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A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers

Antonio Michele Miti, Mauro Spera

TL;DR

The paper develops a hydrodynamical homotopy co-momentum map (HCMM) that transgresses to the standard Arnol'd–Marsden–Weinstein framework and extends to certain Riemannian manifolds. It then provides a covariant phase-space interpretation and connects Rasetti–Regge currents on Brylinski's knot manifold with the HCMM structure, including explicit constructions such as $f_1({\bf b})$ and $f_2$ that enable a full HCMM. A Hamiltonian 1-form for links is formed via Poincaré duals to encode link configurations within the multisymplectic fluid framework, and Massey higher-order linking numbers are reformulated as conserved multisymplectic forms $\Omega_I$, yielding first integrals in involution. Overall, the work offers a multisymplectic reinterpretation of higher-order linking phenomena and proposes a covariant-phase-space viewpoint that suggests new integrability perspectives in fluid dynamics and knot theory.

Abstract

In this article a homotopy co-momentum map (à la Callies-Frégier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids and in particular of Brylinski's manifold of smooth oriented knots is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot theoretic analogues of first integrals in involution are determined.

A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers

TL;DR

The paper develops a hydrodynamical homotopy co-momentum map (HCMM) that transgresses to the standard Arnol'd–Marsden–Weinstein framework and extends to certain Riemannian manifolds. It then provides a covariant phase-space interpretation and connects Rasetti–Regge currents on Brylinski's knot manifold with the HCMM structure, including explicit constructions such as and that enable a full HCMM. A Hamiltonian 1-form for links is formed via Poincaré duals to encode link configurations within the multisymplectic fluid framework, and Massey higher-order linking numbers are reformulated as conserved multisymplectic forms , yielding first integrals in involution. Overall, the work offers a multisymplectic reinterpretation of higher-order linking phenomena and proposes a covariant-phase-space viewpoint that suggests new integrability perspectives in fluid dynamics and knot theory.

Abstract

In this article a homotopy co-momentum map (à la Callies-Frégier-Rogers-Zambon) trangressing to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others is constructed and then generalized to a special class of Riemannian manifolds. Also, a covariant phase space interpretation of the coadjoint orbits associated to the Euler evolution for perfect fluids and in particular of Brylinski's manifold of smooth oriented knots is discussed. As an application of the above homotopy co-momentum map, a reinterpretation of the (Massey) higher order linking numbers in terms of conserved quantities within the multisymplectic framework is provided and knot theoretic analogues of first integrals in involution are determined.

Paper Structure

This paper contains 10 sections, 6 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multisymplectic geometry and hydrodynamics of perfect fluids
  3. Tools in multisymplectic geometry
  4. The hydrodynamical Poisson bracket
  5. A hydrodynamical homotopy co-momentum map
  6. A generalization to Riemannian manifolds
  7. Covariant phase space aspects
  8. A Hamiltonian 1-form for links
  9. A multisymplectic interpretation of Massey products
  10. Conclusions and outlook

Key Result

Theorem 2.1

(i) The map $(f)$ previously given through the above $f_j: \Lambda^j {\mathfrak g} \to \Lambda^{2-j} ({\mathbb R}^3)$, fulfilling (eq:condition1_hccm),(2.5),(2.6), yields a homotopy co-momentum map; explicitly: (ii) The above HCCM transgresses, via the evaluation map ${\rm ev}: L{\mathbb R}^3 \times {\mathbb R} \ni (\gamma, t) \mapsto \gamma(t) \in {\mathbb R}^3$ to the hydrodynamical co-momentu

Figures (3)

  • Figure 1: Tubular neighbourhoods
  • Figure 2: Poincaré duals
  • Figure 3: Starting the Chen procedure

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Proposition 4.1
  • Theorem 4.2