  Even the most simplified models of falling and gliding bodies exhibit rich
nonlinear dynamical behavior. Taking a global view of the dynamics of one such
model, we find an attracting invariant manifold that acts as the dominant
organizing feature of trajectories in velocity space. This attracting manifold
captures the final, slowly changing phase of every passive descent, providing a
higher-dimensional analogue to the concept of terminal velocity, the terminal
velocity manifold. Within the terminal velocity manifold in extended phase
space, there is an equilibrium submanifold with equilibria of alternating
stability type, with different stability basins. In this work, we present
theoretical and numerical methods for approximating the terminal velocity
manifold and discuss ways to approximate falling and gliding motion in terms of
these underlying phase space structures.
