On the solution of linear differential equations in Lie groups

Abstract

The subject matter of this paper is the solution of the linear differential equation y′ = a(t)y,y(0) = y0, where y0 ∈ G, a(.): R+ → g and g is a Lie algebra of the Lie group G. By building upon an earlier work of Wilhelm Magnus, we represent the solution as an infinite series whose terms are indexed by binary trees. This relationship between the infinite series and binary trees leads both to a convergence proof and to a constructive computational algorithm. This numerical method requires the evaluation of a large number of multivariate integrals, but this can be accomplished in a tractable manner by using quadrature schemes in a novel manner and by exploiting the structure of the Lie algebra.