Orateur
Description
Gauss explained in 1818 that the attraction of a body A by another body B moving on a Keplerian orbit is after averaging the attraction of A by a 1-dimensional ring with the shape of the Keplerian ellipse and a mass distribution $dm$ proportional to $dl$ where $l$ is the mean anomaly (and of total mass the mass of B).
Gauss did not invent this averaging but rather invented the 1-dimensional ring which gives a GEOMETRICAL INTUITION on a classical average. However this intuition is insufficient.
Indeed, when two planets move on nearly keplerian orbits around the Sun we get a good approximation of the perturbed motion by doubly averaging the potential on the Keplerian orbits. Such a kind of average appeared much before Gauss, expressed in different forms, e.g. Fourier type expansion followed by a truncature (Clairaut, d'Alembert, Euler, Lagrange, Laplace... or maybe even Newton about the Moon).
PARADOX: in the doubly averaged problem, the same pair of coplanar Keplerian orbits evolve in very different ways if the two planets make their revolutions in the same direction or in opposite directions.
WHY IS THIS A PARADOX? Gauss gives us an intuition of the averaged force, making clear that it does not change if we reverse the direction of revolution. So what does change? The energy is the same, so it should give the same Hamiltonian and the same dynamics.
COMPUTATIONAL SOLUTION OF THIS A PARADOX: A sign changes in the symplectic form, while the Hamiltonian remains the same. As often, the symplectic form looks like the angular momentum. And the angular momentum is sum of the contributions of the two planets. If the revolution of a planet changes direction, a contribution changes sign.
MECHANICAL INTUITION FOR THIS A PARADOX: Gauss explains the attraction of a point by a ring, but not the attraction of a ring by a point. The ring is moved by the force but it reacts according to its inertia.
CONCLUSION: WE SHOULD REPLACE GAUSS' FIXED 1-DIMENSIONAL RING BY A "FLUID" 1-DIMENSIONAL RING WHICH ROTATES either in one direction or in the other.
Then the inertia makes the difference (remember the experiment with a bicycle wheel, taking the axis with your hands).