Journal of Lie Theory 33 (2023), No. 4, 1045--1086
Copyright Heldermann Verlag 2023
ODEs whose Symmetry Groups are not Fiber-Preserving
Boris Kruglikov
Dept. of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, Norway
boris.kruglikov@uit.no
Eivind Schneider
Dept. of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, Norway
eivind.schneider@uit.no
We observe that, up to conjugation, a majority of higher order ODEs (ordinary differential equations) and ODE systems have only fiber-preserving point symmetries. By exploiting Lie's classification of Lie algebras of vector fields, we describe all exceptions to this in the case of scalar ODEs and systems of ODEs on a pair of functions. The scalar ODEs whose symmetry algebras are not fiber preserving can be expressed via absolute and relative scalar differential invariants, while a similar description for ODE systems requires us to also invoke conditional differential invariants and vector-valued relative invariants to deal with singular orbits of the action. Investigating prolongations of the actions, we observe some interesting relations between different realizations of Lie algebras. We also note that it may happen that the prolongation of a finite-dimensional Lie algebra acting on a differential equation never becomes free. An example of an underdetermined ODE system for which this phenomenon occurs shows limitations of the method of moving frames.
Keywords: Point symmetries, contact transformations, differential invariants, relative invariants, conditional invariants.
MSC: 53A55, 17B66, 34C14.
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