[Paper Review] Multisymplectic Theory of Balance Systems and Entropy Principle

Serge Preston|ArXiv.org|Nov 28, 2006

Advanced Differential Geometry Research29 references4 citations

TL;DR

This paper develops a multisymplectic geometric framework for balance systems in continuum thermodynamics using Poincaré-Cartan formalism, generalizing Lagrangian field theory to include constitutive relations as mappings between jet bundles and dual bundles. The key contribution is a unified variational formulation of balance systems, including a Noether-type theorem for symmetries and a geometric characterization of the entropy principle via holonomy of the current component in Rational Extended Thermodynamics (RET).

ABSTRACT

In this paper we are presenting the theory of balance equations of the Continuum Thermodynamics (balance systems) in a geometrical form using Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive relation $\mathcal{C}$ of a balance system $B_{C}$ is realized as a mapping between a (partial) 1-jet bundle of the configurational bundle $π:Y o X$ and the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system $B_{C}$ is presented in three different forms and the space of admissible variations is defined and studied. Action of automorphisms of the bundle $π$ on the constitutive mappings $C$ is studied and it is shown that the symmetry group $Sym(C)$ of the constitutive relation $C$ acts on the space of solutions of the balance system $B_{C}$. Suitable version of Noether Theorem for an action of a symmetry group is presented with the usage of conventional multimomentum mapping. Finally, the geometrical (bundle) picture of the Rational Extended Thermodynamics in terms of Lagrange-Liu fields is developed and the entropy principle is shown to be equivalent to the holonomicy of the current component of the constitutive section.

Motivation & Objective

• To formulate balance equations in continuum thermodynamics using multisymplectic geometry, extending the variational framework of Lagrangian field theory.
• To unify different formulations of irreversible thermodynamics—particularly Rational Extended Thermodynamics (RET)—within a single geometric structure.
• To establish a geometric characterization of the entropy principle as the holonomy condition on the current component of the constitutive section.
• To develop a Noether-type theorem for symmetries of constitutive relations, linking conservation laws to automorphisms of the configuration bundle.
• To provide a canonical formulation of balance systems via a generalized Legendre transformation and multisymplectic forms on jet and dual bundles.

Proposed method

• Formulates balance systems using the Poincaré-Cartan formalism on the 1-jet bundle $J^1(/pi)$ of a configuration bundle $\pi: Y \to X$.
• Introduces a generalized Legendre transformation mapping the 1-jet bundle to an extended dual bundle, analogous to the Legendre map in Lagrangian mechanics.
• Defines the multisymplectic structure via the canonical multisymplectic form $\Omega$ on the bundle $\tilde{Z}$, derived from the Poincaré-Cartan form.
• Introduces the reduced horizontal differential $\hat{d}$, a modified exterior derivative that respects the jet structure and commutes with pullbacks modulo contact forms.
• Uses the $\hat{d}$-operator to define the invariant (variational) form of the balance system in three equivalent formulations: Poincaré-Cartan, Euler-Lagrange, and reduced horizontal differential.
• Applies the theory to RET by introducing Lagrange-Liu dual fields and showing that the entropy principle corresponds to the holonomy of the current component of the constitutive section.

Experimental results

Research questions

• RQ1 How can balance systems in continuum thermodynamics be formulated in a variational, multisymplectic framework analogous to Lagrangian field theory?
• RQ2 What is the geometric role of the constitutive relation $\mathcal{C}$ in relating the state bundle to its dual, and how does it generalize the Legendre transformation?
• RQ3 How does the entropy principle in Rational Extended Thermodynamics manifest geometrically, and what condition on the constitutive section ensures its validity?
• RQ4 How do symmetries of the configuration bundle $\pi$ act on the solutions of the balance system, and what conservation laws arise from such symmetries?
• RQ5 What is the role of the reduced horizontal differential $\hat{d}$ in defining the variational structure of balance systems, and how does it relate to the standard de Rham differential?

Key findings

• The constitutive relation $\mathcal{C}$ is realized as a mapping between the partial 1-jet bundle $J^1_K(\pi)$ and the extended dual bundle $\tilde{Z}$, generalizing the Legendre map.
• The invariant form of the balance system $\mathcal{B}_\mathcal{C}$ is expressed in three equivalent ways: Poincaré-Cartan, Euler-Lagrange, and reduced horizontal differential, all consistent with the multisymplectic structure.
• The entropy principle is geometrically equivalent to the holonomy of the current component of the constitutive section, meaning that the current must be closed under the $\hat{d}$-operator.
• The symmetry group $\mathrm{Sym}(\mathcal{C})$ of the constitutive relation acts on the solution space of $\mathcal{B}_\mathcal{C}$, and a Noether-type theorem is established using the multimomentum map.
• The $\hat{d}$-operator satisfies $\hat{d}^2 = 0$ and commutes with pullbacks modulo contact forms, ensuring consistency of the variational structure under bundle automorphisms.
• For RET, the dual formulation using Lagrange-Liu fields shows that the entropy principle is equivalent to the current component being holonomic, i.e., $\hat{d}$-closed.