  A key aspect of fluid dynamics is the correct definition of the \textit{%
phase-space} Lagrangian dynamics which characterizes arbitrary fluid elements
of an incompressible fluid. Apart being an unsolved theoretical problem of
fundamental importance, the issue is relevant to exhibit the connection between
fluid dynamics and the classical dynamical systems underlying incompressible
and non-isothermal fluid, typically founded either on: a) a
\textit{configuration-space} Lagrangian description of the dynamics of fluid
elements; b) a kinetic description of the molecular dynamics, based on a
discrete representation of the fluid. The goal of this paper is to show that
the exact Lagrangian dynamics can be established based on the inverse kinetic
theory (IKT) for incompressible fluids recently pointed out (Ellero \textit{et
al.}, 2004-2006, \cite{Ellero2004}). The result is reached by adopting an IKT
approach based on a \textit{restricted phase-space representation} of the
fluid, in which the configuration space coincides with the physical fluid
domain. The result appears of potential importance in applied fluid dynamics
and CFD.
