The dynamics of the relativistic Kepler problem

Abstract

We deal with the Hamiltonian system (HS) associated to the Hamiltonian in polar coordinates H=12pr2+pϕ2r2-1r-∊2r2, where ∊ is a small parameter. This Hamiltonain comes from the correction given by the special relativity to the motion of the two-body problem, or by the first order correction to the two-body problem coming from the general relativity. This Hamiltonian system is completely integrable with the angular momentum C and the Hamiltonian H. We have two objectives.First we describe the global dynamics of the Hamiltonian system (HS) in the following sense. Let Sh and Sc are the subset of the phase space where H=h and C=c, respectively. Since C and H are first integrals, the sets Sc,Sh and Shc=Sh∩Sc are invariant by the action of the flow of the Hamiltonian system (HS). We determine the global dynamics on those sets when the values of h and c vary.Second recently Tudoran (2017) provided a criterion which detects when a non-degenerate equilibrium point of a completely integrable system is Lyapunov stable. Every equilibrium point q of the completely integrable Hamiltonian system (HS) is degenerate and has zero angular momentum, so the mentioned criterion cannot be applied to it. But we will show that this criterion is also satisfied when it is applied to the Hamiltonian system (HS) restricted to zero angular momentum.

Keywords

• Kepler problem
• Global dynamics
• Lyapunov
• Stability