Some contributions to k-contact Lagrangian field equations, symmetries and dissipation laws

TL;DR

This paper explores the relationships between solutions, symmetries, and dissipation laws in k-contact Lagrangian field theories, extending geometric frameworks and establishing Noether-like theorems with applications to damped vibrating strings.

Contribution

It introduces new results connecting solutions in different formulations, classifies symmetries based on preserved structures, and relates symmetries to dissipation laws and Newtonoid vector fields.

Findings

New relations between solutions in different k-contact formulations

Classification of symmetries: natural, dynamical, and k-contact

Noether-like theorems linking symmetries and dissipation laws

Abstract

It is well known that k-contact geometry is a suitable framework to deal with non-conservative field theories. In this paper, we study some relations between solutions of the k-contact Euler-Lagrange equations, symmetries, dissipation laws and Newtonoid vector fields. We review the k-contact Euler-Lagrange equations written in terms of k-vector fields and sections and provide new results relating the solutions in both approaches. We also study different kind of symmetries depending on the structures they preserve: natural (preserving the Lagrangian function), dynamical (preserving the solutions), and k-contact (preserving the underlying geometric structures) symmetries. For some of these symmetries, we provide Noether-like theorems relating symmetries and dissipation laws. We also analyse the relation between k-contact symmetries and Newtonoid vector fields. Throughout the paper, we will use the damped vibrating string as our main illustrative example.