Lagrangian Description of the Variational Equations

TL;DR

This paper introduces a modified Lagrangian framework that simultaneously describes a system's equations of motion and their variational equations within an extended configuration space, linking conserved quantities to symmetries.

Contribution

It develops a novel Lagrangian approach that unifies the equations of motion and variational equations using an extended configuration space and relates conserved quantities to Noether symmetries.

Findings

Unified Lagrangian description of equations and variational equations

Introduction of inherited constants of motion

Connection between conserved quantities and Noether symmetries

Abstract

A variant of the usual Lagrangian scheme is developed which describes both the equations of motion and the variational equations of a system. The required (prolonged) Lagrangian is defined in an extended configuration space comprising both the original configurations of the system and all the virtual displacements joining any two integral curves. Our main result establishes that both the Euler-Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the first prolongation of the original LagrangianAfter discussing certain features of the formulation, we introduce the so-called inherited constants of the motion and relate them to the Noether constants of the extended system.