Hamiltonian Formulation Research Articles

A manifold of planar triangular meshes with complete Riemannian metric

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes forms a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.

Some Implications of the Multifractal Informational Entropy

Abstract A new formulation of the informational entropy on multifractal manifolds is proposed. In such a context, some connections with known results from the dynamics of complex systems are presented: correlations of SL(2ℝ) groups with Hamiltonian formalism, simultaneous invariances between two groups (coordinate group and parametric group), linear harmonic oscillator/oscillator ensemble dynamics from the perspective of SL(2ℝ) groups, uncertainty relations on multifractal manifolds etc. Moreover, the existence of a multifractal perspective in the complex dynamics analysis allows direct correspondences with deep learning algorithms.

Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton–Jacobi equation and shadow

The properties of light rays around compact objects surrounded by a plasma are affected by both strong gravitational fields described by a general-relativistic spacetime and by a dispersive and refractive medium, characterized by the density distribution of the plasma. We study these effects employing the relativistic Hamiltonian formalism under the assumption of stationarity and axisymmetry. The necessary and sufficient conditions on the metric and on the plasma frequency are formulated such that the rays can be analytically determined from a fully separated Hamilton–Jacobi equation. We demonstrate how these results allow us to analytically calculate the photon region and the shadow if they exist. Several specific examples are discussed in detail: the “hairy” Kerr black holes, the Hartle–Thorne spacetime metrics, the Melvin universe, and the Teo rotating traversable wormhole. In all of these cases, a plasma medium is present as well.

Global dynamical analysis of an electronic spin–orbit coupling system

By means of a canonical generalized momentum and a canonical conjugate spin variable, a complete canonical Hamiltonian formalism is designed to describe a coulomb field with electronic spin–orbit coupling in a semi-classical and non-relativistic way. After this operation, unlike the existing Lagrange formulation, the concepts of hidden momentum, hidden angular momentum and spin kinetic energy are not used in the canonical formalism. Besides, it is easy to find that there are four first integrals involving the conserved total energy and the conserved total angular momentum vector in an 8-dimensional phase space of the system. In this sense, the global dynamics is typically integrable, regular and non-chaotic, and each orbit in the phase space is a quasi-periodic 4-dimensional Kolmogorov-Arnold-Moser(KAM) torus.

Hamiltonian formulation of the stochastic surface wave problem

We devise a stochastic Hamiltonian formulation of the water wave problem. This stochastic representation is built within the framework of the modelling under location uncertainty. Starting from restriction to the free surface of the general stochastic fluid motion equations, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet–Neumann operator appears explicitly in the energy functional. This, in particular, allows us, in the same way as in deterministic setting, to conduct systematic approximations of the Dirichlet–Neumann operator and to infer different simplified wave models including noise in a natural way.

Cancellation of divergences in the nonprojectable Hořava theory

We perform an analysis of the ultraviolet divergences of the quantum nonprojectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava gravity. We work the quantum-field theory directly in the Hamiltonian formalism provided by the Batalin-Fradkin-Vilkovisky quantization. In this way the second-class constraints can be incorporated into the quantization. A known local gauge-fixing condition leads to a local canonical Lagrangian. Although the canonical fields acquire regular propagators, irregular propagators persist dangerous subdivergences. We show that all these loops cancel exactly between them due to a perfect matching between the propagators and vertices of the fields and ghosts forming the loops. The rest of the divergences behave similarly to the projectable theory, they can be removed by local counterterms. This result points to the renormalization of the nonprojectable Ho\ifmmode \check{r}\else \v{r}\fi{}ava theory.

