This work includes two new results – principally two new constants of motion
for the linearised restricted 3-body problem (e.g. for the Trojan asteroids)
and an important isosceles triangle generalisation of Lagrange’s equilateral
triangle solution of the restricted case leading to hidden constants for
Hildans as well as Trojans. Both of these results are classical, but we also
have included new results on Newtonian quantum gravity emanating from the
asymptotics relevant for WIMPish particles, explaining the origin of systems
like that of the Trojans. The latter result uses a generalisation of our
semi-classical mechanics for Schr\”odinger equations involving vector as well
as scalar potentials, presented here for the first time, thereby providing an
acid test of our ideas in predicting the quantum curvature and torsion of
WIMPish trajectories for our astronomical elliptic states. The combined effect
is to give a new celestial mechanics for WIMPs in gravitational systems as well
as new results for classical problems. As we shall explain, we believe these
results could help to see how spiral galaxies evolve into elliptical ones. A
simple classical consequence of our isosceles triangle result gives a Keplerian
type $4^{\textrm{th}}$ Law for 3-body problems. This is confined to the
Appendix.
