Dynamics of symplectic fluids and point vortices

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D. Ebin’s long-time existence result for geodesics on the symplectomorphism group to metrics not necessarily compatible with the symplectic structure. We also study the dynamics of symplectic point vortices, describe their symmetry groups and integrability. In 1966 V. Arnold showed how the Euler equation describing dynamics of an ideal incompressible fluid on a Riemannian manifold can be viewed as a geodesic equation on the group of volume-preserving diffeomorphisms of this manifold [1]. Consider a similar problem for a symplectic fluid. Let (M2m, ω) be a closed symplectic manifold equipped with a Riemannian metric. A symplectic fluid filling M is an ideal fluid whose motions preserve not only the volume element, but also the symplectic structure ω. (In 2D symplectic and ideal inviscid incompressible fluids coincide.) Such motions are governed by the corresponding Euler-Arnold equation, i.e. the equation describing geodesics on the infinite-dimensional group Sympω(M) of symplectomorphisms of M with respect to the right-invariant L2-metric. The corresponding problem of studying this dynamics was posed in [2] (see Section IV.8). Recently D. Ebin [4] considered the corresponding Euler equation of the symplectic fluid and proved the existence of solutions for all times for compatible metrics and symplectic structures. His proof uses the existence of a pointwise invariant transported by the flow, similar to the vorticity function in 2D. This symplectic vorticity allows one to proceed with the existence proof in the symplectic case similarly to the 2D setting. The purpose of this note is three-fold. First, we describe the Hamiltonian formalism of the Euler-Arnold equation for symplectic fluids, the corresponding dual spaces, inertia operators, and Casimir invariants. This formalism manifests a curious duality to the incompressible case: its natural setting is a quotient space of (n − 1)-forms for symplectic fluids vs. that of 1-forms for incompressible ones. Second, we show the geometric origin of the symplectic vorticity arising from the general approach to ideal fluids. In particular we prove that this quantity is a pointwise invariant for any metric, not only for a metric compatible with the symplectic structure, which allows one to extend the corresponding long-time existence theorem for solutions of the symplectic Euler equation, see [4]. We also present a variational description of the Hamiltonian stationary ∗Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada; e-mail: [email protected]