Hamilton-Jacobi Formalism Research Articles

Adaptive asymptotic solutions of inflationary models in the Hamilton-Jacobi formalism: application to T-models

We develop a method to compute the slow-roll expansion for the Hubble parameter in inflationary models in a flat Friedmann-Lemaître-Robertson-Walker spacetime that is applicable to a wide class of potentials including monomial, polynomial, or rational functions of the inflaton, as well as polynomial or rational functions of the exponential of the inflaton. The method, formulated within the Hamilton-Jacobi formalism, adapts the form of the slow-roll expansion to the analytic form of the inflationary potential, thus allowing a consistent order-by-order computation amenable to Padé summation. Using T-models as an example, we show that Padé summation extends the domain of validity of this adapted slow-roll expansion to the end of inflation. Likewise, Padé summation extends the domain of validity of kinetic-dominance asymptotic expansions of the Hubble parameter into the fast-roll regime, where they can be matched to the aforesaid Padé-summed slow-roll expansions. This matching in turn determines the relation between the expansions for the number N of e-folds and allows us to compute the total amount of inflation as a function of the initial data or, conversely, to select initial data that correspond to a fixed total amount of inflation. Using the slow-roll stage expansions, we also derive expansions for the corresponding spectral index ns accurate to order 1/N2, and tensor-to-scalar ratio r accurate to order 1/N3 for these T-models.

Causal geodesics in cylindrically symmetric vacuum spacetimes using Hamilton-Jacobi formalism

Causal geodesics in cylindrically symmetric vacuum spacetimes using Hamilton-Jacobi formalism

Hamilton–Jacobi formalism for k-inflation

We propose a type of k-inflation under the Hamilton–Jacobi approach. We calculate various observables such as the scalar power spectrum, the tensor-to-scalar ratio, the scalar spectra index for the case where the Hubble parameter is a power-law function of k-field. The model’s parameters are constrained with Planck data and the concrete form of the potential is presented. The results show that the model can be in good agreement with observations.

Motion of test particles and topological interpretation of generic rotating regular black holes coupled to non-linear electrodynamics

This research is devoted to investigate the dynamics of test particles and the intricate topological nature of generic rotating Regular Black Holes (RBHs). By applying the Hamilton–Jacobi formalism, we have presented the paths of test particles as they move around the RBHs, graphically. The dynamics of angular momentum and energy of particles in both counter-rotation and co-rotation are studied. As these particles move around RBH, we observe the interplay of forces shaping their journey. We observe how the effective force, effective potential, and Lyapunov exponent change over time. The Lyapunov exponent, a measure of chaos in their motion, evolves, hinting at the stability of their orbits. Moreover, we study the topological properties of a generic rotating RBH and determine their topological numbers, which are sums of winding numbers around defects and probe the fabric of spacetime itself. The winding number is an integer that indicates how many times a curve encircling a defect wraps around the origin. We find that the total topological number is equal to 0 which suggest a system in balance. As the value of degree of nonlinear electrodynamics parameter (μ) increases, the turning points in particle trajectories multiply and presenting the picture of more complexity. In a twist of topology, the interchange of winding numbers can cause a phase transition, reshaping the order parameter space’s topology.

Optical features of rotating quintessential charged black holes in de-Sitter spacetime

One of the most important and actual issues in relativistic astrophysics is testing gravity theories and obtaining constraint values for the parameters of black holes using observational data. In this research, we aimed to explore the optical features of a Kerr–Newman black hole model in the presence of a quintessential field, which may be a candidate for a dark-energy model with a nonzero cosmological constant. First, we obtain the equations of motion for photons using the Hamilton–Jacobi formalism. We also study the horizons and shapes of the apparent regions of the photon region around the said black hole. In various scenarios, we investigate shadows cast by the black hole using celestial coordinates. Furthermore, we analyze the effects of the quintessential field and black hole charge on the shadow radius and distortion. Furthermore, we look into the constraints on the spin and charge of supermassive black holes M87∗ and Sagittarius A∗ for different values of the quintessential field using their shadow size measured by the Event Horizon Telescope Collaboration. Finally, we study the effects of the quintessential field, black hole spin and charge on its energy emission rate by Hawking radiation and compare our results with those of the available literature.

Efficient and scalable path-planning algorithms for curvature constrained motion in the Hamilton-Jacobi formulation

We present a partial-differential-equation-based optimal path-planning framework for curvature constrained motion, with application to vehicles in 2- and 3-spatial-dimensions. This formulation relies on optimal control theory, dynamic programming, and Hamilton-Jacobi-Bellman equations. We develop efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations which can solve these types of path-planning problems efficiently, even in high dimensions, while maintaining the Hamilton-Jacobi formulation. Because our method is rooted in optimal control theory and has no black box components, it has solid interpretability, and thus averts the tradeoff between interpretability and efficiency for high-dimensional path-planning problems. We demonstrate our method with several examples.

