Hyperbolic Solutions of the Full Two-Body Problem

Journal of Nonlinear Science, 36 (2026), article 46
https://doi.org/10.1007/s00332-026-10265-9

In this work, we study a system of two rigid bodies interacting through Newtonian
gravitation and establish conditions for the occurrence of Hill instability. Specifically,
we prove the existence of forward hyperbolic orbits for an open and unbounded set
of initial conditions. These orbits are characterized by a linear growth in time of the
mutual distance between the bodies. We analyze the asymptotic behavior of both the
rotational and translational motions associated with such trajectories. For the rotational
dynamics, our results rely on the theory of asymptotically autonomous systems.
Regarding the translational motion, we derive an asymptotic formula for the relative
position of the barycenters of the bodies. In the final section, we outline several open
questions concerning forward hyperbolic orbits, which we believe may be of further
interest.