Null Lagrangians and Gauge Functions in Dynamical Systems: Forces and Nonlinearities

TL;DR

This paper explores special null Lagrangians and gauge functions that can transform simple linear dynamical equations into complex nonlinear and driven systems, revealing new roles in dynamics.

Contribution

It constructs general null Lagrangians and gauge functions for second-order ODEs, identifying their ability to induce nonlinearity and forcing in dynamical systems.

Findings

Gauge functions relate to forces and nonlinearities in various systems

Null Lagrangians can convert linear equations into nonlinear driven ones

Novel roles of gauge functions in dynamical behavior are discussed

Abstract

Among different Lagrangians, null Lagrangians are known for having identically zero the Euler-Lagrange equation and, therefore, they have no effects on the resulting equations of motion. However, there is a special family of null Lagrangians that can be used to convert linear and undriven equations of motion into nonlinear and driven ones. To identify this special family, general null Lagrangians and their gauge functions are constructed for second-order ordinary differential equations of motion describing one-dimensional dynamical systems. The gauge functions corresponding to forces and nonlinearities in a variety of known dynamical systems are presented and novel roles of these functions in dynamics are discussed.