A Symplectic Generalization of the Peradzynski Helicity Theorem and Some Applications

TL;DR

This paper extends the Peradzynski helicity theorem using symplectic and geometric methods, providing a unified proof and new conservation laws for MHD superfluid flows, with implications for stability analysis.

Contribution

It introduces a symplectic geometric framework to generalize the helicity theorem for compressible MHD superfluids, offering new conservation laws and a unified proof.

Findings

Unified proof of the helicity theorem using symplectic geometry

Generalization of the theorem for compressible MHD superfluid flows

Discovery of new helicity-type conservation laws for incompressible flows

Abstract

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzynski helicity theorem based on differential - geometric and group-theoretical methods is derived. Having reanalyzed the Peradzynski helicity theorem within the modern symplectic theory of differential- geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.