General Null Lagrangians and Their Novel Role in Classical Dynamics

TL;DR

This paper introduces a method for constructing null Lagrangians and their higher harmonics, revealing their novel role in deriving equations of motion for classical dynamical systems, including dissipative cases.

Contribution

It presents a new approach to construct null Lagrangians and demonstrates their application in obtaining equations of motion for various classical systems.

Findings

Null Lagrangians can produce equations of motion for inertia and dissipative systems.

A necessary condition for deriving equations of motion using null Lagrangians is established.

Null Lagrangians play a role analogous to Euler-Lagrange equations in classical dynamics.

Abstract

A method for constructing general null Lagrangians and their higher harmonics is presented for dynamical systems with one degree of freedom. It is shown that these Lagrangians can be used to obtain non-standard Lagrangians, which give equations of motion for the law of inertia and some dissipative dynamical systems. The necessary condition for deriving equations of motion by using null Lagrangians is presented, and it is demonstrated that this condition plays the same role for null Lagrangians as the Euler-Lagrange equation plays for standard and non-standard Lagrangians. The obtained results and their applications establish a novel role of null Lagrangians in classical dynamics.