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:

[tex]\mathcal{L}=\frac{v^2}{2} + c_e \frac{1}{r} +c_s \frac {(\hat{s} \times \hat{r})\cdot \vec{v}}{r^2} [/tex]

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 [Broken]

From Noether theorem we can find two invariants here:

- energy for time invariance:

[tex]E=\frac{v^2}{2}-\frac{c_e}{r} [/tex]

- only one angular momentum: for rotation around the (fixed) spin axis s:

[tex] L_s = \left(r^2 \dot{\varphi} +\frac{c_s}{r}\right)\sin^2 \theta [/tex]

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?

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:

[tex]\mathcal{L}=\frac{v^2}{2} + c_e \frac{1}{r} +c_s \frac {(\hat{s} \times \hat{r})\cdot \vec{v}}{r^2} [/tex]

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 [Broken]

From Noether theorem we can find two invariants here:

- energy for time invariance:

[tex]E=\frac{v^2}{2}-\frac{c_e}{r} [/tex]

- only one angular momentum: for rotation around the (fixed) spin axis s:

[tex] L_s = \left(r^2 \dot{\varphi} +\frac{c_s}{r}\right)\sin^2 \theta [/tex]

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?

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