Generalized Marsden-Weinstein symplectic structures
Abstract
We define a symplectic structure on singular 1-form densities and a presymplectic structure on augmented triples, a compressible analog of the Marsden--Weinstein symplectic structure on vortex membranes in the incompressible case. Equivalence classes of augmented triples can be regarded as special vorticities in compressible fluids, i.e., singular elements of the dual to the Lie algebra of the group of all diffeomorphisms of a manifold, while Marsden--Weinstein symplectic structure is related to singular vorticities for the group of volume-preserving diffeomorphisms. This structure on singular 1-form densities occupies an intermediate position between two other settings, which we revisit, the Weinstein symplectic structure on weighted isotropic submanifolds of symplectic manifolds and the Marsden--Weinstein symplectic structure on vortex membranes.
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BibTeX
@misc{khesin2026generalized,
title = {{Generalized Marsden-Weinstein symplectic structures}},
author = {Boris Khesin and Klas Modin and Cornelia Vizman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01401},
eprint = {2610.01401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01401},
}
RIS
TY - UNPB
TI - Generalized Marsden-Weinstein symplectic structures
AU - Boris Khesin
AU - Klas Modin
AU - Cornelia Vizman
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01401
UR - https://arxiv.org/abs/2610.01401
AB - We define a symplectic structure on singular 1-form densities and a presymplectic structure on augmented triples, a compressible analog of the Marsden--Weinstein symplectic structure on vortex membranes in the incompressible case. Equivalence classes of augmented triples can be regarded as special vorticities in compressible fluids, i.e., singular elements of the dual to the Lie algebra of the group of all diffeomorphisms of a manifold, while Marsden--Weinstein symplectic structure is related to singular vorticities for the group of volume-preserving diffeomorphisms. This structure on singular 1-form densities occupies an intermediate position between two other settings, which we revisit, the Weinstein symplectic structure on weighted isotropic submanifolds of symplectic manifolds and the Marsden--Weinstein symplectic structure on vortex membranes.
ER -