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It has been suggested that a possible classical remnant of the phenomenon oftarget-space duality (T-duality) would be the equivalence of the classicalstring Hamiltonian systems. Given a simple compact Lie group $G$ with abi-invariant metric and a generating function $\Gamma$ suggested in the physicsliterature, we follow the above line of thought and work out the canonicaltransformation $\Phi$ generated by $\Gamma$ together with an $\Ad$-invariantmetric 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 underthe Hamiltonian formalism. In this article, some general features of thisHamiltonian setting are discussed. We study properties of the canonicaltransformation $\Phi$ including a careful analysis of its domain and image. Thegeometry of the T-dual structure on $\frak g$ is lightly touched.Comment: Two references and related comments added, also some typos corrected.  LaTeX and epsf.tex, 36 pages, 4 EPS figures included in a uuencoded fil

Target-space duality between simple compact Lie groups and Lie algebras under the Hamiltonian formalism: I. Remnants of duality at the classical level