Poisson bracket ≈ Poisson bracket
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GEMPIC: geometric electromagnetic particle-in-cell methods Open
We present a novel framework for finite element particle-in-cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov–Maxwell system. We derive a semi-discrete Poisson bracket, which retains the definin…
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Covariant phase space with boundaries Open
The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically t…
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Deformed integrable<i>σ</i>-models, classical<i>R</i>-matrices and classical exchange algebra on Drinfel’d doubles Open
We describe a unifying framework for the systematic construction of\nintegrable deformations of integrable $\\sigma$-models within the Hamiltonian\nformalism. It applies equally to both the `Yang-Baxter' type as well as `gauged\nWZW' type …
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Structure and structure-preserving algorithms for plasma physics Open
Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms. Geometric structures associated with and emergent from HAP formulations are discuss…
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Generalized BMS charge algebra Open
It has been argued that the symmetries of gravity at null infinity should\ninclude a Diff$(S^2)$ factor associated to diffeomorphisms on the celestial\nsphere. However, the standard phase space of gravity does not support the\naction of su…
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Field theories on null manifolds Open
A bstract We argue that generic field theories defined on null manifolds should have an emergent BMS or conformal Carrollian structure. We then focus on a simple interacting conformal Carrollian theory, viz. Carrollian scalar electrodynami…
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Simultaneous deformations and poisson geometry Open
We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an $L_{\infty }$ -algebra, which we construct explicitly. Our machinery is based on Voronov’s derived bracket co…
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A general framework for gravitational charges and holographic renormalization Open
We develop a general framework for constructing charges associated with diffeomorphisms in gravitational theories using covariant phase space techniques. This framework encompasses both localized charges associated with space–time subregio…
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Hamiltonian formulation of mimetic gravity Open
The Hamiltonian formulation of Mimetic Gravity is formulated. Although there\nare two more equations than those of general relativity, these are proved to be\nthe constraint equation and the conservation of energy-momentum tensor. The\nPoi…
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3D gravity in a box Open
The quantization of pure 3D gravity with Dirichlet boundary conditions on a finite boundary is of interest both as a model of quantum gravity in which one can compute quantities which are ``more local" than S-matrices or asymptotic boundar…
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Remarkable connections between extended magnetohydrodynamics models Open
Through the use of suitable variable transformations, the commonality of all extended magnetohydrodynamics (MHD) models is established. Remarkable correspondences between the Poisson brackets of inertialess Hall MHD and inertial MHD (which…
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Parameters of noncommutativity in Lie-algebraic noncommutative space Open
We find condition on the parameters of noncommutativity on which a list of\nimportant results can be obtained in a space with Lie-algebraic\nnoncommutativity. Namely, we show that the weak equivalence principle is\nrecovered in the space, …
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The κ-(A)dS quantum algebra in (3 + 1) dimensions Open
The quantum duality principle is used to obtain explicitly the Poisson\nanalogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the\ncorresponding Poisson-Lie structure on the dual solvable Lie group. The\nconstruction is fully …
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A multisymplectic approach to defects in integrable classical field theory Open
We introduce the concept of multisymplectic formalism, familiar in covariant\nfield theory, for the study of integrable defects in 1+1 classical field\ntheory. The main idea is the coexistence of two Poisson brackets, one for each\nspaceti…
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Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry Open
We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of al…
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The Sklyanin bracket and cluster adjacency at all multiplicity Open
We argue that the Sklyanin Poisson bracket on Gr(4,n) can be used to efficiently test whether an amplitude in planar ${\\cal{N}}=4$ supersymmetric Yang-Mills theory satisfies cluster adjacency. We use this test to show that cluster adjacen…
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Drinfeld double of GL <sub>n</sub> and generalized cluster structures Open
We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group $GL_n$ and derive from it a generalized cluster structure on $GL_n$ compatible with the push-forward …
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Lifts of Symmetric Tensors: Fluids, Plasma, and Grad Hierarchy Open
Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establis…
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A general metriplectic framework with application to dissipative extended magnetohydrodynamics Open
General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that …
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Structure-preserving reduced basis methods for Poisson systems Open
We develop structure-preserving reduced basis methods for a large class of nondissipative problems by resorting to their formulation as Hamiltonian dynamical systems. With this perspective, the phase space is naturally endowed with a Poiss…
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Adler–Gelfand–Dickey Approach to Classical 𝒲-Algebras Within the Theory of Poisson Vertex Algebras Open
We put the Adler–Gelfand–Dickey approach to classical $\\mathcal {W}$-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the Kadomtsev–Petviashvili (KP) hierarchy, together with its gen…
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Symplectic realization of electric charge in fields of monopole distributions Open
We construct a symplectic realisation of the twisted Poisson structure on the\nphase space of an electric charge in the background of an arbitrary smooth\nmagnetic monopole density in three dimensions. We use the extended phase space\nvari…
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Manifest Covariant Hamiltonian Theory of General Relativity Open
The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called “DeDonder-Weyl” formalism to the treatment of classical fields in curved space…
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Hamiltonian closures for fluid models with four moments by dimensional analysis Open
Fluid reductions of the Vlasov-Amp{è}re equations that preserve the Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian closures using the first four moments of the Vlasov distribution are obtained, and all clos…
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Bounds on spectral norms and barcodes Open
We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered co…
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The algebraic and geometric classification of transposed Poisson algebras Open
The algebraic and geometric classification of all complex 3-dimensional transposed Poisson algebras is obtained. Also we discuss special 3-dimensional transposed Poisson algebras.
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Quantization of BMS3 orbits: A perturbative approach Open
We compute characters of the BMS group in three dimensions. The approach is\nthe same as that performed by Witten in the case of coadjoint orbits of the\nVirasoro group in the eighties, within the large central charge approximation.\nThe p…
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Towards an exact frame formulation of conformal higher spins in three dimensions Open
In this note we discuss some aspects of the frame formulation of conformal higher spins in three dimensions. We give some exact formulae for the coupled spin two - spin three part of the full higher spin theory and propose a star product L…
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Topology and Bifurcations in Nonholonomic Mechanics Open
This paper develops topological methods for qualitative analysis of the behavior of nonholonomic dynamical systems. Their application is illustrated by considering a new integrable system of nonholonomic mechanics, called a nonholonomic hi…
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New bi-Hamiltonian systems on the plane Open
We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and …