Symplectic geometry ≈ Symplectic geometry
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REBOUNDx: a library for adding conservative and dissipative forces to otherwise symplectic N-body integrations Open
Symplectic methods, in particular the Wisdom–Holman map, have revolutionized our ability to model the long-term, conservative dynamics of planetary systems. However, many astrophysically important effects are dissipative. The consequences …
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Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class Open
Non-Hermitian Hamiltonians are generally sensitive to boundary conditions, and their spectra and wave functions under open boundary conditions are not necessarily predicted by the Bloch band theory for periodic boundary conditions. To eluc…
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Bordered Heegaard Floer homology Open
We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes…
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Functors and Computations in Floer homology with Applications Part II Open
The results in this paper concern computations of Floer cohomology using generating functions. The first part proves the isomorphism between Floer cohomology and Generating function cohomology introduced by Lisa Traynor. The second part pr…
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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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Hybrid symplectic integrators for planetary dynamics Open
Hybrid symplectic integrators such as MERCURY are widely used to simulate complex dynamical phenomena in planetary dynamics that could otherwise not be investigated. A hybrid integrator achieves high accuracy during close encounters by usi…
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A Symplectic Instantaneous Optimal Control for Robot Trajectory Tracking With Differential-Algebraic Equation Models Open
Robot trajectory tracking control based on differential-algebraic equation (DAE) models is still a thorny issue, because the DAEs of such systems are inherently complex and unstable, such as the high-index problem. In this paper, a symplec…
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Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons Open
The set of solutions to the AdS3 Einstein gravity with Brown-Henneaux boundary conditions is known to be a family of metrics labeled by two arbitrary periodic functions, respectively left and right-moving. It turns out that there exists an…
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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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Spectral form factors and late time quantum chaos Open
This is a collection of notes about spectral form factors of standard ensembles in random matrix theory, written for the practical usage of the current study of late time quantum chaos. More precisely, we consider the Gaussian unitary ense…
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Explicit symplectic approximation of nonseparable Hamiltonians: Algorithm and long time performance Open
Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. This article proposes for arbitrary Hamiltonians similar integrators, which are explicit…
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Symplectic Runge--Kutta Schemes for Adjoint Equations, Automatic Differentiation, Optimal Control, and More Open
The study of the sensitivity of the solution of a system of differential equations with respect to changes in the initial conditions leads to the introduction of an adjoint system, whose discretization is related to reverse accumulation in…
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Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control Open
In this paper, we introduce Symplectic ODE-Net (SymODEN), a deep learning framework which can infer the dynamics of a physical system, given by an ordinary differential equation (ODE), from observed state trajectories. To achieve better ge…
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Kähler geometry and Chern insulators: Relations between topology and the quantum metric Open
We study Chern insulators from the point of view of K\\"ahler geometry, i.e.\nthe geometry of smooth manifolds equipped with a compatible triple consisting\nof a symplectic form, an integrable almost complex structure and a Riemannian\nmet…
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Microwave Realization of the Gaussian Symplectic Ensemble Open
Following an idea by Joyner et al. [Europhys. Lett. 107, 50004 (2014)], a microwave graph with an antiunitary symmetry T obeying T^{2}=-1 is realized. The Kramers doublets expected for such systems are clearly identified and can be lifted …
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Symplectic integration and physical interpretation of time-dependent coupled-cluster theory Open
The formulation of the time-dependent Schrödinger equation in terms of coupled-cluster theory is outlined, with emphasis on the bivariational framework and its classical Hamiltonian structure. An indefinite inner product is introduced, ind…
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Of bulk and boundaries: Generalized transfer matrices for tight-binding models Open
We construct a generalized transfer matrix corresponding to noninteracting tight-binding lattice models, which can subsequently be used to compute the bulk bands as well as the edge states. Crucially, our formalism works even in cases wher…
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Symplectic Recurrent Neural Networks Open
We propose Symplectic Recurrent Neural Networks (SRNNs) as learning algorithms that capture the dynamics of physical systems from observed trajectories. An SRNN models the Hamiltonian function of the system by a neural network and furtherm…
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Persistent homology and Floer–Novikov theory Open
We construct “barcodes” for the chain complexes over Novikov rings that arise in Novikov’s Morse theory for closed one-forms and in Floer theory on not-necessarily-monotone symplectic manifolds. In the case of classical Morse theory these …
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Physical Symmetries Embedded in Neural Networks Open
Neural networks are a central technique in machine learning. Recent years have seen a wave of interest in applying neural networks to physical systems for which the governing dynamics are known and expressed through differential equations.…
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Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control Open
In this paper, we introduce Symplectic ODE-Net (SymODEN), a deep learning framework which can infer the dynamics of a physical system from observed state trajectories. To achieve better generalization with fewer training samples, SymODEN i…
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Construction of Explicit Symplectic Integrators in General Relativity. II. Reissner–Nordström Black Holes Open
In a previous paper, second- and fourth-order explicit symplectic integrators were designed for a Hamiltonian of the Schwarzschild black hole. Following this work, we continue to trace the possibility of construction of explicit symplectic…
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Loop-corrected subleading soft theorem and the celestial stress tensor Open
A bstract We demonstrate that the one-loop exact subleading soft graviton theorem automatically follows from conservation of the BMS charges, provided that the hard and soft fluxes separately represent the extended BMS algebra at null infi…
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Explicit symplectic algorithms based on generating functions for charged particle dynamics Open
Dynamics of a charged particle in the canonical coordinates is a Hamiltonian system, and the well-known symplectic algorithm has been regarded as the de facto method for numerical integration of Hamiltonian systems due to its long-term acc…
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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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Geometric formulation of the covariant phase space methods with boundaries Open
We analyze in full detail the geometric structure of the covariant phase space (CPS) of any local field theory defined over a space-time with boundary. To this end, we introduce a new frame: the "relative bicomplex framework." It is the re…
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Combinatorial constructions of derived equivalences Open
Given a certain kind of linear representation of a reductive group, referred\nto as a quasi-symmetric representation in recent work of \\v{S}penko and Van den\nBergh, we construct equivalences between the derived categories of coherent\nsh…
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A study of symplectic integrators for planetary system problems: error analysis and comparisons Open
The symplectic Wisdom–Holman map revolutionized long-term integrations of planetary systems. There is freedom in such methods of how to split the Hamiltonian and which coordinate system to employ, and several options have been proposed in …
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Higher order explicit symmetric integrators for inseparable forms of coordinates and momenta Open
Pihajoki proposed the extended phase-space second-order explicit symmetric leapfrog methods for inseparable Hamiltonian systems. On the basis of this work, we survey a critical problem on how to mix the variables in the extended phase spac…
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A slow-down time-transformed symplectic integrator for solving the few-body problem Open
An accurate and efficient method dealing with the few-body dynamics is important for simulating collisional N-body systems like star clusters and to follow the formation and evolution of compact binaries. We describe such a method which co…