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No Access Submitted: 17 April 2019 Accepted: 28 August 2019 Published Online: 20 September 2019

  • On a global Lagrangian construction for ordinary variational equations on 2-manifolds

J. Math. Phys. 60, 092902 (2019); https://doi.org/10.1063/1.5100351
Zbyněk Urbana) and Jana Volnáb)
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  • Zbyněk Urban
  • Jana Volná
ABSTRACT
Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, however, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of the solvability of the exactness equation for the Lepage 2-forms and the top-cohomology theorems. Examples from geometry and mechanics are discussed.

ACKNOWLEDGMENTS

This work has been completed—thanks to the financial support provided to the VSB-Technical University of Ostrava by the Czech Ministry of Education, Youth and Sports from the budget for the conceptual development of science, research, and innovations for the year 2019, Project No. IP2309921/2104. Z.U. also appreciates support from the Visegrad Grant No. 51810810 at the University of Prešov. The authors are thankful to the anonymous referee for valuable comments improving this paper.

REFERENCES
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  1. © 2019 Author(s). Published under license by AIP Publishing.

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