On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet

Motivated by the work of previous authors on vortex sheets and their applications, the inviscid evolution equations of a closed vortex sheet in a plane, separating two piecewise constant density fluids, and their Hamiltonian form are investigated. The model has potential applications to bubble dynamics. A Poisson bracket is obtained containing the curve-tangential derivative. A Lagrangian invariant of the sheet motion by its self-induced velocity--the Cauchy principal value of the Biot-Savart integral--is derived.