  For the classical $N$-body problem in $\mathbb{R}^d$ with $d\ge2$,
Maderna-Venturelli in their remarkable paper [Ann. Math. 2020] proved the
existence of hyperbolic motions with any positive energy constant, starting
from any configuration and along any non-collision configuration. Their
original proof relies on the long time behavior of solutions by Chazy 1922 and
Marchal-Saari 1976, on the H\"{o}lder estimate for Ma\~{n}\'{e}'s potential by
Maderna 2012, and on the weak KAM theory.
  We give a new and completely different proof for the above existence of
hyperbolic motions. The central idea is that, via some geometric observation,
we build up uniform estimates for Euclidean length and angle of geodesics of
Ma\~{n}\'{e}'s potential starting from a given configuration and ending at the
ray along a given non-collision configuration. Note that we do not need any of
the above previous studies used in Maderna-Venturelli's proof.
  Moreover, our geometric approach works for Hamiltonians
$\frac12\|p\|^2-F(x)$, where $F(x)\ge 0$ is lower semicontinuous and decreases
very slowly to $0$ faraway from collisions. We therefore obtain the existence
of hyperbolic motions to such Hamiltonians with any positive energy constant,
starting from any admissible configuration and along any non-collision
configuration. Consequently, for several important potentials $F\in
C^{2}(\Omega)$, we get similar existence of hyperbolic motions to the
generalized $N$-body system $\ddot{x} = \nabla_x F(x)$, which is an extension
of Maderna-Venturelli [Ann. Math. 2020].
