  Variational integrators for Euler--Lagrange equations and Hamilton's
equations are a class of structure-preserving numerical methods that respect
the conservative properties of such systems. Lie group variational integrators
are a particular class of these integrators that apply to systems which evolve
over the tangent bundle and cotangent bundle of Lie groups. Traditionally,
these are constructed from a variational principle which assumes fixed position
endpoints. In this paper, we instead construct Lie group variational
integrators with a novel Type II variational principle on the cotangent bundle
of a Lie group which allows for Type II boundary conditions, i.e., fixed
initial position and final momenta; these boundary conditions are particularly
important for adjoint sensitivity analysis, which is the motivating application
in our paper. In general, such Type II variational principles are only globally
defined on vector spaces or locally defined on general manifolds; however, by
left translation, we are able to define this variational principle globally on
cotangent bundles of Lie groups. By developing the continuous and discrete Type
II variational principles over Lie groups, we construct a structure-preserving
Lie group variational integrator that is both symplectic and
momentum-preserving. Subsequently, we introduce adjoint systems on Lie groups,
and show how these adjoint systems can be used to perform geometric adjoint
sensitivity analysis for optimization problems on Lie groups. Finally, we
conclude with two numerical examples to show how adjoint sensitivity analysis
can be used to solve initial-value optimization problems and optimal control
problems on Lie groups.
