  Explicit, momentum-based dynamics that optimize functions defined on Lie
groups can be constructed via variational optimization and momentum
trivialization. Structure preserving time discretizations can then turn this
dynamics into optimization algorithms. This article investigates two types of
discretization, Lie Heavy-Ball, which is a known splitting scheme, and Lie
NAG-SC, which is newly proposed. Their convergence rates are explicitly
quantified under $L$-smoothness and local strong convexity assumptions. Lie
NAG-SC provides acceleration over the momentumless case, i.e. Riemannian
gradient descent, but Lie Heavy-Ball does not. When compared to existing
accelerated optimizers for general manifolds, both Lie Heavy-Ball and Lie
NAG-SC are computationally cheaper and easier to implement, thanks to their
utilization of group structure. Only gradient oracle and exponential map are
required, but not logarithm map or parallel transport which are computational
costly.
