Variational reduction of homogenous Lagrangian systems

In this paper we show that a variational reduction procedure can be defined for Lagrangian systems subject to scaling symmetries (i.e. Lagrangian systems defined by a homogenous Lagrangian function), in such a way that the trajectories of the system can be reconstructed up to quadratures from the critical points of the reduced variational principle. Also, we characterize the mentioned critical points in terms of a set of ordinary differential equations which are the scaling analogue of the Lagrange-Poincar\'e equations. Finally, we study if the homogeneous Lagrangian systems are naturally related or not with the Herglotz variational principle.