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A HYDRODYNAMICAL HOMOTOPY CO-MOMENTUM MAP AND A MULTISYMPLECTIC INTERPRETATION OF HIGHER-ORDER LINKING NUMBERS

Published online by Cambridge University Press:  11 February 2021

ANTONIO MICHELE MITI*
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy and Department of Mathematics, KU Leuven, Celestijnenlaan 200B Box 2400, 3001 Leuven, Belgium
MAURO SPERA
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121 Brescia, Italy e-mail: mauro.spera@unicatt.it
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Abstract

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In this paper a homotopy co-momentum map (à la Callies, Frégier, Rogers and Zambon) transgressing to the standard hydrodynamical co-momentum map of Arnol’d, Marsden, 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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by Robert Yuncken

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A HYDRODYNAMICAL HOMOTOPY CO-MOMENTUM MAP AND A MULTISYMPLECTIC INTERPRETATION OF HIGHER-ORDER LINKING NUMBERS
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