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Un-reduction of systems of second-order ordinary differential equations

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Abstract
In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) "primary un-reduced SODE", and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.
Keywords
reduction, symmetry, principal connection, second-order ordinary, differential equations, Lagrangian system

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Chicago
García-Toraño Andrés, Eduardo, and Tom Mestdag. 2016. “Un-reduction of Systems of Second-order Ordinary Differential Equations.” Symmetry Integrability and Geometry-methods and Applications 12.
APA
García-Toraño Andrés, E., & Mestdag, T. (2016). Un-reduction of systems of second-order ordinary differential equations. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS, 12.
Vancouver
1.
García-Toraño Andrés E, Mestdag T. Un-reduction of systems of second-order ordinary differential equations. SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 2016;12.
MLA
García-Toraño Andrés, Eduardo, and Tom Mestdag. “Un-reduction of Systems of Second-order Ordinary Differential Equations.” SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS 12 (2016): n. pag. Print.
@article{8508885,
  abstract     = {In this paper we consider an alternative approach to {\textacutedbl}un-reduction{\textacutedbl}. This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) {\textacutedbl}primary un-reduced SODE{\textacutedbl}, and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.},
  articleno    = {115},
  author       = {Garc{\'i}a-Tora{\~n}o Andr{\'e}s, Eduardo and Mestdag, Tom},
  issn         = {1815-0659},
  journal      = {SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS},
  keyword      = {reduction,symmetry,principal connection,second-order ordinary,differential equations,Lagrangian system},
  language     = {eng},
  pages        = {20},
  title        = {Un-reduction of systems of second-order ordinary differential equations},
  url          = {http://dx.doi.org/10.3842/SIGMA.2016.115},
  volume       = {12},
  year         = {2016},
}

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