On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top

The aim of this work is to show how the moving frames method can be applied for reducing and solving two nonlinear mechanical systems: a bead on a rotating wire hoop and a spinning top. Once both problems are adequately formulated, we explicitly determine the corresponding moving frames associated to the symmetry group of transformations admitted by the systems. The knowledge of the moving frames for the action of the corresponding symmetry groups permits to perform order reductions. Furthermore, we are able to compute the general solutions to each problem from the general solutions of the corresponding reduced systems. Finally, we also discuss the connection of the presented approach with the classical method provided by the celebrated Noether's Theorem.