Null Lagrangians and Invariance of Action: Exact Gauge Functions

TL;DR

This paper derives and generalizes null Lagrangians and gauge functions to ensure action invariance, introducing a new family of exact null Lagrangians applicable to any ODE with a known Lagrangian.

Contribution

It presents new non-standard null Lagrangians and establishes conditions for action invariance, expanding the theoretical framework of null Lagrangians and gauge functions.

Findings

Generalized standard null Lagrangians

Introduced new non-standard null Lagrangians

Derived conditions for action invariance

Abstract

Null Lagrangians and their gauge functions are derived for given standard and non-standard Lagrangians. The obtained standard null Lagrangians generalize those previously found but the non-standard null Lagrangians are new. The gauge functions are used to make the action invariant and introduce the exact null Lagrangians, which form a new family of null Lagrangians. The conditions required for the action to be invariant are derived for all null Lagrangians presented in this paper, thus, making them exact. As a specific application, the exact null Lagrangians are derived for a second-order ordinary differential equation. The application shows that the exact null Lagrangians can be obtained for any ordinary differential equation with known Lagrangian.