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

TL;DR

This paper develops a new homotopy co-momentum map framework for hydrodynamics, linking higher order linking numbers to conserved quantities in a multisymplectic setting, with implications for knot theory and fluid dynamics.

Contribution

It introduces a generalized homotopy co-momentum map for hydrodynamics and interprets higher order linking numbers as conserved quantities within multisymplectic geometry.

Findings

Reinterpretation of Massey higher order linking numbers as conserved quantities.

Construction of a covariant phase space for Euler fluids.

Identification of knot invariants as first integrals in involution.

Abstract

In this article a homotopy co-momentum map (\`a la Callies-Fr\'egier-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.