Hamilton-Jacobi Equation Research Articles

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

Dynamics of Particles with Electric Charge and Magnetic Dipole Moment near Schwarzschild-MOG Black Hole

The study of electromagnetic interactions among test particles with electric charges and magnetic dipole moments is of great significance when examining the dynamics of particles within strong gravitational fields surrounding black holes. In this work, we focus on investigating the dynamics of particles possessing both electric charges and magnetic dipole moments in the spacetime of a Schwarzschild black hole within the framework of modified gravity (MOG), denoted as a Schwarzschild-MOG black hole. Our approach begins by offering a solution to Maxwell’s equations for the angular component of the electromagnetic four potentials within Schwarzschild-MOG spacetime. Subsequently, we derive the equations of motion and establish the effective potential for particles engaged in circular motion. This is achieved using a hybrid formulation of the Hamilton–Jacobi equation, encompassing interactions between electric charges and magnetic dipole moments, the external magnetic field (assumed to be asymptotically uniform), and interactions between the particles and the MOG field. Furthermore, we investigate the impacts of these three types of interactions on critical parameters, including the radius of innermost stable circular orbits (ISCOs), as well as the energy and angular momentum of particles when situated at their respective ISCOs. Finally, a detailed analysis concerning the effects of these interactions on the center-of-mass energy is presented in collisions involving neutral, electrically charged, and magnetized particles.

Modified Hawking radiation of Schwarzschild-like black hole in bumblebee gravity model

In this article, we study the Hawking radiation of the Schwarzschild black hole within the bumblebee gravity model (SBHBGM). Considering classical approaches involving Killing vectors and the standard Hamilton-Jacobi method, the Hawking radiation of SBHBGM is computed. The Painlevé-Gullstrand, ingoing Eddington-Finkelstein, and Kruskal-Szekeres coordinate systems are introduced as alternatives to the naive coordinates, providing insights into gravitational behavior around massive objects like black holes. We thus examine whether Hawking radiation’s temperature depends on the chosen coordinate system or not. Incorporating the Generalized Uncertainty Principle (GUP) into the Hamilton-Jacobi equation, a modified equation characterizing particle behavior near the event horizon is obtained. By calculating the tunneling probability using the modified action, the GUP-induced modifications to the emitted particle’s behavior are considered, resulting in the derivation of the modified temperature of the SBHBGM. In conclusion, we explore the quantum-adjusted entropy of SBHBGM and its associated temperature and assess the findings we have acquired.

Convergence of the solutions of the nonlinear discounted Hamilton–Jacobi equation: The central role of Mather measures

Given a continuous Hamiltonian H:(x,p,u)↦H(x,p,u) defined on T⁎M×R, where M is a closed connected manifold, we study viscosity solutions, uλ:M→R, of discounted equations:H(x,dxuλ,λuλ(x))=cin M where λ>0 is called a discount factor and c is the critical value of H(⋅,⋅,0).When the Hamiltonian H is convex and superlinear in p and non–decreasing in u, under an additional non–degeneracy condition, we obtain existence and uniqueness (with comparison principles) results of solutions and we prove that the family of solutions (uλ)λ>0 converges to a specific solution u0 ofH(x,dxu0,0)=cin M. Our non–degeneracy condition requires H to be increasing (in u) on localized regions linked to the support of Mather measures, whereas usual similar results are obtained for Hamiltonians that are everywhere increasing in u.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

Magnetized Baryonic layer and a novel BPS bound in the gauged-non-linear-sigma-model-Maxwell theory in (3+1)-dimensions through Hamilton-Jacobi equation

It is show that one can derive a novel BPS bound for the gauged Non-Linear-Sigma-Model (NLSM) Maxwell theory in (3+1) dimensions which can actually be saturated. Such novel bound is constructed using Hamilton-Jacobi equation from classical mechanics. The configurations saturating the bound represent Hadronic layers possessing both Baryonic charge and magnetic flux. However, unlike what happens in the more common situations, the topological charge which appears naturally in the BPS bound is a non-linear function of the Baryonic charge. This BPS bound can be saturated when the surface area of the layer is quantized. The far-reaching implications of these results are discussed. In particular, we determine the exact relation between the magnetic flux and the Baryonic charge as well as the critical value of the Baryonic chemical potential beyond which these configurations become thermodynamically unstable.

Charged spinning and magnetized test particles orbiting quantum improved charged black holes

In the present work, we aimed to investigate the dynamics of spinning charged and magnetized test particles around both electrically and magnetically charged quantum-improved black holes. We derive the equations of motion for charged spinning test particles using the Mathisson-Papapetrou-Dixon ***equations with the Lorentz coupling term. The radius of innermost stable circular orbits (ISCOs), specific angular momentum, and energy for charged spinless, uncharged spinning, and charged spinning test particles around the charged and non-charged quantum-improved black holes are analyzed separately. We found that the quantum parameter increases the maximum spin value, smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document}, which leads to the nonphysical motion (superluminal motion) of the charged spinning test particle. In contrast, the black hole charge decreases its value. We also found that, in contrast to the Reissner Nordström black hole, spinning charged test particles in the quantum-improved charged black hole have higher smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document}; moreover, positively charged spinning particles can have higher values of smax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_\ extrm{max}$$\\end{document} near the extreme black hole cases when compared with uncharged spinning particles. Finally, we investigate the magnetized test particle’s dynamics in the spacetime of a quantum-improved magnetically charged black hole in Quantum Einstein Gravity using the Hamilton–Jacobi equation. We show that the presence of ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} increases the maximum value of the effective potential and decreases the minimum energy and angular momentum of magnetized particles at their circular orbits. We found an upper constraint in the black hole charge at the ISCO.

