Generalized Lie Symmetries and Almost Regular Lagrangians: A Link Between Symmetry and Dynamics

TL;DR

This paper explores the relationship between generalized Lie symmetries and almost regular Lagrangians, revealing how symmetries influence system dynamics and the structure of constraints, with specific examples illustrating key points.

Contribution

It establishes the connection between generalized Lie symmetries of the action and the Euler-Lagrange equations, and shows how these symmetries affect the evolution and constraints of dynamical systems.

Findings

Generalized Lie symmetry implies a singular Lagrangian.

Symmetries of the action form a subgroup of symmetries of equations of motion.

Each symmetry adds an arbitrary function to system evolution.

Abstract

The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined. It is found that if the action has a generalized Lie symmetry, then the Lagrangian is necessarily singular; the converse is not true, as we show with a specific example. It is also found that the generalized Lie symmetry of the action is a Lie subgroup of the generalized Lie symmetry of the Euler-Lagrange equations of motion. The converse is once again not true, and there are systems for which the Euler-Lagrange equations of motion have a generalized Lie symmetry while the action does not, as we once again show through a specific example. Most importantly, it is shown that each generalized Lie symmetry of the action contributes one arbitrary function to the evolution of the dynamical system. The number of such symmetries gives a lower bound to the dimensionality of the family of curves emanating from any set of allowed initial data in the Lagrangian phase space. Moreover, if second- or higher-order Lagrangian constraints are introduced during the application of the Lagrangian constraint algorithm, these additional constraints could not have been due to the generalized Lie symmetry of the action.