Generalised Symmetries and the Ermakov-Lewis Invariant

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous ErmakovLewis invariant, but in fact reveals a richer structure. The origins of the linear second-order ordinary differential equation known as the timedependent linear oscillator are disparately manifold. A classical source is the lengthening pendulum described in the normal approximation by θ̈ + ω(t)θ = 0. (0.1) (The pendulum has to be one of increasing length. Otherwise the approximation sin θ ≈ θ breaks down [36, 35].) At the first Solvay Conference in 1911 Lorentz proposed an adiabatic invariant for (0.1) based on its Hamiltonian representation as Iadiabatic = 1 2ω(t) (