Lie group integrators for mechanical systems

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