Hamilton-Jacobi Method Research Articles

An action function for a higher step Grushin operator

The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.

Hawking radiation of Schwarzschild–de Sitter black hole by Hamilton–Jacobi method

We investigate the Hawking radiation of Schwarzschild–de Sitter (SdS) black hole by massive particles tunneling method. We consider the spacetime background to be dynamical, incorporate the self-gravitation effect of the emitted particles and show that the tunneling rate is related to the change of Bekenstein–Hawking entropy and the derived emission spectrum deviates from the pure thermal spectrum when energy and angular momentum are conserved. Our result is also in accordance with Parikh and Wilczekʼs opinion and gives a correction to the Hawking radiation of SdS black hole.

An extension of the Hamilton-Jacobi method

Abstract A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.

QUANTUM TUNNELING AND SPECTROSCOPY OF NONCOMMUTATIVE INSPIRED KERR BLACK HOLE

We discuss the thermodynamics of the noncommutative inspired Kerr black hole by means of a reformulated Hamilton–Jacobi method and a dimensional reduction technique. In order to investigate the effect of the angular momentum of the tunneling particle, we calculate the wave function to the first order of the WKB ansatz. Then, using a density matrix technique we derive the radiation spectrum from which the radiation temperature can be read out. Our results show that the radiation of this noncommutative inspired black hole corresponds to a modified temperature which involves the effect of noncommutativity. However, the angular momentum of the tunneling particle has no influence on the radiation temperature. Moreover, we analyze the entropy spectrum and verify that its quantization is modified neither by the noncommutativity of spacetime nor by the quantum correction of wave functions.

Hamilton-Jacobi renormalization for Lifshitz spacetime

Just like AdS spacetimes, Lifshitz spacetimes require counterterms in order to make the on-shell value of the bulk action finite. We study these counterterms using the Hamilton-Jacobi method. Rather than imposing boundary conditions from the start, we will derive suitable boundary conditions by requiring that divergences can be canceled using only local counterterms. We will demonstrate in examples that this procedure indeed leads to a finite bulk action while at the same time it determines the asymptotic behavior of the fields. This puts more substance to the belief that Lifshitz spacetimes are dual to well-behaved field theories. As a byproduct, we will find the analogue of the conformal anomaly for Lifshitz spacetimes.

Hawking radiation of Dirac particles from black strings

Hawking radiation has been studied as a phenomenon ofquantum tunneling in different black holes. In this paper we extendthis semi-classical approach to cylindrically symmetric black holes.Using the Hamilton-Jacobi method and WKB approximation we calculatethe tunneling probabilities of incoming and outgoing Dirac particlesfrom the event horizon and find the Hawking temperature of theseblack holes. We obtain results both for uncharged as well as chargedparticles.

Hawking radiation as tunneling from charged black holes in 0A string theory

There has been much work on explaining Hawking radiation as a quantum tunneling process through horizons. Basically, this intuitive picture requires the calculation of the imaginary part of the action for outgoing particle. And two ways are known for achieving this goal: the null-geodesic method and the Hamilton–Jacobi method. We apply these methods to the charged black holes in 2D dilaton gravity which is originated from the low energy effective theory of type 0A string theory. We derive the correct Hawking temperature of the black holes including the effect of the back reaction of the radiation, and obtain the entropy by using the 1st law of black hole thermodynamics. For fixed-charge ensemble, the 0A black holes are free of phase transition and thermodynamically stable regardless of mass-charge ratio. We show this by interpreting the back reaction term as the inverse of the heat capacity of the black holes. Finally, the possibility of the phase transition in the fixed-potential ensemble is discussed.

Lemaitre-Tolman-Bondi Model of the Universe and Hawking Radiation of a Dynamical Horizon

The paper deals with anisotropic spherically symmetric Lemaitre-Tolman-Bondi model of the universe bounded by the apparent horizon. Using Hamilton-Jacobi method for both massive and massless test particles, we are able to show that the temperature associated with the apparent horizon due to Hawking radiation is the usual Hawking Temperature. Also the usual Lemaitre-Tolman-Bondi model and the transformed r-gauge Lemaitre-Tolman-Bondi model are shown to be equivalent. Finally, the results of Tunneling approach agree with the Hamiltonian-Jacobi method.

