The Equivalence Problem for Systems of Second‐Order Ordinary Differential Equations

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.