  We consider paths of Hamiltonian diffeomorphism preserving a given compact
monotone Lagrangian in a symplectic manifold that extend to an
$S^1$--Hamiltonian action. We compute the leading term of the associated
Lagrangian Seidel element. We show that such paths minimize the Lagrangian
Hofer length. Finally we apply these computations to Lagrangian uniruledness
and to give a nice presentation of the Quantum cohomology of real lagrangians
in Fano symplectic toric manifolds.
