  Since they were introduced in the 1990s, Lie group integrators have become a
method of choice in many application areas. These include multibody dynamics,
shape analysis, data science, image registration and biophysical simulations.
Two important classes of intrinsic Lie group integrators are the
Runge--Kutta--Munthe--Kaas methods and the commutator free Lie group
integrators.
  We give a short introduction to these classes of methods. The Hamiltonian
framework is attractive for many mechanical problems, and in particular we
shall consider Lie group integrators for problems on cotangent bundles of Lie
groups where a number of different formulations are possible. There is a
natural symplectic structure on such manifolds and through variational
principles one may derive symplectic Lie group integrators. We also consider
the practical aspects of the implementation of Lie group integrators, such as
adaptive time stepping. The theory is illustrated by applying the methods to
two nontrivial applications in mechanics. One is the N-fold spherical pendulum
where we introduce the restriction of the adjoint action of the group $SE(3)$
to $TS^2$, the tangent bundle of the two-dimensional sphere. Finally, we show
how Lie group integrators can be applied to model the controlled path of a
payload being transported by two rotors. This problem is modeled on
$\mathbb{R}^6\times \left(SO(3)\times \mathfrak{so}(3)\right)^2\times (TS^2)^2$
and put in a format where Lie group integrators can be applied.
