Trivial Second-Order Lagrangians in Classial Field Theory

TL;DR

This paper provides a comprehensive analysis of trivial second-order Lagrangians in classical field theory, detailing their dependence on second derivatives and exploring polynomial structures called hyper-Jacobians.

Contribution

It extends previous work by offering a complete description of second-order derivative dependence and clarifies linear dependencies among hyper-Jacobians.

Findings

Complete characterization of trivial second-order Lagrangians

Identification of polynomial hyper-Jacobians and their dependencies

Extension of prior theoretical frameworks in classical field theory

Abstract

Trivial second-order Lagrangians are studied and a complete description of the dependence on the second-order derivatives is given. This extends previous work of Olver and others. In particular, this description involves some polynomial expressions called hyper-Jacobians. There exists some linear dependencies between these polynomials which are elucidated for the (second-order) hyper-Jacobians.