On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet
Abstract
Motivated by the work of previous authors on vortex sheets and their applications, the intrinsic 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 problems involving the dynamics of interfaces of two immiscible fluids. A boundary Poisson bracket, which appears to be new and related to the KdV bracket, is obtained containing the curve-tangential derivative . Lagrangian invariants of the sheet motion by its self-induced velocity--the Cauchy principal value of the Biot-Savart integral--are also derived.
Keywords
Cite
@article{arxiv.2408.13462,
title = {On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet},
author = {Banavara N. Shashikanth},
journal= {arXiv preprint arXiv:2408.13462},
year = {2025}
}
Comments
The previous version (v2) had some typos and omissions, which have been corrected. A new observation has been made: the vortex sheet bracket has the form of the KdV bracket. An appendix on Poisson brackets on infinite-dimensional manifolds has been added