Iterated Commutators, Lie's Reduction Method and Ordinary Di erential Equations on Matrix Lie Groups

In the context of devising geometrical integrators that retain qualitative features of the underlying solution, we present a family of numerical methods (the method of iterated commutators, [5, 13]) to integrate ordinary di erential equations that evolve on matrix Lie groups. The schemes apply to the problem of nding a numerical approximation to the solution of Y 0 = A(t;Y )Y; Y (0) = Y0; whereby the exact solution Y evolves in a matrix Lie group G and A is a matrix function on the associated Lie algebra g. We show that the method of iterated commutators, in a linear setting, is intrinsically related to Lie's reduction method for nding the fundamental solution of the Lie-group equation Y 0 = A(t)Y .