Lagrangian and Hamiltonian formalism in Field Theory: a simple model

TL;DR

This paper develops a simplified model of classical field theory using the Lagrangian and Hamiltonian formalisms, constructing the Tulczyjew triple and discussing generalizations to higher dimensions.

Contribution

It introduces a straightforward model for field theory and constructs the Tulczyjew triple, providing a foundation for extending these methods to arbitrary dimensions.

Findings

Constructed the Tulczyjew triple for the simplified model

Described the Lagrangian and Hamiltonian formalisms in this context

Outlined generalizations to higher-dimensional field theories

Abstract

The static of smooth maps from the two-dimensional disc to a smooth manifold can be regarded as a simplified version of the Classical Field Theory. In this paper we construct the Tulczyjew triple for the problem and describe the Lagrangian and Hamiltonian formalism. We outline also natural generalizations of this approach to arbitrary dimensions.