  A new description, different by the classical theory of Hamiltonian
Mechanics, in the general framework of generalized Lie algebroids is presented.
In the particular case of Lie algebroids, new and important results are
obtained. We present the \emph{dual mechanical systems} called by use,
\emph{dual mechanical}$(\rho,\eta) $\emph{-systems, Hamilton
mechanical}$(\rho,\eta) $\emph{-systems} or \emph{% Cartan
mechanical}$(\rho,\eta) $\emph{-systems.} and we develop their geometries. We
obtain the canonical $(\rho,\eta) $\emph{-}semi(spray) associated to a dual
mechanical $(\rho,\eta) $-system. The Hamilton mechanical $(\rho,\eta)$-systems
are the spaces necessary to develop a Hamiltonian formalism. We obtain the
$(\rho,\eta)$-semispray associated to a regular Hamiltonian $H$ and external
force $F_{e}$ and we derive the equations of Hamilton-Jacobi type.
