## Abstract

Long ago it was stated [7,5] that quantum vortices in superfluid helium can be studied either as open lines with their ends terminating on free surfaces of walls of the container or as closed curves. Nowadays the closed vortices are treated as topological objects equivalent to circles. The existence of structures such as knotted and linked vertex lines in the turbulent phase is almost obvious [12] and has forced researchers to develop new mathematical tools for their detailed investigation. In this proposed direction Z. Peradzýnski [8] proved a new version of the Helicity theorem, based on differential-geometric methods applied to the description of the collective motion in the incompressible superfluid. The Peradzýnski helicity theorem describes in a unique way, both the superfluid equations and the related helicity invariants, which are, in the conservative case, very important for studying the topological structure of vortices. By reanalyzing the Peradzýnski helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, we propose a new unified proof and give a magneto-hydrodynamic generalization of this theorem for the case of an incompressible superfluid flow. As a by-product, in the conservative case we construct a sequence of nontrivial helicity type conservation laws, which play a crucial role in studying the stability problem of a superfluid under suitable boundary conditions.

Original language | English (US) |
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Pages | 491-498 |

Number of pages | 8 |

State | Published - Jan 1 2012 |

Event | 5th International Conference on Chaotic Modeling and Simulation, CHAOS 2012 - Athens, Greece Duration: Jun 12 2012 → Jun 15 2012 |

### Conference

Conference | 5th International Conference on Chaotic Modeling and Simulation, CHAOS 2012 |
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Country | Greece |

City | Athens |

Period | 6/12/12 → 6/15/12 |

## All Science Journal Classification (ASJC) codes

- Modeling and Simulation