  Hamiltonian systems that are either open, leaking, or contain holes in the
phase space possess solutions that eventually escape the system's domain. The
motion described by such escape orbits before crossing the escape threshold can
be understood as a transient behavior. In this work, we introduce a numerical
method to visually illustrate and quantify the transient motion in Hamiltonian
systems based on the transient measure, a finite-time version of the natural
measure. We apply this method to two physical systems: the single-null divertor
tokamak, described by a symplectic map; and the Earth-Moon system, as modeled
by the planar circular restricted three-body problem. Our results portray how
different locations for the ensemble of initial conditions may lead to
different transient dynamical scenarios in both systems. We show that these
scenarios can be properly quantified from a geometrical aspect, the transient
correlation dimension, and a dynamical aspect, the transient complexity
coefficient.
