Variational reduction of homogenous Lagrangian systems

TL;DR

This paper develops a variational reduction method for homogeneous Lagrangian systems with scaling symmetries, enabling trajectory reconstruction from reduced critical points and relating to the Herglotz principle.

Contribution

It introduces a novel reduction procedure for homogeneous Lagrangian systems and characterizes critical points via scaling analogues of Lagrange-Poincaré equations.

Findings

Trajectories can be reconstructed from reduced variational principles.

Critical points satisfy a set of scaling-analogous differential equations.

Explores the relation between homogeneous Lagrangian systems and the Herglotz principle.

Abstract

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.