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Classification of Second‐Order Ordinary Differential Equations Admitting Lie Groups of Fibre‐Preserving Point Symmetries
Author(s) -
Hsu L.,
Kamran N.
Publication year - 1989
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-58.2.387
Subject(s) - mathematics , homogeneous space , ordinary differential equation , transitive relation , differential equation , symmetry (geometry) , separable partial differential equation , lie group , group (periodic table) , order (exchange) , mathematical analysis , symmetry group , lie theory , equivalence (formal languages) , pure mathematics , differential algebraic equation , combinatorics , adjoint representation of a lie algebra , physics , geometry , lie conformal algebra , finance , quantum mechanics , economics
We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second‐order ordinary differential equations admitting Lie groups of fibre‐preserving point symmetries. We then apply our results to the determination of all second‐order equations which are equivalent, under fibre‐preserving transformations, to the free particle equation. In addition we present those equations of Painlevé type which admit a transitive symmetry group. Finally we determine the symmetry group of some equations of physical interest, such as the Duffing and Holmes‐Rand equations, which arise as models of non‐linear oscillators.