4D Einstein-Gauss-Bonnet black hole in Power-Yang-Mills field: a shadow study

We consider a static black hole immersed in the Power-Yang-Mills field in four-dimensional Einstein-Gauss-Bonnet gravity and investigate the effect of various parameters on the radius of the photon sphere. The modified form of the Newman-Janis algorithm is used for obtaining a rotating black hole solution in this gravity. Further, we try to explore the influence of the Yang-Mills magnetic charge Q with power q, Gauss-Bonnet parameter α, and spin a on the horizon radius. The geodesic equations are constructed by incorporating the Hamilton-Jacobi formalism. The radial component of the geodesic equations gives the effective potential which is further used in deriving the mathematical structure for the shadows by using Bardeen's procedure for a fixed observer at infinity. The shadows are calculated and plotted in terms of two celestial coordinates for an equatorial observer. It is observed that all the parameters have a very significant effect on the shadow and related physical observables. We also obtain the constraint values for the spin, magnetic charge, and Gauss-Bonnet parameters, using the shadow size of supermassive black holes Sagittarius A* and M87* from the EHT observations for the cases of q = 0.6 and q = 0.9. It is shown that there are upper and lower bounds for the charge and spin of M87* at q = 0.6, while only the upper bounds for the charge and spin of Sagittarius A*. Finally, we investigate the energy emission rate in the Hawking radiation around the 4D Einstein-Gauss-Bonnet black hole in the Power-Yang-Mills field.

Classical Evolution without Evolution

The well-known argument of Page and Wootters demonstrates how to “derive" the usual quantum dynamics of a subsystem in a global state which is an eigenstate of the total Hamiltonian. I show how the same argument can be made in classical physics by using a formalism that closely resembles the quantum one. This is not surprising, since the Hamilton–Jacobi formulation of classical dynamics is precisely motivated by the logic of timeless dynamics. Ultimately, the key to obtaining dynamics without dynamics is the principle of energy conservation which leads to correlations between times pertaining to different subsystems. The same can, of course, be said about all other conserved quantities, and we show how to address this problem in its full generality so as to realise rotation without rotation, translation without translation and so on. The classical and quantum interpretations do, however, have one major difference and this is the Church of the Higher Hilbert Space interpretation of mixtures, which only exists in quantum physics. We discuss a few consequences of this point.

Analysis of linearized Weyl gravity via the Hamilton–Jacobi method

The Hamilton–Jacobi formalism is used to analyze the Weyl theory in the weak-field limit. The complete set of involutive Hamiltonians is obtained, which are classified into involutive and non-involutive. The counting of degrees of freedom is performed. Additionally, the generalized brackets and gauge symmetries are reported.

An extension of quantum Hamilton-Jacobi formalism to N-spatial dimensions and its applications

An extension of quantum Hamilton-Jacobi formalism to N-spatial dimensions and its applications

Generalized Hamilton-Jacobi Formulation of Damped Harmonic System Using Fractional Derivatives

Generalized Hamilton-Jacobi Formulation of Damped Harmonic System Using Fractional Derivatives

Fractional Quantization of Podolsky Electrodynamics Using Fractional Hamilton-Jacobi Formulation

Fractional Quantization of Podolsky Electrodynamics Using Fractional Hamilton-Jacobi Formulation

Holographic renormalization in the Hamilton-Jacobi formulation with exact ansatz generation

In $\mathrm{AdS}/\mathrm{CFT}$ corresponding, the UV divergence of generating functional on the field theory can be removed as the IR divergence in the gravity. This geometric process is well known as holographic renormalization. The standard method of holographic renormalization is based on the Fefferman-Graham expansion, which is strict and universal but technically cumbersome. To improve the technique, different methods have been proposed. Here we develop an alternative approach to holographic renormalization based on the Hamilton-Jacobi formulation of gravity. Compared to previous approaches, its distinguishing feature is the generation of exact ansatz of counterterms. We apply this approach to several typical holographic models, which consistently performs well.

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.

SWKB quantization condition for conditionally exactly solvable systems and the residual corrections

The SWKB quantization condition is an exact quantization condition for the conventional shape-invariant potentials. On the other hand, this condition equation does not hold for other known solvable systems. The origin of the (non-)exactness is understood in the context of the quantum Hamilton–Jacobi formalism. First we confirm the statement and show inexplicit properties numerically for the case of the conditionally exactly solvable systems by Junker and Roy. The SWKB condition clearly breaks for this case, but the condition equation is restored within a certain degree of accuracy. We propose a novel approach to evaluate the residual by perturbation, intending to explore what the correction terms for the SWKB condition equation look like.

