49 research outputs found
3-Body Problems, Hidden Constants, Trojans and WIMPs
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 Law for 3-body problems. This is confined to the
Appendix.Comment: 43 pages, no figure
The Divine Clockwork: Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state
We consider the Bohr correspondence limit of the Schrodinger wave function
for an atomic elliptic state. We analyse this limit in the context of Nelson's
stochastic mechanics, exposing an underlying deterministic dynamical system in
which trajectories converge to Keplerian motion on an ellipse. This solves the
long standing problem of obtaining Kepler's laws of planetary motion in a
quantum mechanical setting. In this quantum mechanical setting, local mild
instabilities occur in the Kelperian orbit for eccentricities greater than
1/\sqrt{2} which do not occur classically.Comment: 42 pages, 18 figures, with typos corrected, updated abstract and
updated section 6.
Generalized Ito formulae and space-time Lebesgue-Stieltjes integrals of local times
Generalised Ito formulae are proved for time dependent functions of
continuous real valued semi-martingales.The conditions involve left space and time
first derivatives, with the left space derivative required to have locally bounded
2-dimensional variation. In particular a class of functions with discontinuous first
derivative is included. An estimate of Krylov allows further weakening of these
conditions when the semi-martingale is a diffusion