Hamiltonian formulation and loop quantization of a recent extension of the Kruskal spacetime

We study the Hamiltonian formulation of the Ashtekar-Olmedo-Singh model for the description of the interior geometry of non-rotating, uncharged black holes. This model incorporates loop quantum effects through the introduction of two regularization parameters. We consider an extended phase space formalism proposed by the creators of the model that includes such parameters as configuration variables, constrained to be functions of the black hole mass. We generalize this restriction, allowing for an off-shell phase space dependence. We then introduce a gauge fixing procedure and reduce the system, proving that the reduced symplectic structure cannot reproduce the standard relativistic one in terms of the densitized triad and the Ashtekar-Barbero connection. Actually, the reduced structure precisely compensates the modifications that arise in the Hamilton equations when the regularization parameters are treated as phase space functions, rather than as numbers, attaining a consistent Hamiltonian derivation of the dynamics. We then choose the extended phase space formalism as starting point to address the loop quantization of the model. Taking the definition of certain geometric operators as the only basic ingredient and adopting prescriptions that have proven successful in loop quantum cosmology, we construct a polymer representation of all the constraints and deduce the formal expression of the physical states, assuming reasonable spectral properties for the constraint operators. The physical states turn out to be characterized by a wave function of the black hole mass with support on a very specific set. We finally discuss conditions that guarantee the existence of physical states in the region of large black hole masses. This is a first step in the development of a new loop quantum theory of black holes.

Optimal control strategies for a computer network under virus threat

This study proposes different optimal control strategies to eliminate the damage of virus propagation in a computer network with a minimum cost of installing anti-virus software. The model discussed to be developed is a fractional-order SEIR epidemiological model. Unlike the existing studies on the model, it has been considered that the recovered computers may malfunction due to any mechanical reason, and the model has been developed accordingly. In addition, unit consistency for the model is provided. Although there are a limited number of studies in the literature, the control variable affects the equilibrium points and thus the stability of the controlled system. Therefore, the reproduction number is recalculated for the controlled system. Before the optimal control problem is formulated, the existence of optimal control is proved. Then, the optimal system is obtained using Hamiltonian formalism. The numerical solutions of the optimal system are achieved by the fractional Euler method combined with the forward–backward sweep algorithm. The graphs drawn with MATLAB software show the efficiency of the fractional parameter for different control scenarios. Comparing the three proposed control strategies, it is clear that Strategy 1 is the most effective anti-virus installation strategy. In other words, if an anti-virus software is installed on both infected computers and susceptible computers connected to exposed and infected computers, virus propagation in the network can be prevented quickly as intended. Moreover, this is achieved by minimizing the cost of installing an antivirus program, thanks to the optimal control strategy.

Hamiltonian and Lagrangian BRST Quantization in Riemann Manifold II

We have previously developed the BRST quantization on the hypersurface V N − 1 embedded in N -dimensional Euclidean space R N in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L -dimensional manifold V L embedded in R N with 1 ≤ L < N . The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.

Analytical results for phase bunching in the pendulum model of wave-particle interactions

Radiation belt electrons are strongly affected by resonant interactions with cyclotron-resonant waves. In the case of a particle passing through resonance with a single, coherent wave, a Hamiltonian formulation is advantageous. With certain approximations, the Hamiltonian has the same form as that for a plane pendulum, leading to estimates of the change at resonance of the first adiabatic invariant I, energy, and pitch angle. In the case of large wave amplitude (relative to the spatial variation of the background magnetic field), the resonant change in I and its conjugate phase angle ξ are not diffusive but determined by nonlinear dynamics. A general analytical treatment of slow separatrix crossing has long been available and can be used to give the changes in I associated with “phase bunching,” including the detailed dependence on ξ, in the nonlinear regime. Here we review this treatment, evaluate it numerically, and relate it to previous analytical results for nonlinear wave-particle interactions. “Positive phase bunching” can occur for some particles even in the pendulum Hamiltonian approximation, though the fraction of such particles may be exponentially small.

Unified one-electron Hamiltonian formalism of spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in tetrahedral and octahedral symmetries.

