Transformation groups on real plane and their differential invariants

Abstract

Complete sets of bases of differential invariants, operators of invariant differentiation, and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.

Group classification of second-order delay ordinary differential equations

“…DODEs do not posses an equivalence transformation related with the change of the dependent and independent variables. Hence, to find invariants of second-order DODEs one needs to consider a representation of a Lie algebra which is equivalent (with respect to change of the dependent and independent variables) to one of Lie algebras of Table 1 in [6]. Let Since x and y are considered as arbitrary variables, for convenience the symbols x and y will be used instead from here on.…”

“…As explained in Section 1, there is a complete description of all finite dimensional Lie algebras on the real plane [6]. This classification is obtained up to a nonsingular change of the variables x and y, and consists of a list of 56 Lie algebras (see Table 1 in [6]).…”

“…This classification is obtained up to a nonsingular change of the variables x and y, and consists of a list of 56 Lie algebras (see Table 1 in [6]). Since the set of generators admitted by a second-order DODE composes a Lie algebra, then after some change of the dependent and independent variables 3 this algebra is equivalent to one of these 56 Lie algebras.…”

“…Here the notation L n j is used to denote the n-dimensional Lie algebra of the number j from Table 1 in [6]. Example 1.…”

“…. ; g r form a fundamental system of solutions of an r-order ordinary differential equation Table 1 in [6]. Consider r ¼ 2, the Lie algebra is defined by the generators…”