Closed trajectories for Kepler problem with classical spin-orbit corrections?
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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 (http://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:
L=v22+ce1r+cs(s^×r^)⋅v⃗ r2L=v22+ce1r+cs(s^×r^)⋅v→r2
Where hat means vector normalized to 1.
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 (http://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:
L=v22+ce1r+cs(s^×r^)⋅v⃗ r2L=v22+ce1r+cs(s^×r^)⋅v→r2
Where hat means vector normalized to 1.
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noted, and the corrections to the orbits of nonrelativistic particles are sometimes ... THE MECHANICAL PROBLEM .... only relativistic classical mechanics, not quantum mechanics. IV.
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