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

2018/12/11 by Zbyněk Urban, Jana Volná · 1 citation
Mathematics · #math.DG #math.AT #msc:58E30 #msc:58A15 #msc:14F40 #msc:70H99

paper · pdf · doi:10.1063/1.5100351

published as J. Math. Phys. 60, 092902 (2019)

arxiv created 2018/12/11 · arxiv updated 2020/04/01

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, how-ever, 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 solvability of exactness equation for Lepage 2-forms, and the top-cohomology theorems. Examples from geometry and mechanics are discussed.

Cited by

• Exactness of Lepage 2-forms and globally variational differential equations

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