied to systems of interest. This method recovers classicLidov-Kozai and Laplace-Lagrange theory in the test particle limit to anyorder, but with fewer assumptions, and reveals a complete analytic solution forthe averaged planetary pericentre precession in coplanar circular circumbinarysystems to at least the first three nonzero orders in semimajor axis ratio.Finally, I show how the unaveraged equations may be used to express resonantangle evolution in an explicit manner that is not subject to expansions ofeccentricity and inclination about small nor any other values.

Explicit evolution relations with orbital elements for eccentric, inclined, elliptic and hyperbolic restricted few-body problems