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Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids

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Eulerian dynamical equations in a three-dimensional domain are used to construct a formal symplectic structure on time-extended space. Symmetries, invariants, and conservation laws are related to this geometric structure. The symplectic structure incorporates dynamics of helicities as identities. | Turk J Math (2016) 40: 925 – 940 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1410-38 Research Article Lagrangian description, symplectization, and Eulerian dynamics of incompressible fluids ∗ ¨ Hasan GUMRAL Department of Mathematics, School of Engineering, Australian College of Kuwait, Kuwait Received: 16.10.2014 • Accepted/Published Online: 18.12.2015 • Final Version: 21.10.2016 Abstract: Eulerian dynamical equations in a three-dimensional domain are used to construct a formal symplectic structure on time-extended space. Symmetries, invariants, and conservation laws are related to this geometric structure. The symplectic structure incorporates dynamics of helicities as identities. The generator of the infinitesimal dilation for symplectic two-form can be interpreted as a current vector for helicity. Symplectic dilation implies the existence of contact hypersurfaces. In particular, these include contact structures on the space of streamlines and on the Bernoulli surfaces. Key words: Incompressible fluid, symplectic and contact structures, symplectic dilation, helicity conservation, Lagrangian description 1. Introduction 1.1. Motivations The Euler equation for steady flow of incompressible fluid makes the construction of a contact structure on threedimensional space of Lagrangian trajectories of velocity vector field possible provided the (time-independent) helicity density is nonvanishing. One natural question is to ask to what extent these Eulerian equations characterize the space of integral curves of the velocity field? In this work, we shall be concerned with the relations between Lagrangian description and Eulerian equations of incompressible fluid and exploit the Eulerian evolution equations to obtain geometric structures relevant to a qualitative study of the Lagrangian description of motion. Our aim is, first, to show that a construction of symplectic structure on the time extended space R