Hamilton--Jacobi theory for non-conservative field theories in the $k$-contact framework

TL;DR

This paper extends Hamilton--Jacobi theory to non-conservative field theories within the k-contact geometric framework, addressing dissipative systems and providing new approaches for both z-independent and z-dependent cases.

Contribution

It introduces evolution k-contact k-vector fields and develops two Hamilton--Jacobi theories, broadening applications to dissipative field models and recovering classical cases.

Findings

Developed z-independent and z-dependent Hamilton--Jacobi approaches.

Applied theory to examples like Klein--Gordon and thermodynamic models.

Extended contact Hamilton--Jacobi theory to dissipative field systems.

Abstract

This article develops a Hamilton--Jacobi theory for non-conservative classical field theories, with particular emphasis on dissipative systems, in the framework of co-oriented k-contact geometry. We introduce evolution k-contact k-vector fields, extending the contact evolution formalism to field theories, and analyse the corresponding Hamilton--De Donder--Weyl equations. Moreover, we develop two distinct families of Hamilton--Jacobi theories: a z-independent approach, based on the reconstruction of the dynamics from an integrable k-vector field defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q$, and a z-dependent approach, where the integrable k-vector field is defined on the base manifold of $(\bigoplus^kT^*Q)\times\mathbb{R}^k\to Q\times\mathbb{R}^k$. We develop in detail the important case of Hamiltonian functions with affine dependence on the dissipative variables, show how quadratic dependence on these variables can be used structurally to enlarge the range of applications, and recover the ordinary contact Hamilton--Jacobi theory as the particular case k=1, while removing some technical assumptions appearing in previous formulations. Our theory is illustrated through several representative examples, including the telegrapher/Klein--Gordon equation, a dissipative Hunter--Saxton equation, a simple dissipative non-regular first-order field model, and a relativistic thermodynamic model.