  Due to the emergence of symplectic geometry, the geometric treatment of
mechanics underwent a great development during the last century. In this
scenario the pressence of symmetries in Hamiltonian systems leads naturally to
the existence of conserved quantities. This integrals of motions are described
by the well-known momentum mapping. Furthermore, the equations of motion can be
simplified by a process known as reduction, if the system is invariant under
the action of a certain group. This process can also be considered in the
framework of contact geometry, a much more recent field of study. This kind of
geometry has proven to be valuable in areas as different as thermodynamics,
control theory, or neurogeometry. Our knowledge about contact geometry is much
smaller than that of symplectic, and so a process known as symplectificacion is
extremely useful, since it allows to study contact problems in the symplectic
frame.
  The aim of the text is to study the commutativity relations between the
processes of reduction and symplectification. To do so, we first introduce the
basis of symplectic and contact geometry and we show how to perform the
reduction via the momentum map in both scenarios. The well-known coisotropic
reduction theorem will be crucial in the description. Finally, the
symplectification process is analyzed in detail, and its relation with
symplectic and contact reduction is studied.
