The origin of variational principles

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.Comment: 35 page

A variational formulation of analytical mechanics in an affine space

Variational formulations of statics and dynamics of mechanical systems controlled by external forces are presented as examples of variational principles.Comment: 17 pages, corrected typos, accepted for publication in Rep. Math. Phy

The category of local algebras and points proches

Categorial methods for generating new local algebras from old ones are presented. A direct proof of the differential structure of the prolongations of a manifold is proposed