Li(e)nearity

International audienceThis article brings to light the fact that linearity is by itself a meaningful symmetry in the senses of Lie and Noether. First, the role played by that ‘linearity symmetry’ in the quadrature of linear second-order differential equations is revisited through the use of adapted variables and the identification of a conserved quantity as Lie invariant. Second, the celebrated Caldirola–Kanai Lagrangian—from which the differential equation is deducible—is shown to be naturally generated by a Jacobi last multiplier inherited from the linearity symmetry. Then, the latter is recognised to be also a Noether one. Finally, the study is extended to higher-order linear differential equations, derivable or not from an action principle. Incidentally, this work can serve as an introduction to the central question of continuous symmetries in physics and mathematics. It has the advantage of being approachable to undergraduate students since the linearity symmetry is by its very nature sufficiently simple to be treatable without any use of Lie generators