Journal of Nonlinear Mathematical Physics

Volume 12, Issue 1, February 2005, Pages 15 - 26

Generalised Symmetries and the Ermakov-Lewis Invariant

Authors
R. Goodall, P.G.L. Leach
Corresponding Author
R. Goodall
Received 21 July 2004, Accepted 15 September 2004, Available Online 1 February 2005.
DOI
https://doi.org/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 =
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
12 - 1
Pages
15 - 26
Publication Date
2005/02
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.2005.12.1.3How to use a DOI?
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

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  -