Balance Systems and the Variational Bicomplex

TL;DR

This paper demonstrates that balance systems in continuum thermodynamics can be naturally integrated into the variational bicomplex framework, providing a decomposition into Lagrangian and non-Lagrangian parts, especially when derivatives are absent from constitutive relations.

Contribution

It introduces a novel application of the variational bicomplex to balance systems, including a local splitting into Lagrangian and non-Lagrangian components, extending previous formulations.

Findings

Balance systems are embedded in the variational bicomplex formalism.

A local splitting of balance systems into Lagrangian and non-Lagrangian parts is achieved.

Non-Lagrangian systems align with Godunov's systems when derivatives are absent from constitutive relations.

Abstract

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental "pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the "pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947-948] and, later, asserted as the canonical hyperbolic form of balance systems in [M\"uller I., Ruggeri T., Rational extended thermodynamics, 2nd ed., Springer Tracts in Natural Philosophy, Vol. 37, Springer-Verlag, New York, 1998].