  In this article, we present new computational realizations of principal
geodesic analysis for the unit sphere $S^2$ and the special orthogonal group
$SO(3)$. In particular, we address the construction of long-time smooth lifts
across branches of the respective logarithm maps. To this end, we pay special
attention to certain critical numerical aspects such as singularities and their
consequences on the numerical precision. Moreover, we apply principal geodesic
analysis to investigate the behavior of several mechanical systems that are
very rich in dynamics. The examples chosen are computationally modeled by
employing a director-based formulation for rigid and flexible mechanical
systems. Such a formulation allows to investigate our algortihms in a direct
manner while avoiding the introduction of additional sources of error that are
unrelated to principal geodesic analysis. Finally, we test our numerical
machinery with the examples and, at the same time, we gain deeper insight into
their dynamical beahvior.
