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The Equivalence Problem for Systems of Second‐Order Ordinary Differential Equations
Author(s) -
Fels M. E.
Publication year - 1995
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/s3-71.1.221
Subject(s) - mathematics , ordinary differential equation , pure mathematics , equivalence (formal languages) , dimension (graph theory) , differential equation , mathematical analysis , separable partial differential equation , differential algebraic equation
The equivalence problem for systems of second‐order differential equations under point transformations is found to give rise to an { e }‐structure of dimension n 2 + 4 n + 3. It is then shown that the structure function for this { e }‐structure is a differential function of two fundamental tensor invariants. The parametric forms of the fundamental invariants are given and their vanishing characterizes the trivial equationx ¨ i = 0 . We also show that the vanishing of the fundamental invariants characterizes the unique system of second‐order ordinary differential equations admitting a maximal‐dimension Lie symmetry group. Thus, equations not equivalent tox ¨ i = 0 admit symmetry groups of dimension strictly less than n 2 + 4 n + 3.