Volume 12, Issue 1, February 2005, Pages 15 - 26
Generalised Symmetries and the Ermakov-Lewis Invariant
R. Goodall, P.G.L. Leach
Received 21 July 2004, Accepted 15 September 2004, Available Online 1 February 2005.
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- Generalised symmetries and point symmetries coincide for systems of first-order odinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the sytem of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous ErmakoLewis invariant, but in fact reveals a richer structure. The origins of the linear second-order ordinary differential equation known as the timdependent linear oscillator are disparately manifold. A classical source is the lengthening pendulum described in the normal approximation by ¨ + 2 (t) = 0. (0.1) (The pendulum has to be one of increasing length. Otherwise the approximation sin 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 =
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TY - JOUR AU - R. Goodall AU - P.G.L. Leach PY - 2005 DA - 2005/02 TI - Generalised Symmetries and the Ermakov-Lewis Invariant JO - Journal of Nonlinear Mathematical Physics SP - 15 EP - 26 VL - 12 IS - 1 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2005.12.1.3 DO - https://doi.org/10.2991/jnmp.2005.12.1.3 ID - Goodall2005 ER -