  Whereas in a coordinate-dependent setting the Euler-Lagrange equations
establish necessary conditions for solving variational problems in which both
the integrands of functionals and the resulting paths are assumed to be
sufficiently smooth, uniqueness and global optimality are generally hard to
prove in the absence of convexity conditions, and often times they may not even
exist. This is also true for variational problems on Lie groups, with the
Euler-Poincar\'{e} equation establishing necessary conditions. The difficulties
compound when integrands and/or the optimal paths are not sufficiently regular,
since in this case the classical necessary conditions no longer apply. This
article therefore reviews several nonstandard cases where unique globally
optimal solutions can be guaranteed, and establishes a ``bootstrapping''
process to build new globally optimal variational solutions on larger spaces
from existing ones on smaller spaces. Surprisingly, it is possible to prove
global optimality in some nonconvex cases where even the regularity conditions
required for classical necessary conditions do not hold. This general theory is
then applied to several topics such as optimal framing of curves in
three-dimensional Euclidean space, optimal motion interpolation, and optimal
reparametrization of video sequences to compare salient actions.
