Tulczyjew triples: from statics to field theory

TL;DR

This paper develops a geometric framework using Tulczyjew triples to unify the formulation of dynamical equations across statics, mechanics, and classical field theories, providing a comprehensive theoretical approach.

Contribution

It extends the Tulczyjew triple formalism from mechanics to classical field theories, offering a unified geometric perspective.

Findings

Unified geometric framework for statics, mechanics, and field theories

Illustrated with key examples demonstrating the approach

Provides a foundation for future theoretical developments

Abstract

We propose a geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple. We show the evolution of these concepts, starting with the roots lying in the variational calculus for statics, through Lagrangian and Hamiltonian mechanics, and concluding with Tulczyjew triples for classical field theories, illustrated with a numer of important examples.