## Abstract

A novel procedure to reduce by four the order of Euler-Lagrange equations associated to nth order variational problems involving single variable integrals is presented. In preparation, a new formula for the commutator of two C^{∞}-symmetries is established. The method is based on a pair of variational C^{∞}-symmetries whose commutators satisfy a certain solvability condition. It allows one to recover a (2n - 2)-parameter family of solutions for the original 2nth order Euler-Lagrange equation by solving two successive first order ordinary differential equations from the solution of the reduced Euler-Lagrange equation. The procedure is illustrated by two different examples.

Original language | English (US) |
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Article number | 145202 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 51 |

Issue number | 14 |

DOIs | |

State | Published - Mar 8 2018 |

### Bibliographical note

Funding Information:A R and C M acknowledge financial support from the University of Cádiz and from Junta de Andalucía to the research group FQM 377.

Funding Information:

A R acknowledges financial support from the Ministry of Education, Culture and Sport of Spain (FPU grant FPU15/02872) and from a grant of the University of Cádiz program ‘Movilidad Internacional, Becas UCA-Internacional Posgrado 2016–2017’ during his stay at the University of Minnesota.

Publisher Copyright:

© 2018 IOP Publishing Ltd.

## Keywords

- Euler-Lagrange equation
- Lagrangian
- commutator
- solvability condition
- variational C-symmetry
- variational problem

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