  A new relation among a class of optimal control systems and Lagrangian
systems with symmetry is discussed. It will be shown that a family of solutions
of optimal control systems whose control equation are obtained by means of a
group action are in correspondence with the solutions of a mechanical
Lagrangian system with symmetry. This result also explains the equivalence of
the class of Lagrangian systems with symmetry and optimal control problems
discussed in \cite{Bl98}, \cite{Bl00}.
  The explicit realization of this correspondence is obtained by a judicious
use of Clebsch variables and Lin constraints, a technique originally developed
to provide simple realizations of Lagrangian systems with symmetry. It is
noteworthy to point out that this correspondence exchanges the role of state
and control variables for control systems with the configuration and Clebsch
variables for the corresponding Lagrangian system.
  These results are illustrated with various simple applications.
