Composition of Lie Group Elements from Basis Lie Algebra Elements

It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions.