Null Lagrangians and Gauge Functions in Physics: Applications and Recent Developments

The Lagrangian formalism has provided a powerful and elegant framework for obtaining governing equations for classical and quantum systems. It is based on the concept of action, which involves Lagrangians, whose a priori knowledge is required. There are different methods to obtain Lagrangians for given equations of motion, and a brief review of these methods is presented. However, the main purpose of this review paper is to describe the so-called null Lagrangians and their gauge functions, and discuss their physical applications. The paper also reviews some recent results, which demonstrate that gauge functions play the most fundamental roles in classical dynamics as they can be used to predict the future states of dynamical systems, without solving the equations of motion, as well as to construct their Lagrangians.