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 modelled on R 6 × (SO(3) × so(3)) 2 × (TS 2 ) 2 and put in a format where Lie group integrators can be applied.

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Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined by one scalar function, the Hamiltonian. The solution trajectories are often constrained to evolve on a submanifold of a linear vector space. In this work, we propose new approaches for the accurate approximation of the Hamiltonian function of constrained mechanical systems given sample data information of their solutions. We focus on the importance of the preservation of the constraints in the learning strategy by using both explicit Lie group integrators and other classical schemes.

Since their introduction, Lie group integrators have become a method of choice in many application areas. Various formulations of these integrators exist, and in this work we focus on Runge-Kutta-Munthe-Kaas methods. First, we briefly introduce this class of integrators, considering some of the practical aspects of their implementation, such as adaptive time stepping. We then present some mathematical background that allows us to apply them to some families of Lagrangian mechanical systems. We conclude with an application to a nontrivial mechanical system: the N-fold 3D pendulum.