Variational equations of Lagrangian systems and Hamilton's principle

TL;DR

This paper introduces a new variational principle for deriving both the equations of motion and their variations in Lagrangian systems, using an extended configuration space and Hamilton's principle, with applications to symmetries and perturbations.

Contribution

It proposes a novel variational principle with a new Lagrangian in an extended space, enabling direct derivation of variational equations and constants of motion.

Findings

Derived constants of motion related to symmetries.

Formulated an intrinsic, coordinate-free approach.

Potential for reducing equations in control systems.

Abstract

We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- comprising both the original coordinates and the compatible ``virtual displacements'' joining two solutions of the original system. The variational principle is Hamilton's with the new Lagrangian. We use this formulation to obtain constants of motion in the Jacobi equations of any Lagrangian system with symmetries. These constants are related to constants in the original system and so with symmetries of the original Lagrangian. We cast our approach in an intrinsic coordinate free formulation. Our results can be of interest for reducing the dimensions of the equations that characterize perturbations in a Lagrangian control system.