Hamilton-Jacobi Equation Research Articles

On the vanishing discount approximation for compactly supported perturbations of periodic Hamiltonians: the 1d case

We study the asymptotic behavior of the viscosity solutions of the Hamilton-Jacobi (HJ) equation as the positive discount factor λ tends to 0, where is the perturbation of a Hamiltonian –periodic in the space variable and convex and coercive in the momentum, by a compactly supported potential The constant c(G) appearing above is defined as the infimum of values for which the HJ equation in admits bounded viscosity subsolutions. We prove that the functions locally uniformly converge, for to a specific solution of the critical equation We identify in terms of projected Mather measures for G and of the limit to the unperturbed periodic problem. Our work also includes a qualitative analysis of the critical equation with a weak KAM theoretic flavor.

Revisiting the second law and weak cosmic censorship conjecture in high-dimensional charged-AdS black hole: an additional assumption

The verification of the second law of black hole mechanics and the WCCC in the context of enthalpy as mass of the black hole and its related thermodynamic properties has not been tested through a vast number of literature in the recent past. Such studies are of great physical importance as they provide us with a large number of information regarding the thermodynamics and the dynamics of AdS black hole systems. We invest the prior limited surveys of such analysis to investigate the WCCC for the D- dimensional asymptotically AdS-charged black holes characterized by its mass (M), electric charge (Q), and AdS radius (l) under the absorption of scalar particles of charge q. We examine the WCCC by analyzing the energy-momentum condition of the electrically charged particles as absorbed by the black holes. We prove that the conjecture is well verified irrespective of whether the initial black hole configurations are extremal or non-extremal by changing its charge, the AdS radius, and their variations. We show that the first law and the WCCC are valid for all spacetime dimensions (D) independent of the choice of the parameters characterizing the black holes. But to verify the second law in the extremal and non- extremal configurations one has to be very cautious as it gets strongly affected by the choices of the values of the black hole parameters and their variations. In other words, we use charged particle dynamics as described by the Hamilton-Jacobi equation to obtain the energy-momentum relation as the charged particle dropped into the higher dimensional charged AdS black hole and verify the thermodynamic laws when the scalar charged particle gets absorbed by the black holes and correspondingly the black hole neutralization in different manners. Additionally, we further probe the validity of WCCC in such a black hole background. In the context of the extended phase space, taking the grand canonical potential into account allow us to obtain the missing information about the variation of the cosmological constant necessary to construct the extended phase space, namely the notion of the black hole pressure, and which is absent in the previous literature so far.

Rate of convergence for singular perturbations of Hamilton-Jacobi equations in unbounded spaces

We prove rate of convergence results for singular perturbations of Hamilton-Jacobi equations in unbounded spaces where the fast operator is linear, uniformly elliptic and has an Ornstein-Uhlenbeck-type drift. The slow operator is a fully nonlinear elliptic operator while the source term is assumed only locally Hölder continuous in both fast and slow variables. We obtain several rates of convergence according to the regularity of the source term.

Degenerate linear parabolic equations in divergence form on the upper half space

We study a class of second-order degenerate linear parabolic equations in divergence form in ( − ∞ , T ) × R + d (-\infty , T) \times {\mathbb {R}}^d_+ with homogeneous Dirichlet boundary condition on ( − ∞ , T ) × ∂ R + d (-\infty , T) \times \partial {\mathbb {R}}^d_+ , where R + d = { x ∈ R d : x d > 0 } {\mathbb {R}}^d_+ = \{x \in {\mathbb {R}}^d: x_d>0\} and T ∈ ( − ∞ , ∞ ] T\in {(-\infty , \infty ]} is given. The coefficient matrices of the equations are the product of μ ( x d ) \mu (x_d) and bounded uniformly elliptic matrices, where μ ( x d ) \mu (x_d) behaves like x d α x_d^\alpha for some given α ∈ ( 0 , 2 ) \alpha \in (0,2) , which are degenerate on the boundary { x d = 0 } \{x_d=0\} of the domain. Our main motivation comes from the analysis of degenerate viscous Hamilton-Jacobi equations. Under a partially VMO assumption on the coefficients, we obtain the well-posedness and regularity of solutions in weighted Sobolev spaces. Our results can be readily extended to systems.

