Lie-point symmetries of the Lagrangian system on time scales

TL;DR

This paper explores Lie point symmetries and conserved quantities of Lagrangian systems on time scales, unifying continuous and discrete cases through symmetry analysis and providing illustrative examples.

Contribution

It establishes the determining equations and structure of Lie symmetries on time scales, extending symmetry analysis to unified continuous-discrete Lagrangian systems.

Findings

Derived the determining equations for Lie symmetries on time scales.

Obtained the structure equations and conserved quantities involving delta derivatives.

Provided examples illustrating the application of the theoretical results.

Abstract

This letter investigates the Lie point symmetries and conserved quantities of the Lagrangian systems on time scales, which unify the Lie symmetries of the two cases for the continuous and the discrete Lagrangian systems. By defining the infinitesimal transformations' generators and using the invariance of differential equations under infinitesimal transformations, the determining equations of the Lie symmetries on time scales are established. Then the structure equations and the form of conserved quantities with delta derivatives are obtained. The letter also gives brief discussion on the Lie symmetries for the discrete systems. Finally, several examples are designed to illustrate these results.