  This paper deals with the Lagrangian analogue of symplectic or point
reduction by stages. We develop Routh reduction as a reduction technique that
preserves the Lagrangian nature of the dynamics. To do so we heavily rely on
the relation between Routh reduction and cotangent symplectic reduction. The
main results in this paper are: (i) we develop a class of so called magnetic
Lagrangian systems and this class has the property that it is closed under
Routh reduction; (ii) we construct a transformation relating the magnetic
Lagrangian system obtained after two subsequent Routh reductions and the
magnetic Lagrangian system obtained after Routh reduction w.r.t. to the full
symmetry group.