Shadow of the 5D Reissner–Nordström AdS black hole

In this paper, we discuss the shadow cast by the charged Reissner–Nordström (RN) anti-de Sitter (AdS) black hole. With the help of the Killing equation and Hamilton–Jacobi equation, we calculate the geodesic equations for null particles. With the help of geodesics of null particle, we then determine the celestial coordinates ([Formula: see text], [Formula: see text]) and the shadow radius of the RN AdS black hole. We present a graphical analysis of the black hole shadow and find that shadow is a perfectly dark circle. The impacts of charge and cosmological constant of the RN AdS black hole on the radius of shadow are also presented. In this connection, the radius of the shadow is a decreasing function of the charge. Furthermore, we study the effects of plasma medium on the RN AdS black hole shadow. Here, we find that radius of circular shadow increases with increasing plasma parameter. We study the shadow radius for the constrained values of charge and cosmological constant from the [Formula: see text] and [Formula: see text] black holes. In addition, we also discuss the energy emission the rate of RN black hole. The effects of parameters like charge, cosmological constant and plasma parameter on energy emission rate are analyzed graphically.

Dispersion chain of quantum mechanics equations

Based on the dispersion chain of the Vlasov equations, the paper considers the construction of a new chain of equations of quantum mechanics of high kinematical values. The proposed approach can be applied to consideration of classical and quantum systems with radiation. A number of theorems are proved on the form of extensions of the Hamilton operators, Lagrange functions, Hamilton–Jacobi equations, and Maxwell equations to the case of a generalized phase space. In some special cases of lower dimensions, the dispersion chain of quantum mechanics is reduced to quantum mechanics in phase space (the Wigner function) and the de Broglie–Bohm «pilot wave» theory. An example of solving the Schrödinger equation of the second rank (for the phase space) is analyzed, which, in contrast to the Wigner function, gives a positive distribution density function.

An Eikonal Equation with Vanishing Lagrangian Arising in Global Optimization

We show a connection between global unconstrained optimization of a continuous function f and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution v of the critical Hamilton–Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then v is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of v. Such trajectories are proved to converge to the set of minima of f, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.

A mean field model for the interactions between firms on the markets of their inputs

We consider an economy made of competing firms which are heterogeneous in their capital and use several inputs for producing goods. Their consumption policy is fixed rationally by maximizing a utility and their capital cannot fall below a given threshold (state constraint). We aim at modeling the interactions between firms on the markets of the different inputs on the long term. The stationary equlibria are described by a system of coupled non-linear differential equations: a Hamilton–Jacobi equation describing the optimal control problem of a single atomistic firm; a continuity equation describing the distribution of the individual state variable (the capital) in the population of firms; the equilibria on the markets of the production factors. We prove the existence of equilibria under suitable assumptions.

Action functional gradient descent algorithm for estimating escape paths in stochastic chemical reaction networks.

We first derive the Hamilton-Jacobi theory underlying continuous-time Markov processes, and then we use the construction to develop a variational algorithm for estimating escape (least improbable or first passage) paths for a generic stochastic chemical reaction network that exhibits multiple fixed points. The design of our algorithm is such that it is independent of the underlying dimensionality of the system, the discretization control parameters are updated toward the continuum limit, and there is an easy-to-calculate measure for the correctness of its solution. We consider several applications of the algorithm and verify them against computationally expensive means such as the shooting method and stochastic simulation. While we employ theoretical techniques from mathematical physics, numerical optimization and chemical reaction network theory, we hope that our work finds practical applications with an inter-disciplinary audience including chemists, biologists, optimal control theorists and game theorists.

Hamilton–Jacobi approach to thermodynamic transformations

In this note, we formulate and study a Hamilton–Jacobi approach for describing thermodynamic transformations. The thermodynamic phase space assumes the structure of a contact manifold with the points representing equilibrium states being restricted to certain submanifolds of this phase space. We demonstrate that Hamilton–Jacobi theory consistently describes thermodynamic transformations on the space of externally controllable parameters or equivalently the space of equilibrium states. It turns out that in the Hamilton–Jacobi description, the choice of the principal function is not unique but, the resultant dynamical description for a given transformation remains the same irrespective of this choice. Some examples involving thermodynamic transformations of the ideal gas are discussed where the characteristic curves on the space of equilibrium states completely describe the dynamics. The geometric Hamilton–Jacobi formulation which has emerged recently is also discussed in the context of thermodynamics.

Event-triggered constrained neural critic control of nonlinear continuous-time multiplayer nonzero-sum games

In this article, by exploiting the event-triggered neural critic control scheme, the optimal control question of nonlinear continuous-time multiplayer nonzero-sum games with asymmetric control constraints is investigated. First, to satisfy the asymmetric constraints, we propose an appropriate nonquadratic function. Also, the time-based coupled Hamilton-Jacobi (HJ) equations and the optimal control policies are deduced. To save communication resources and improve control efficiency, the event-driven mechanism is innovated. Further, the event-triggered optimal control strategies and the event-triggered coupled HJ equations are given. Afterwards, a novel triggering condition is established, and the critic neural networks are built to evaluate the optimal cost functions, thereby obtaining the event-triggered near-optimal controls. During neural critic learning, a new weight adjustment rule is proposed. Moreover, by exploiting the Lyapunov approach, the stability of the system state and the weight estimation errors is guaranteed. Finally, two examples are given to verify the efficacy of the established event-triggered neural critic control scheme.

