Abstract
This chapter discusses Lie groups applicable to linear and nonlinear ordinary differential equations (ODEs). Technically, a Lie group is a topological group (that is, a group that is also a topological space), which is also an analytic manifold on which the group operations are analytic. The tangent space to that manifold is Lie algebra, which is a linear vector space. Lie groups yield invariants and symmetries of a differential equation. Often, these can be used to solve a differential equation. By determining the transformation group under which a given differential equation is invariant, the information can be obtained about the invariants and symmetries of a differential equation. Sometimes these can be used to solve a given differential equation. Using Lie groups to find symmetries of differential equations can be computationally intensive. Algorithms have been developed for computerized handling of the calculations.
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