Embedded geodesic problems and optimal control for matrix lie groups

This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic ows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called n-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these ows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.