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

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 int...

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Published inJournal of the Australian Mathematical Society (2001) Vol. 112; no. 3; pp. 335 - 354
Main Authors MITI, ANTONIO MICHELE, SPERA, MAURO
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2022
Subjects
Knot theory
Momentum
Riemann manifold
symplectic and multisymplectic geometry
higher-order linking numbers
37J39
55S30
53D99
53D20
hydrodynamics
57K10
homotopy co-momentum maps
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ISSN1446-7887
1446-8107
DOI10.1017/S1446788720000518

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Summary: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.
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ISSN:1446-7887
1446-8107
DOI:10.1017/S1446788720000518