A survey on geometric frameworks for action-dependent classical field theories and their relationship

TL;DR

This survey reviews three geometric frameworks—k-contact, k-cocontact, and multicontact—for classical action-dependent field theories, detailing their structures, formalisms, and interrelations.

Contribution

It provides a comprehensive comparison and analysis of recent geometric approaches to action-dependent field theories, clarifying their relationships and differences.

Findings

Introduces and explains k-contact, k-cocontact, and multicontact structures.

Develops Lagrangian and Hamiltonian formalisms for these frameworks.

Analyzes relationships among the structures and compares with existing definitions.

Abstract

This work presents a comprehensive overview of three recently developed geometric frameworks for the study of classical action-dependent field theories. Specifically, the three underlying geometric structures - namely, k-contact, k-cocontact, and multicontact - are first introduced, and then used to develop the Lagrangian and Hamiltonian formalisms of the aforementioned theories. Finally, the relationship among these three types of structures is analyzed in the case of trivial bundles; as well as the comparison with other alternative definitions of multicontact structure presented in the literature.