An optimal control problem for the continuity equation arising in smart charging

This paper is focused on the mathematical modeling and solution of the optimal charging of a large population of identical plug-in electric vehicles (PEVs) with mixed state variables (continuous and discrete). A mean field assumption is formulated to describe the evolution interaction of the PEVs population. The optimal control of the resulting continuity equation of the mixed system under state constraints is investigated. We prove the existence of a minimizer. We then characterize the solution as the weak solution of a system of two coupled PDEs: a continuity equation and of a Hamilton-Jacobi equation. We provide regularity results of the optimal feedback control.

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations

Hydrodynamical formulation of Wigner–Dunkl quantum mechanics

The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.

On the symmetric rearrangement of the gradient of a Sobolev function

In this paper, we generalize a classical comparison result for solutions to Hamilton–Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton–Jacobi equations with non-zero boundary trace. As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions.

A nonlocal finite-dimensional integrable system related to the nonlocal nonlinear Schrödinger equation hierarchy

Based on the Lenard gradient sequence, a hierarchy of the nonlocal nonlinear Schrödinger (NNLS) equations is obtained. Using the Lax representation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure is presented. Then, under coordinate transformation, the nonlocal finite-dimensional integrable system with Lie–Poisson structure can be expressed as the canonical Hamiltonian system of the standard symplectic structures. Moreover, the parametric representation of the NNLS equation and the nonlocal complex modified Korteweg–de Vries (NcmKdV) equation are constructed. Finally, according to the Hamilton–Jacobi theory, the action–angle variables are built and the inversion problem related to the Lie–Poisson Hamiltonian systems is discussed.

Research on the Dynamic Control Method of CFETR Multi-Purpose Overload Robot

The CFETR multi-purpose overload robot (CMOR) is a key subsystem of the remote handling system of the China fusion engineering test reactor (CFETR). This paper first establishes the kinematic and dynamic models of CMOR and analyzes the working process in the vacuum chamber. Based on the uncertainty of rigid-flexible coupling, a CMOR adaptive robust sliding mode controller (ARSMC) is designed based on the Hamilton-Jacobi equation to enhance the robustness of the control system. In addition, to compensate the influence of non-geometric factors on position accuracy, an error compensation method is designed. Based on the matrix differentiation method, the CMOR coupling parameter errors are decoupled, and then the gridded workspace principle is used to identify the parameter errors and improve the motion control accuracy. Finally, the CMOR rigid-flexible coupling simulation system is established by ADAMS-MATLAB/Simulink to analyze the dynamic control effect of ARSMC. The simulation results show that the CMOR end position error exceeds 0.1 m for single joint motion. The average value of CMOR end position error is less than 0.025 m after compensation, and the absolute error value is reduced by 4 times, improves the dynamic control accuracy of CMOR.

Multiple asymptotic behaviors of solutions in the generalized vanishing discount problem

Consider the generalized discounted Hamilton-Jacobi equation \[ λ a ( x ) u + H ( x , D u ) = c ( H ) , \lambda a(x)u+H(x,Du)=c(H), \] where a ( x ) a(x) may vanish or change the signs. Two examples are given in this paper showing that the viscosity solutions of the above equation may not converge as λ \lambda tends to zero.

Lagrangian and Hamiltonian Formalisms for Relativistic Mechanics with Lorentz-Invariant Evolution Parameters in 1 + 1 Dimensions

This article presents alternative Hamiltonian and Lagrangian formalisms for relativistic mechanics using proper time and proper Lagrangian coordinates in 1 + 1 dimensions as parameters of evolution. The Lagrangian and Hamiltonian formalisms for a hypothetical particle with and without charge are considered based on the relativistic equation for the dynamics and integrals of particle motion. A relativistic invariant law for the conservation of energy and momentum in the Lorentz representation is given. To select various generalized coordinates and momenta, it is possible to modify the Lagrange equations of the second kind due to the relativistic laws of conservation of energy and momentum. An action function is obtained with an explicit dependence on the velocity of the relativistic particles. The angular integral of the particle motion is derived from Hamiltonian mechanics, and the displacement Hamiltonian is obtained from the Hamilton–Jacobi equation. The angular integral of the particle motion θ is an invariant form of the conservation law. It appears only at relativistic intensities and is constant only in a specific case. The Hamilton–Jacobi–Lagrange equation is derived from the Hamilton–Jacobi equation and the Lagrange equation of the second kind. Using relativistic Hamiltonian mechanics, the Euler–Hamilton equation is obtained by expressing the energy balance through the angular integral of the particle motion θ. The given conservation laws show that the angular integral of the particle motion reflects the relativistic Doppler effect for particles in 1 + 1 dimensions. The connection between the integrals of the particle motion and the doubly special theory of relativity is shown. As an example of the applicability of the proposed invariant method, analyses of the motion of relativistic particles in circularly polarized, monochromatic, spatially modulated electromagnetic plane waves and plane laser pulses are given, and comparisons are made with calculations based on the Landau and Lifshitz method. To allow for the analysis of the oscillation of a particle in various fields, a phase-plane method is presented.

Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

Rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales

We study the rate of convergence in periodic homogenization for convex Hamilton–Jacobi equations with multiscales, where the Hamiltonian depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is by optimal control formulas, scale separations and curve cutting techniques. We also show the rate of homogenization for the static problem based on the same idea. Additionally, we provide examples that illustrate the rate of convergence for the Cauchy problem is optimal for and .

Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations

In the paper, we consider a path-dependent Hamilton–Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton–Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of inequalities for coinvariant sub- and supergradients. We prove that these two notions are equivalent, which is the main result of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton–Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein–Preiss variational principle with an appropriate gauge-type functional.

Global existence and convergence results for a class of nonlinear time fractional diffusion equation

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−.