Applications of the Lie symmetries to complete solution of a bead on a rotating wire hoop

  • Cláudio H. C. Costa Basquerotto
  • Edison Righetto
  • Samuel da Silva
Technical Paper


The existence of Lie symmetries in differential equations can generate transformations in the dependent and independent variables and obtain new equations that may be easier to integrate. In particular, in some situations, one can reduce the order and it is possible to obtain first integrals. Thus, this article presents the application of the fundamental Lie theorem to obtain the complete solution of a classical nonlinear problem of the dynamics of mechanical systems: the bead on a rotating wire hoop. From the first integral obtained with the Lie symmetry generators, the exact solution can be found with the aid of the Jacobi elliptic functions.


Lie symmetries Classical mechanics Jacobi elliptic functions 



The first author would like to thank his scholarship from CAPES and the financial support provided during his stay at the University of Minnesota (USA). The authors are also grateful for the financial support provided by National Council for Scientific and Technological Development (CNPq - Brazil) Grant number 404463/2016-9 that intends to popularize the Lie theory in the engineering community. Additionally, the authors would like to thank the anonymous reviewers and the Associate Editor for their relevant comments and useful suggestions. The authors also thank the comments and revision performed by Prof. Peter J. Olver and we would like to thank his kind hospitality during the stay of the first author in Minnesota.

Compliance with ethical standards

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  • Cláudio H. C. Costa Basquerotto
    • 1
  • Edison Righetto
    • 1
  • Samuel da Silva
    • 1
  1. 1.Departamento de Engenharia MecânicaUniversidade Estadual Paulista, UNESP, Faculdade de Engenharia de Ilha SolteiraIlha SolteiraBrazil

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