Gravitational energy seen by quasilocal observers

There have been many attempts to define quasilocal energy/mass for a spacelike 2-surface in a spacetime by the Hamilton–Jacobi method. The main difficulty in this approach is the subtle choice of the background configuration to be subtracted from the physical Hamiltonian. Quasilocal mass should be positive for general surfaces, but on the other hand should be zero for surfaces in the flat spacetime. In this paper, we survey the work in a series of papers [6, 25–27] in which a new notion of quasilocal mass/energy–momentum is proposed and investigated. In particular, the notion of energy observed by a ‘quasilocal observer’ will be discussed.

Geodesic motions in extraordinary string geometry

The geodesic properties of the extraordinary vacuum string solution in (4+1) dimensions are analyzed by using Hamilton-Jacobi method. The geodesic motions show distinct properties from those of the static one. Especially, any freely falling particle can not arrive at the horizon or singularity. There exist stable null circular orbits and bouncing timelike and null geodesics. To get into the horizon {or singularity}, a particle need to follow a non-geodesic trajectory. We also analyze the orbit precession to show that the precession angle has distinct features for each geometry such as naked singularity, black string, and wormhole.

Particle motion around black hole in Hořava-Lifshitz gravity

Analytical solutions of Maxwell equations around black hole immersed in external uniform magnetic field in the background of the Kehagias-Sfetsos (KS) asymptotically flat black hole solution of Ho\v{r}ava-Lifshitz gravity have been found. Influence of magnetic field on effective potential of the radial motion of charged test particle around black hole immersed in external magnetic field in Ho\v{r}ava-Lifshitz gravity has been investigated by using Hamilton-Jacobi method. Exact analytical solution for dependence of the minimal radius of the circular orbits $r_{\rm mc}$ from KS parameter $\omega$ for motion of test particle around spherical symmetric black hole in Ho\v{r}ava-Lifshitz gravity has been derived. The critical values of the particle's angular momentum for captured particles by black hole in Ho\v{r}ava-Lifshitz gravity have been obtained numerically. The comparison of the obtained numerical results with the astrophysical observational data on radii of the innermost stable circular orbits gives us the estimation of the parameter as $\omega \simeq 3.6 \cdot 10^{-24} {\rm cm}^{-2}$.

Applications of hybrid reachability analysis to robotic aerial vehicles

The control of complex non-linear systems can be aided by modeling each system as a collection of simplified hybrid modes, with each mode representing a particular operating regime defined by the system dynamics or by a region of the state space in which the system operates. Guarantees on the safety and performance of such hybrid systems can still be challenging to generate, however. Reachability analysis using a dynamic game formulation with Hamilton—Jacobi methods provides a useful way to generate these types of guarantees, and the technique is flexible enough to analyze a wide variety of systems. This paper presents two applications of reachable sets, both focused on guaranteeing the safety and performance of robotic aerial vehicles. In the first example, reachable sets are used to design and implement a backflip maneuver for a quadrotor helicopter. In the second, reachability analysis is used to design a decentralized collision avoidance algorithm for multiple quadrotors. The theory for both examples is explained, and successful experimental results are presented from flight tests on the STARMAC quadrotor helicopter platform.

Minimizing orbits in the discrete Aubry–Mather model

We consider a generalization of the Frenkel–Kontorova model in higher dimension leading to a new theory of configurations with minimal energy, as in Aubry's theory or in Mather's twist approach in the periodic case. We consider a one-dimensional chain of particles and their minimizing configurations and we allow the state of each particle to possess many degrees of freedom. We assume that the Hamiltonian of the system satisfies some twist condition. The usual ‘total ordering’ of minimizing configurations does not exist any more and new tools need to be developed. The main mathematical tool is to cast the study of the minimizing configurations into the framework of discrete Lagrangian theory. We introduce forward and backward Lax–Oleinik problems and interpret their solutions as discrete viscosity solutions as in Hamilton–Jacobi methods. We give a fairly complete description of a particular class of minimizing configurations: the calibrated class. These configurations may be thought of as ‘ground states’ obtained in the thermodynamic limit at temperature zero. We obtain, in particular, Mather's graph property or the noncrossing property of two calibrated configurations and the existence of a rotation number for most of the calibrated configurations.