Heavy element compounds with high symmetries often feature both spin-orbit coupling and vibronic coupling. This is especially true for systems with tetrahedral and octahedral symmetries, whose electronic states may be threefold degenerate and experience complicated Jahn-Teller and pseudo-Jahn-Teller interactions. To accurately describe these interactions, high quality spin-orbit vibronic Hamiltonian operators are needed. In this study, we present a unified one-electron Hamiltonian formalism for spin-orbit vibronic interactions for systems in all tetrahedral and octahedral symmetries. The formalism covers all spin-orbit Jahn-Teller and pseudo-Jahn-Teller problems in the symmetries with arbitrary types and arbitrary numbers of vibrational modes and generates Hamiltonian expansion formulas of arbitrarily high order.

Comment on the subtlety of defining a real-time path integral in lattice gauge theories

Abstract Recently, Hoshina, Fujii, and Kikukawa pointed out that the naive lattice gauge theory action in the Minkowski signature does not result in a unitary theory in the continuum limit, and Kanwar and Wagman proposed alternative lattice actions to the Wilson action without divergences. We show here that the subtlety can be understood from the asymptotic expansion of the modified Bessel function, which has been discussed for the path integral of compact variables in nonrelativistic quantum mechanics. The essential ingredient for defining the appropriate continuum theory is the iε prescription, and with the proper implementation of this we show that the Wilson action can be used for real-time path integrals. It is important that iε should be implemented for both timelike and spacelike plaquettes. We also suggest why iε is required for the Wilson action from the Hamiltonian formalism: it is needed to manifestly suppress the contributions from singular paths, for which the Wilson action can give different values from those of the actual continuum action.

How to Choose a Gauge? The Case of Hamiltonian Electromagnetism

We develop some ideas about gauge symmetry in the context of Maxwell’s theory of electromagnetism in the Hamiltonian formalism. One great benefit of this formalism is that it pairs momentum and configurational degrees of freedom, so that a decomposition of one side into subsets can be translated into a decomposition of the other. In the case of electromagnetism, this enables us to pair degrees of freedom of the electric field with degrees of freedom of the vector potential. Another benefit is that the formalism algorithmically identifies subsets of the equations of motion that represent time-dependent symmetries. For electromagnetism, these two benefits allow us to define gauge-fixing in parallel to special decompositions of the electric field. More specifically, we apply the Helmholtz decomposition theorem to split the electric field into its Coulombic and radiative parts, and show how this gives a special role to the Coulomb gauge (i.e. div(A)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\mathbf{A}) = 0$$\\end{document}). We relate this argument to Maudlin’s (Entropy, 2018. https://doi.org/10.3390/e20060465) discussion, which advocated the Coulomb gauge.

Peter Bergmann on observables in Hamiltonian General Relativity: A historical-critical investigation

The problem of observables and their supposed lack of change has been significant in Hamiltonian quantum gravity since the 1950s. This paper considers the unrecognized variety of ideas about observables in the thought of constrained Hamiltonian dynamics co-founder Peter Bergmann, who trained many students at Syracuse and invented observables. Whereas initially Bergmann required a constrained Hamiltonian formalism to be mathematically equivalent to the Lagrangian, in 1953 Bergmann and Schiller introduced a novel postulate, motivated by facilitating quantum gravity. This postulate held that observables were invariant under transformations generated by each individual first-class constraint. While modern works rely on Bergmann's authority and sometimes speak of “Bergmann observables,” he had much to say about observables, generally interesting and plausible but not all mutually consistent and much of it neglected.On occasion he required observables to be locally defined (not changeless and global); at times he wanted observables to be independent of the Hamiltonian formalism (implicitly contrary to a definition involving separate first-class constraints). But typically he took observables to have vanishing Poisson bracket with each first-class constraint and took this result to be justified by the example of electrodynamics. He expected observables to be analogous to the transverse true degrees of freedom of electromagnetism. Given these premises, there is no coherent concept of observables which he reliably endorsed, much less established.A revised definition of observables that satisfies the requirement that equivalent theories should have equivalent observables using the Rosenfeld–Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints that changes the canonical action ∫dt(pq̇−H) by a boundary term. Bootstrapping from theory formulations with no first-class constraints, one finds that the “external” coordinate gauge symmetry of GR calls for covariance (a transformation rule and hence a 4-dimensional Lie derivative for the Poisson bracket), not invariance (0 Poisson bracket), under G (not each first-class constraint separately).

