The origin of variational principles

Details The origin of variational principles The origin of variational principles

2004-05-14

arxiv.org/abs/math-ph/0405041v1

Classical and Quantum Integrability, Banach Center Publications, vol. 59, Warsaw 2003

Physics

Mathematical Physics

35 pages

Scientific paper

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.

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