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In mathematics, a Lie transform or Lie transformation is a specific type of infinitesimal transform/transformation (arbitrarily small transformations) of manifolds, all such transforms of a particular manifold forming a type of transformation group termed a Lie group, which has an associated algebra (Lie algebra). Lie transforms can be used in the development of perturbation theories, using abstract algebra concepts and methods to handle the differential equations of Hamiltonian mechanical systems, which are used for analysis of orbits in celestial mechanics as well as in electron orbits. Lie transforms, Lie groups and Lie algebras are based on underlying work by the mathematician Sophus Lie and named for him.