  We present several results on the inverse problem and equivalent
contactLagrangian systems. These problems naturally lead to consider smooth
transformations on the z variable (i.e., reparametrizations of the action). We
present the extended contact Lagrangian systems to formalize this notion. With
this structure we define horizontal equivalence of Lagrangians, which
generalizes the symplectic case. We also present some results on the inverse
problem for extended contact systems.
