Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 013, 18 pages      arXiv:2309.01594      https://doi.org/10.3842/SIGMA.2024.013
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Lepage Equivalents and the Variational Bicomplex

David Saunders
Lepage Research Institute, Prešov, Slovakia

Received October 05, 2023, in final form January 30, 2024; Published online February 09, 2024

Abstract
We show how to construct, for a Lagrangian of arbitrary order, a Lepage equivalent satisfying the closure property: that the Lepage equivalent vanishes precisely when the Lagrangian is null. The construction uses a homotopy operator for the horizontal differential of the variational bicomplex. A choice of symmetric linear connection on the manifold of independent variables, and a global homotopy operator constructed using that connection, may then be used to extend any global Lepage equivalent to one satisfying the closure property. In the second part of the paper we investigate the rôle of vertical endomorphisms in constructing such Lepage equivalents. These endomorphisms may be used directly to construct local homotopy operators. Together with a symmetric linear connection they may also be used to construct global vertical tensors, and these define infinitesimal nonholonomic projections which in turn may be used to construct Lepage equivalents. We conjecture that these global vertical tensors may also be used to define global homotopy operators.

Key words: jet bundle; Poincaré-Cartan form; Lepage equivalent of a Lagrangian; variational bicomplex.

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