Abstract
AbstractIn 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.
Highlights
In this paper we discuss some applications of multisymplectic techniques in a hydrodynamical context
In the present section we freely use basic material on symplectic and multisymplectic geometry tailored to our subsequent needs, prominently referring, for additional details, to [37, 48, 49] for the former and to [45, 46] for the latter
In the present subsection we briefly review, for motivation and further applications, the symplectic geometrical portrait underlying the theory of perfect fluids, in its simplest instance
Summary
In this paper we discuss some applications of multisymplectic techniques in a hydrodynamical context. The possibility of applying symplectic techniques therein comes from Arnol’d’s pioneering work culminating in the geometrization of fluid mechanics [1,2,3, 32] In this connection we may mention the paper [40], with its symplectic reinterpretation [36,37,38], and the general portrait depicted in [6]. This paper is an improved version of part of the preprint [34]
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