The Tulczyjew triple for classical fields

TL;DR

This paper develops a covariant, comprehensive Tulczyjew triple framework for first-order classical field theory, deriving all components from variational calculus without ad hoc assumptions, and incorporating affine geometry and duality.

Contribution

It constructs the Tulczyjew triple for field theory directly from variational principles, integrating Lagrangian, Hamiltonian, and phase space structures in a covariant affine geometric setting.

Findings

Derived the Tulczyjew triple for first-order field theory from variational calculus.

Unified Lagrangian and Hamiltonian formalisms within a covariant affine geometric framework.

Included the phase space, phase dynamics, and Legendre transformation in the formulation.

Abstract

The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of variational calculus, we construct the Tulczyjew triple for first-order Field Theory. The important feature of our approach is that we do not postulate {\it ad hoc} the ingredients of the theory, but obtain them as unavoidable consequences of the variational calculus. This picture of Field Theory is covariant and complete, containing not only the Lagrangian formalism and Euler-Lagrange equations but also the phase space, the phase dynamics and the Hamiltonian formalism. Since the configuration space turns out to be an affine bundle, we have to use affine geometry, in particular the notion of the affine duality. In our formulation, the two maps $\alpha$ and $\beta$ which constitute the Tulczyjew triple are morphisms of double structures of affine-vector bundles. We discuss also the Legendre transformation, i.e. the transition between the Lagrangian and the Hamiltonian formulation of the first-order field theory