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Jacobi equation for field theories and a geometric variational description of dissipation

David Martin de Diego, Najma Mosadegh

TL;DR

This work develops a geometric framework for Jacobi-type field equations by prolonging a first-order Lagrangian $L$ to a $k$-cosymplectic manifold on $R^k\times T^1_kQ$, so Jacobi equations arise as Euler-Lagrange equations for a prolonged Lagrangian $L_C$ on $R^k\times T^1_kTQ$ (denoted $\tilde{L}$). It then extends the approach to dissipative field theories by introducing dissipation data $F_i,F^\mu_i$ and a modified Lagrangian $\tilde{L}_F$, yielding dissipative Euler-Lagrange equations within a purely variational description. Key contributions include explicit construction of the prolonged Lagrangian via the canonical isomorphism $TT^1_kQ\cong T^1_kTQ$, and a variational framework for dissipative dynamics compatible with the $k$-cosymplectic formalism. The results pave the way for multisymplectic extensions and variational integrators for dissipative field theories, enabling energy-based analyses and structure-preserving numerics.

Abstract

In this paper we give a geometric description of the Jacobi equations associated to a first-order Lagrangian field theory using a prolongation of the Lagrangian $L$ on a $k$-cosymplectic formulation. Moreover, using an appropriate modification of the prolonged Lagrangian, we obtain a variational formulation of field theories with dissipation.

Jacobi equation for field theories and a geometric variational description of dissipation

TL;DR

This work develops a geometric framework for Jacobi-type field equations by prolonging a first-order Lagrangian to a -cosymplectic manifold on , so Jacobi equations arise as Euler-Lagrange equations for a prolonged Lagrangian on (denoted ). It then extends the approach to dissipative field theories by introducing dissipation data and a modified Lagrangian , yielding dissipative Euler-Lagrange equations within a purely variational description. Key contributions include explicit construction of the prolonged Lagrangian via the canonical isomorphism , and a variational framework for dissipative dynamics compatible with the -cosymplectic formalism. The results pave the way for multisymplectic extensions and variational integrators for dissipative field theories, enabling energy-based analyses and structure-preserving numerics.

Abstract

In this paper we give a geometric description of the Jacobi equations associated to a first-order Lagrangian field theory using a prolongation of the Lagrangian on a -cosymplectic formulation. Moreover, using an appropriate modification of the prolonged Lagrangian, we obtain a variational formulation of field theories with dissipation.

Paper Structure

This paper contains 11 sections, 1 theorem, 61 equations.

Table of Contents

  1. Introduction
  2. Geometric preliminaries
  3. The manifold ${\mathbb R}^{k} \times T^1_kQ$
  4. $k$-vector fields and integral sections
  5. Canonical structures in ${\mathbb R}^{k}\times T^1_kQ$
  6. $k$-cosymplectic structures
  7. Field equations for a Lagrangian system
  8. The canonical isomorphism between $TT^1_kQ$ and $T^1_kTQ$
  9. The prolongation of the Lagrangian $L$ to the $k$-cosymplectic manifold ${\mathbb R}^{k} \times T^1_k TQ$
  10. A geometric variational description of dissipative field theories
  11. Conclusions and Future work

Key Result

Proposition 4

For the Lagrangian $\tilde{L}_F: {\mathbb R}^k\times T^1_kTQ\rightarrow {\mathbb R}$ defined by the corresponding Euler-Lagrange field equations are and the last equation is given exactly by equation (eq:forced).

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Example 1