In the latter part of the 19th century Sophus Lie introduced the notion of continuous groups, now known as Lie groups, in order to unify and extend various specialized solution methods for ordinary differential equations. Lie was inspired by lectures of Sylow given at Christiania (present-day Oslo) on Galois theory and Abel’s related works. [In 1881 Sylow and Lie collaborated in a careful editing of Abel’s complete works.] Lie showed that the order of an ordinary differential equation can be reduced by one, constructively, if it is invariant under a one-parameter Lie group of point transformations.
KeywordsInfinitesimal Generator Determine Equation Invariant Solution Continuous Group Point Transformation
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