Generalised Symmetries and the Ermakov-Lewis Invariant

DOI

10.2991/jnmp.2005.12.1.3How to use a DOI?

Abstract

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 =

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - R. Goodall
AU  - P.G.L. Leach
PY  - 2005
DA  - 2005/02/01
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  - 10.2991/jnmp.2005.12.1.3
ID  - Goodall2005
ER  -