Closed trajectories for Kepler with spin-orbit corrections?

[jarekduda]

Kepler problem explains closed elliptic trajectories for planetary systems or in Bohr's classical atomic model - let say two approximately point objects, the central one has practically fixed position, they attract through 1/r^2 Newton's or Coulomb force.

Kind of the best motivated expansion we could think of is considering that one of the two objects has also a magnetic dipole moment (leading to additional Lorentz force): for example intrinsic one due to being e.g. electron, or just a magnet, or a spinning charge.
Analogously, it could be a spinning mass in gravitational considerations: the first correction of general relativity, directly tested by Gravity Probe B, is gravitomagnetism ( https://en.wikipedia.org/wiki/Gravitoelectromagnetism ): making Newton law Lorentz-invariant in analogy to Coulomb - adding gravitational analogue of magnetism and second set of Maxwell's equations (for gravity).
So in this approximation of GR, a spinning mass gets gravitomagnetic moment - also leading to Lorentz force corrections to Kepler problem (frame-dragging), especially for a millisecond pulsar or spinning black hole

The Lagrangian for such Kepler problem with one of the two objects having also (gravito)magnetic dipole moment (the question which one chooses the sign in magnetic term due to 3rd Newton) with simplified constants and assuming fixed spin(dipole) direction (s) becomes:
$$\mathcal{L}=\frac{v^2}{2} + c_e \frac{1}{r} +c_s \frac {(\hat{s} \times \hat{r})\cdot \vec{v}}{r^2}$$
Where hat means vector normalized to 1.
Here is a simple Mathematica simulator: http://demonstrations.wolfram.com/KeplerProblemWithClassicalSpinOrbitInteraction/
Some example trajectories (for much stronger magnetic dipole moment than in nature):
https://dl.dropboxusercontent.com/u/12405967/traje.png

From Noether theorem we can find two invariants here:
- energy for time invariance:
$$E=\frac{v^2}{2}-\frac{c_e}{r}$$
- only one angular momentum: for rotation around the (fixed) spin axis s:
$$L_s = \left(r^2 \dot{\varphi} +\frac{c_s}{r}\right)\sin^2 \theta$$
There is missing one invariant to make it integrable (maybe there is?)

It is an extremely interesting question to understand and characterize especially the closed trajectories here, like for repeating electron-nucleus scatterings.
How to search for closed trajectories of such well motivated but mathematically far nontrivial system?

 

[mfb]

Circular orbits should be closed and stable if the spins and the orbit are all aligned, but those are boring.

Studying the evolution of the eccentricity and the argument of periapsis (evaluated at each point assuming a Kepler problem) could be interesting. In the two-dimensional system, the orbit will look somewhat closed (even if it doesn't have to be exactly closed) if those two parameters visit the same point in phase space again. It is a two-dimensional space - the orbits should either visit some points again (at least approximately) or diverge.

 

[jarekduda]

So let me summarize my experience.
The magnetic constant in real systems is much smaller - the demonstration ends with c_s = 10^-3, while for electron+proton it would be rather ~10^-6 (there are numerical issues with tiny c_s).
This v/r^3 correction is nearly negligible unless the electron nearly misses the proton (large v, small r) - the scattering situations.
If it starts in the plane perpendicular to the spin, it should remain in this plane (numerically not necessarily).
Just free falling in this plane (zero angular momentum), surprisingly we get exactly 120deg scattering (for any charge and magnetic dipole!):
https://dl.dropboxusercontent.com/u/12405967/triang.png
So Lorentz force is a nice explanation why electron cannot fall on the nucleus - instead, it should e.g. travel between three vertices of equilateral triangle.
The minimal distance is ~10^-13m here.
Starting with nonzero angular momentum in this plane, we can get e.g. nearly back-scattering trajectories, like:
https://dl.dropboxusercontent.com/u/12405967/traj.png
These seem interesting from the point of view of fusion: if another proton would approach from the orbit direction, electron could stay between them, screening the Coulomb barrier.

Going out of this plane, there is also supposed to be tetrahedral trajectory - between e.g. (1,1,1), (1,-1,-1), (-1,-1,1) and (-1,1,-1) vertices.
But searching for other closed trajectories seems difficult (?)

And sure, the question is when (initial distance?) we should rather go to quantum description of electron-proton scattering?