Abstract
Lie symmetry analysis provides a systematic method of obtaining exact solutions of nonlinear (systems of) differential equations, whether partial or ordinary. Of special interest is the procedure that Lie developed to transform scalar nonlinear second‐order ordinary differential equations to linear form. Not much work was done in this direction to start with, but recently there have been various developments. Here, first the original work of Lie (and the early developments on it), and then more recent developments based on geometry and complex analysis, apart from Lie’s own method of algebra (namely, Lie group theory), are reviewed. It is relevant to mention that much of the work is not linearization but uses the base of linearization.
Highlights
Symmetry has been one of the criteria of aesthetics and beauty but has repeatedly proved extremely useful
Inspired by Galois’ success with algebraic polynomial equations, Lie tried to replicate it for differential equations
Galois’ development seems more fundamental than Lie’s. It led to more definitive results, namely, the non solvability of quintic and higher order equations by means of radicals
Summary
Symmetry has been one of the criteria of aesthetics and beauty but has repeatedly proved extremely useful. One method Lie adopted was a generalization of the methods for some specific first-order ODEs, changing them to linear form by using an invertible transformation of the dependent and independent variables. Lie proved that the necessary and sufficient condition for a scalar nonlinear ODE to be linearizable is that it must have 8 Lie point symmetries He exploited the fact that all scalar linear second-order ODEs are equivalent under point transformations; that is every linearizable scalar ODE is reducible to the free particle equation. All the work mentioned so far has been for scalar ODEs. Nothing has been said about systems of ODEs. In 1988 systems of two second-order ODEs, linearizable to constant coefficient systems, were proved to have three equivalence classes 15 with 7, 8, or 15 dimensional Lie algebras.
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