Influence of integer design variables in topology optimization of incompressible turbulent flow

• Optimize incompressible turbulent flow at bidimensional and tridimensional domains employing the continuous adjoint approach • Analyze the influence of integer design variables in topology optimization of incompressible turbulent fluids • Develop the adjoint near wall distance calculation to treat the adjoint spalart allmaras turbulence model Turbulent fluid flow is a challenging physical problem to be addressed with topology optimization due to the vortex generation and the near wall distance calculation that difficult the calibration of topology optimization parameters when continuous design variables based-optimizers are used. The intrinsic nature of topology optimization inverse permeability interpolation clashes with the accuracy requirements from turbulent fluid flow dynamics. To solve the mentioned challenge, the present paper investigates the influence of an integer continuous variable-based-optimizer, i.e. a [0, 1] binary behaviour, that avoids grey regions during the optimization process. Its influence on the near wall distance calculation is analyzed as it affects the permeability of the fluid domain. The turbulence phenomenon is treated via the Spalart-Allmaras model and its near wall distance calculation is approached by the Hamilton-Jacobi formulation. Permeability independence on the topology optimization formulation constraints is proposed firstly at the continuous adjoint approach to avoid miscalculations in the fluid cells. The finite volume method is employed to solve the Reynolds Averaged Navier-Stokes governing equations and 2D and 3D tests are developed, where discussions on the solid-fluid boundaries definition, the material penalization dependence, objective function comparisons and accuracy of the CFD analysis and the wall distance calculation are presented.

Epistemic Uncertainty from an Averaged Hamilton–Jacobi Formalism

In recent years, the non-relativistic quantum dynamics derived from three assumptions; (i) probability current conservation, (ii) average energy conservation, and (iii) an epistemic momentum uncertainty (Budiyono and Rohrlich in Nat Commun 8:1306, 2017). Here we show that, these assumptions can be derived from a natural extension of classical statistical mechanics.

A Real Scalar Field Unifying the Early Inflation and the Late Accelerating Expansion of the Universe through a Quadratic Equation of State: The Vacuumon

In a previous paper we introduced a cosmological model describing the early inflation, the intermediate decelerated expansion, and the late accelerating expansion of the universe in terms of a single barotropic fluid characterized by a quadratic equation of state. We obtained a scalar field representation of this fluid and determined the potential V(ϕ) connecting the inflaton potential in the early universe to the quintessence potential in the late universe. This scalar field has later been called the ‘vacuumon’ by other authors, in the context of the Running Vacuum model. In this paper, we study how the scalar field potential is modified by the presence of other cosmic components such as stiff matter, black-body radiation, baryonic matter, and dark matter. We also determine the mass m and the self-interaction constant λ of the scalar field given by the second and fourth derivatives of the potential at its extrema. We find that its mass is imaginary in the early universe with a modulus of the order of the Planck mass MP=(ℏc/G)1/2=1.22×1019GeV/c2 and real in the late universe with a value of the order of the cosmon mass mΛ=(Λℏ2/c4)1/2=2.08×10−33eV/c2 predicted by string theory. Although our model is able to describe the evolution of the homogeneous background for all times, it cannot account for the spectrum of fluctuations in the early universe. Indeed, by applying the Hamilton–Jacobi formalism to our model of early inflation, we find that the Hubble hierarchy parameters and the spectral indices lead to severe discrepancies with the observations. This suggests that the vacuumon potential is just an effective classical potential that cannot be directly used to compute the fluctuations in the early universe. A fully quantum field theory may be required to achieve that goal. Finally, we discuss the connection between our model based on a quadratic equation of state and the Running Vacuum model which assumes a variation of the cosmological constant with the Hubble parameter.

Path Integral Quantization of Doubly Supersymmetric Model

The Hamilton-Jacobi formalism is used to discuss the path integral quantization of the double supersymmetric models with the spinning superparticle in the component and superfield form. The equations of motion are obtained as total differential equations in many variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.

Asymptotic Solutions of a Generalized Starobinski Model: Kinetic Dominance, Slow Roll and Separatrices

We consider a generalized Starobinski inflationary model. We present a method for computing solutions as generalized asymptotic expansions, both in the kinetic dominance stage (psi series solutions) and in the slow roll stage (asymptotic expansions of the separatrix solutions). These asymptotic expansions are derived in the framework of the Hamilton-Jacobi formalism where the Hubble parameter is written as a function of the inflaton field. They are applied to determine the values of the inflaton field when the inflation period starts and ends as well as to estimate the corresponding amount of inflation. As a consequence, they can be used to select the appropriate initial conditions for determining a solution with a previously fixed amount of inflation.