On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top
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O R I G I NA L PA P E R
C. H. C. C. Basquerotto
· A. Ruiz
On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top
Received: 6 June 2020 / Revised: 21 July 2020 / Accepted: 9 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract 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. Keywords Moving frames · Mechanical systems · Lie symmetries · Noether symmetries 1 Introduction Many problems found in mechanical engineering can be described by differential equations that need to be solved by analytical methods or approximated by numerical methods of integration [1,40,44]. Thus, in the middle of the nineteenth century, the Norwegian mathematician Sophus Lie presented an ingenious alternative to solve differential equations using symmetries that can be associated with invariants or conserved quantities [4,35]. The great original contribution of Lie was to advance in the understanding of the concept of symmetry group of transformations, which helped to set up the stage for the fundamental advances in physics and mathematics that took place in the twentieth century. As a result of Lie’s work, several integration methods for solving special classes of differential equations have been elegantly developed or unified. It is important to highlight that before Lie’s work, the known methods for solving differential equations were extremely dispersed and specific with almost no generality or unification of concepts. Moreover, the importance of the knowledge of a symmetry group of transformations for a differential equation lies in the possibility of reducing the order or, in some cases, to reduce the equation to a quadrature [3,12,30,33]. Later on, early in the twentieth century, Emmy Noether established her celebrated theorem, which concerns the invariance of a variational problem under the action of a Lie group with a finite number of independent infinitesimal generators (called Noether symmetries) and associates to each one of these generators a conservation law of the corresponding Euler–Lagrange equation [11,25,29,43]. This is the typical situation that C. H. C. C. Basquerotto (B) Instituto de Geociências e Engenharias, Faculdade de Engenharia Mecânica, Universidade Federal do Sul e Sudeste do Pará, Marabá, Pará 68505-080, Brazil E-m
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