THE HAMILTON–JACOBI ANALYSIS OF OPTICAL WAVEGUIDES

In this paper, the Hamilton–Jacobi method for solving differential equations of wave propagation is investigated. It is shown that when a wave propagates through a background medium that varies slowly on the scale of the wavelength, the trajectory of the light particle follows the usual rule of refraction.

Hawking radiation from a Vaidya black hole by Hamilton-Jacobi method

Using the Hamilton-Jacobi method, Hawking radiation from the apparent horizon of a dynamical Vaidya black hole is calculated. The black hole thermodynamics can be built successfully on the apparent horizon. If a relativistic perturbation is given to the apparent horizon, a similar calculation can also lead to a purely thermal spectrum, which corresponds to a modified temperature from the former. The first law of thermodynamics can also be constructed successfully at a new supersurface which has a small deviation from the apparent horizon. When the event horizon is thought as such a deviation from the apparent horizon, the expressions of the characteristic position and temperature are consistent with the previous result that asserts that thermodynamics should be built on the event horizon. It is concluded that the thermodynamics should be constructed on the apparent horizon exactly while the event horizon thermodynamics is just one of the perturbations near the apparent horizon.

Fermion Tunneling from an Axis-Symmetric Black Hole beyond the Semiclassical Approximation

Based on the Hamilton-Jacobi method beyond the semiclassical approximation proposed by R. Baberjee and B.R. Majhi, Hawking radiation of Dirac particle as tunneling through the event horizon is calculated. It is shown that all quantum corrections in the fermion particle action are proportional to the usual semiclassical contribution. Under the conception of irreducible mass and the first law of thermodynamics, the modifications to Hawking temperature and Bekenstein-Hawking entropy are given for a Kerr-Newmann black hole.

Stochastic deformation of integrable dynamical systems and random time symmetry

We present a deformation of a class of elementary classical integrable systems using stochastic diffusion processes. This deformation applies to the solution of the associated classical Newtonian, Hamiltonian, Lagrangian, and variational problems and to the Hamilton–Jacobi method of characteristics. The underlying stochastic action functionals involve dual random times, whose expectations are connected to the new variables of the system after a canonical transformation.

Hamilton–Jacobi counterterms for Einstein–Gauss–Bonnet gravity

The on-shell gravitational action and the boundary stress tensor are essential ingredients in the study of black hole thermodynamics. We employ the Hamilton–Jacobi method to calculate the boundary counterterms necessary to remove the divergences and allow the study of the thermodynamics of Einstein–Gauss–Bonnet black holes.

Hawking radiation due to photon and gravitino tunneling

Applying the Hamilton–Jacobi method we investigate the tunneling of photon across the event horizon of a static spherically symmetric black hole. The necessity of the gauge condition on the photon field, to derive the semiclassical Hawking temperature, is explicitly shown. Also, the tunneling of photon and gravitino beyond this semiclassical approximation are presented separately. Quantum corrections of the action for both cases are found to be proportional to the semiclassical contribution. Modifications to the Hawking temperature and Bekenstein–Hawking area law are thereby obtained. Using this corrected temperature and Hawking’s periodicity argument, the modified metric for the Schwarzschild black hole is given. This corrected version of the metric, up to ℏ order is equivalent to the metric obtained by including one loop back reaction effect. Finally, the coefficient of the leading order correction of entropy is shown to be related to the trace anomaly.

Electrostatic ion traps with separation of variables in parabolic coordinates

Electrostatic traps with ideal spatiotemporal focusing in one of the directions of ion motion and the motion in transverse directions integrated using the Hamilton-Jacobi method are considered. Conditions for the ion confinement in the trap field are determined. The finite character of ion motion under confinement conditions is independently checked by the numerical integration of the Lagrange equations.