Li(e)nearity

TL;DR

This paper explores the significance of linearity as a symmetry in differential equations, demonstrating its role in solving, conserved quantities, and deriving Lagrangians through Lie and Noether symmetries.

Contribution

It provides a comprehensive analysis of linearity symmetry in differential equations, linking Lie invariants, conserved quantities, and Lagrangian derivation, extending to higher-order cases.

Findings

Linearity symmetry corresponds to a Lie invariant and conserved quantity.

The Caldirola-Kanai Lagrangian can be derived from symmetry principles.

Linearity symmetry extends to higher-order differential equations.

Abstract

We demonstrate the fact that linearity is a meaningful symmetry in the sense of Lie and Noether. The role played by that `linearity symmetry' in the quadrature of linear ordinary second-order differential equations is reviewed, by the use of canonical coordinates and the identification of a Wronskian-like conserved quantity as Lie invariant. The Jacobi last multiplier associated with two independent linearity symmetries is applied to derive the Caldirola-Kanai Lagrangian from symmetry principles. Then the symmetry is recognized to be also a Noether one. Finally, the study is extended to higher-order linear ordinary differential equations, derivable or not from an action principle.