An appearance of classical matter from the self-organizing process of quantum systems

We present a quantum effect where matter follows the classical Hamilton-Jacobi equation, which emerges from quantum systems with Riemannian structures, as in standard quantum systems such as semiconductor heterostructures, quantum plasmas, and quantum dots. The proposed effect is derived from solving a standard elliptic partial differential equation of the radial part of the wave function, which is equivalent to a vanished quantum potential of the system. We then analyze such an effect and examine how the classical matter tends to be denser at the boundary region of the system when the quantum system is given in a finite region in space. While the proposed effect is derived from the hydrodynamical formulation of quantum mechanics, the results are free from any interpretation of quantum mechanics.

New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast with the case without vector potential, the corresponding integrals may no longer be connected to separation of variables in the Hamilton–Jacobi equation and can have more general leading order terms. We focus on two cases extending the physically relevant cylindrical- and spherical-type integrals. We find three new integrable systems in the generalized cylindrical case but none in the spherical one. We conjecture that this is related to the presence, respectively absence, of maximal abelian Lie subalgebra of the three-dimensional Euclidean algebra generated by first order integrals in the limit of vanishing magnetic field. By investigating superintegrability, we find only one (minimally) superintegrable system among the integrable ones. It does not separate in any orthogonal coordinate system. This system provides a mathematical model of a helical undulator placed in an infinite solenoid.

Horizontally quasiconvex envelope in the Heisenberg group

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex, for short) envelope of a given continuous function in the Heisenberg group.We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton– Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton–Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well.

Quantitative homogenization for combustion in random media

We obtain the first quantitative stochastic homogenization result for reaction–diffusion equations, for ignition reactions in dimensions d\le 3 that either have finite ranges of dependence or are close enough to such reactions, and for solutions with initial data that approximate characteristic functions of general convex sets. We show an algebraic rate of convergence of these solutions to their homogenized limits, which are (discontinuous) viscosity solutions of certain related Hamilton–Jacobi equations.

Convex sets with obstacle and gradient constraints

For 1<p<∞, given g and ψ non-negative functions belonging to L∞(Ω) and W01,p(Ω)∩C(Ω‾), respectively, we show that there exists a pseudo-metric L‾g defined in Ω‾ such that u(x):=miny∈Ω‾⁡{ψ(y)+L‾g(x,y)} is a subsolution of the Hamilton-Jacobi equation with obstacle max⁡{|∇u|−g,u−ψ}=0. Besides, for Kg,ψ={v∈W01,p(Ω):|∇v|≤g,v≤ψ}, we have u(x)=⋁{v(x):v∈Kg,ψ}.As a consequence, we prove the Mosco convergence of Kgn,ψn to Kg,ψ, as long as gn converges to g in L∞(Ω) and ψn to ψ in C(Ω‾). As an application, we prove a stability result for the solutions un of variational inequalities defined in the convex sets Kgn,ψn.

“A call to action”: Schrödinger's representation of quantum mechanics via Hamilton's principle

A few years ago, one of the former Editors of this journal launched “a call to action” (E. F. Taylor, Am. J. Phys. 71, 423–425 (2003)) for a revision of teaching methods in physics in order to emphasize the importance of the principle of least action. In response, we suggest the use of Hamilton's principle of stationary action to introduce the Schrödinger equation. When considering the geometric interpretation of the Hamilton–Jacobi theory, the real part of the action S defines the phase of the wave function exp iS/ℏ, and requiring the Hamilton–Jacobi wave function to obey wave-front propagation (i.e., Re(S) is a constant of the motion) yields the Schrödinger equation.

Hamilton–Jacobi equations for controlled gradient flows: The comparison principle

Motivated by recent developments in the fields of large deviations for interacting particle systems and mean field control, we establish a comparison principle for the Hamilton–Jacobi equation corresponding to linearly controlled gradient flows of an energy function E defined on a metric space (E,d). Our analysis is based on a systematic use of the regularizing properties of gradient flows in evolutional variational inequality (EVI) formulation, that we exploit for constructing rigorous upper and lower bounds for the formal Hamiltonian at hand and, in combination with the use of the Tataru's distance, for establishing the key estimates needed to bound the difference of the Hamiltonians in the proof of the comparison principle. Our abstract results apply to a large class of examples only partially covered by the existing theory, including gradient flows on Hilbert spaces and the Wasserstein space equipped with a displacement convex energy functional E satisfying McCann's condition.

Paradigm for the creation of scales and phases in nonlinear evolution equations

The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.

Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal

In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated.