  It has been suggested that a possible classical remnant of the phenomenon of
target-space duality (T-duality) would be the equivalence of the classical
string Hamiltonian systems. Given a simple compact Lie group $G$ with a
bi-invariant metric and a generating function $\Gamma$ suggested in the physics
literature, we follow the above line of thought and work out the canonical
transformation $\Phi$ generated by $\Gamma$ together with an $\Ad$-invariant
metric and a B-field on the associated Lie algebra $\frak g$ of $G$ so that $G$
and $\frak g$ form a string target-space dual pair at the classical level under
the Hamiltonian formalism. In this article, some general features of this
Hamiltonian setting are discussed. We study properties of the canonical
transformation $\Phi$ including a careful analysis of its domain and image. The
geometry of the T-dual structure on $\frak g$ is lightly touched.
