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キーワード:
Mathematics, Symplectic Geometry, Mathematical Physics, Analysis of PDEs
要旨:
In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics
in which the motion of an inviscid incompressible fluid is described as the
geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler-Arnold equations for geodesics on groupoids equipped with one-sided invariant metrics.
