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The Herglotz variational principle for dissipative field theories

Jordi Gaset, Manuel Lainz, Arnau Mas, Xavier Rivas

TL;DR

This work generalizes the Herglotz variational principle to first-order and higher-order field theories within a k-contact/multicontact geometric framework to model dissipative dynamics. It develops two non-equivalent field formulations—nonholonomic and vakonomic—and identifies a precise closed-action-dependence condition under which they coincide; without this condition, the two approaches can yield different equations. The authors illustrate the framework with examples including a damped vibrating string and a KdV-type dissipative system, showing how dissipation terms enter the field equations. The results clarify the connections between action-dependent variational principles and dissipative field theories and point to future work on unifying the geometric formalisms and addressing singular Lagrangians.

Abstract

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg-De Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

The Herglotz variational principle for dissipative field theories

TL;DR

This work generalizes the Herglotz variational principle to first-order and higher-order field theories within a k-contact/multicontact geometric framework to model dissipative dynamics. It develops two non-equivalent field formulations—nonholonomic and vakonomic—and identifies a precise closed-action-dependence condition under which they coincide; without this condition, the two approaches can yield different equations. The authors illustrate the framework with examples including a damped vibrating string and a KdV-type dissipative system, showing how dissipation terms enter the field equations. The results clarify the connections between action-dependent variational principles and dissipative field theories and point to future work on unifying the geometric formalisms and addressing singular Lagrangians.

Abstract

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg-De Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.
Paper Structure (16 sections, 10 theorems, 101 equations)

This paper contains 16 sections, 10 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. The Herglotz principle in mechanics
  3. Herglotz principle: implicit version
  4. Herglotz principle: vakonomic version
  5. Herglotz principle: nonholonomic version
  6. The Herglotz principle for fields
  7. Geometric structures
  8. Herglotz principle for fields: non-holonomic version
  9. Herglotz principle for fields: vakonomic version
  10. Relations between both approaches
  11. Higher-order Lagrangian densities
  12. Examples
  13. Vibrating string with damping
  14. The non-holonomic and the vakonomic principles are not equivalent
  15. The Korteweg--De Vries Lagrangian
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $L: \mathrm{T} Q \times \mathbb{R} \to \mathbb{R}$ be a Lagrangian function and consider $c\in \Omega(q_0, q_1)$ and $z_0 \in \mathbb{R}$. Then, $(c,\dot{c}, \mathcal{Z}_{z_0}(c))$ satisfies the Herglotz equations if and only if $c$ is a critical point of the contact action functional $\mathcal{A}_{z_0}$.

Theorems & Definitions (21)

  • Theorem 2.1: Herglotz variational principle, implicit version
  • proof
  • Theorem 2.2: Herglotz variational principle, vakonomic version
  • Theorem 2.3: Lagrange multiplier Theorem
  • proof : Proof of Herglotz variational principle, vakonomic version
  • Definition 2.4
  • Theorem 2.5: Herglotz's variational principle, nonholonomic version
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 11 more