Stability of smooth solitary waves in the [formula omitted]-Camassa–Holm equation

We derive the precise stability criterion for smooth solitary waves in the b-family of Camassa–Holm equations. The smooth solitary waves exist on the constant background. In the integrable cases b=2 and b=3, we show analytically that the stability criterion is satisfied and smooth solitary waves are orbitally stable with respect to perturbations in H3(R). In the non-integrable cases, we show numerically and asymptotically that the stability criterion is satisfied for every b>1. The orbital stability theory relies on a Hamiltonian formulation of the b-family which is different from the Hamiltonian formulations available for b=2 and b=3.

On the Validity of the Tight-Binding Method for Describing Systems of Subwavelength Resonators

The goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We first study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbor approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbor approximations are not generally appropriate.

Dynamics of co-orbital exoplanets in a first-order resonance chain with tidal dissipation

Co-orbital planets (in a 1: 1 mean motion resonance) can be formed within a Laplace resonance chain. We develop a secular model tc study the dynamics of the resonance chain p: p : p + 1, where the co-orbital pair is in a first-order mean motion resonance with the outermost third planet. Our model takes into account tidal dissipation through the use of a Hamiltonian version of the constant time-lag model, which extends the Hamiltonian formalism of the point-mass case. We show the existence of several families of equilibria, anc how these equilibria extend to the complete system. In one family, which we call the main branch, a secular resonance between the libration frequency of the co-orbitals and the precession frequency of the pericentres has unexpected dynamical consequences when tidal dissipation is added. We report the existence of two distinct mechanisms that make co-orbital planets much more stable within the p : p : p + 1 resonance chain rather than outside it. The first is due to negative real parts of the eigenvalues of the linearised system with tides, in the region of the secular resonance mentioned above. The second comes from non-linear contributions of the vector fielt and is due to eccentricity damping. These two stabilising mechanisms increase the chances of a future detection of exoplanets in the co-orbital configuration.

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

The paper is devoted to real Hamiltonian forms of 2-dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral theory of the associated Lax operators. Real Hamiltonian forms are a special type of "reductions" of Hamiltonian systems, similar to real forms of semi-simple Lie algebras. Examples of real Hamiltonian forms of affine Toda field theories related to exceptional complex untwisted affine Kac-Moody algebras are studied. Along with the associated Lax representations, we also formulate the relevant Riemann-Hilbert problems and derive the minimal sets of scattering data that determine uniquely the scattering matrices and the potentials of the Lax operators.

Вещественные гамильтоновы формы аффинных теорий поля Тоды: спектральные аспекты

Рассматриваются вещественные гамильтоновы формы двумерных теорий поля Тоды, связанных с исключительными простыми алгебрами Ли, и спектральная теория соответствующих операторов Лакса. Эти вещественные гамильтоновы формы представляют собой особый тип "редукций" гамильтоновых систем и аналогичны вещественным формам полупростых алгебр Ли. Рассмотрены примеры вещественных гамильтоновых форм аффинных теорий поля Тоды, связанных с исключительными комплексными нескрученными аффинными алгебрами Каца-Муди. Наряду с представлениями Лакса сформулированы соответствующие задачи Римана-Гильберта и найдены минимальные наборы данных рассеяния, которые однозначно определяют матрицы рассеяния и потенциалы операторов Лакса.

The Contour Gauge in Use: Telling Untold

In Quantum Field Theory, we discuss the main features of the (non-local) contour gauge which extends the local axial-type gauge used in most approaches. Based on the gluon geometry, we demonstrate that the contour gauge does not suffer from the residual gauge. We discuss the useful correspondence between the contour gauge conception and the Hamiltonian (Lagrangian) formalism. Having compared the local and non-local gauges, we again advocate the advantage of the contour gauge use.