Hyperbolic motions in the $N$-body problem with homogeneous potentials

In the $N$-body problem, a motion is called hyperbolic, when the mutual distances between the bodies go to infinity with non-zero limiting velocities as time goes to infinity. For Newtonian potential, in \cite{MV20} Maderna and Venturelli proved that starting from any initial position there is a hyperbolic motion with any prescribed limiting velocities at infinity. Recently based on a different approach, Liu, Yan and Zhou \cite{LYZ21} generalized this result to a larger class of $N$-body problem. As the proof in \cite{LYZ21} is quite long and technical, we give a simplified proof for homogeneous potentials following the approach given in the latter paper.