The origin of variational principles

TL;DR

This paper explores the foundational role of the principle of virtual work in deriving variational principles in classical physics, analyzing static and dynamic systems with a focus on configuration spaces.

Contribution

It offers a conceptual framework linking virtual work to variational principles and examines configuration spaces that are not differential manifolds.

Findings

Principle of virtual work as the basis for variational formulations

Analysis of local equilibria in finite-dimensional systems

Construction of non-manifold configuration spaces for dynamics and statics

Abstract

This note presents an attempt to provide a conceptual framework for variational formulations of classical physics. Variational principles of physics have all a common source in the {\it principle of virtual work} well known in statics of mechanical systems. This principle is presented here as the first step in characterizing local stable equilibria of static systems. An extended analysis of local equilibria is given for systems with configuration manifolds of finite dimensions. Numerous examples of the principle of virtual work and the Legendre transformation applied to static mechanical systems are provided. Configuration spaces for the dynamics of autonomous mechanical systems and for statics of continua are constructed in the final sections. These configuration spaces are not differential manifolds.