Jacobi Formalism Research Articles

Hamilton–Jacobi formalism for linearized gravity

In this work, we study the theory of linearized gravity via the Hamilton–Jacobi formalism. We make a brief review of this theory and its Lagrangian description, as well as a review of the Hamilton–Jacobi approach for singular systems. Then, we apply this formalism to analyze the constraint structure of the linearized gravity in instant- and front-form dynamics.

Hamilton–Jacobi Formulation for Reach–Avoid Differential Games

A new framework for formulating reachability problems with competing inputs, nonlinear dynamics, and state constraints as optimal control problems is developed. Such reach-avoid problems arise in, among others, the study of safety problems in hybrid systems. Earlier approaches to reach-avoid computations are either restricted to linear systems, or face numerical difficulties due to possible discontinuities in the Hamiltonian of the optimal control problem. The main advantage of the approach proposed in this paper is that it can be applied to a general class of target-hitting continuous dynamic games with nonlinear dynamics, and has very good properties in terms of its numerical solution, since the value function and the Hamiltonian of the system are both continuous. The performance of the proposed method is demonstrated by applying it to a case study, which involves the target-hitting problem of an underactuated underwater vehicle in the presence of obstacles.

MAGNETIZED PARTICLE MOTION AROUND BLACK HOLE IN BRANEWORLD

We investigate the motion of a magnetized particle orbiting around a black hole in braneworld immersed in asymptotically uniform magnetic field. The influence of brane parameter on effective potential of the radial motion of magnetized spinning particle around the braneworld black hole using Hamilton–Jacobi formalism is studied. It is found that circular orbits for photons and slowly moving particles may become stable near r = 3M. It was argued that the radii of the innermost stable circular orbits are sensitive on the change of brane parameter. Similar discussion without Weil parameter has been considered by de Felice et al. in Refs. 1 and 2.

Hamilton–Jacobi Formulation for Systems in Terms of Riesz’s Fractional Derivatives

The paper presents fractional Hamilton–Jacobi formulations for systems containing Riesz fractional derivatives (RFD’s). The Hamilton–Jacobi equations of motion are obtained. An illustrative example for simple harmonic oscillator (SHO) has been discussed. It was observed that the classical results are recovered for integer order derivatives.

Hamilton–Jacobi and fractional like action with time scaling

This paper represents the Hamilton–Jacobi formulation for fractional variational problem with fractional like action written as an integration over a time scaling parameter. Also we developed the fractional Hamiltonian formulation for the fractional like action. In all the given calculations, the most popular Riemann–Liouville (RL) and Caputo fractional derivatives are employed. An example illustrates our approach.

Optimality Conditions for the Guaranteed Result in Time-Delay Control Problems⋆

AbstractControl under disturbances or counteraction is considered for time-delay dynamical systems with Lipschitz data. The value of the optimal guaranteed result of control is defined within the game-theoretical approach of N.N. Krasovskii and A.I. Subbotin as a nonanticipating functional of the motion realization. An appropriate Lipschitz continuity of this functional is proved, and its nonlocal and infinitesimal characterization including the Hamilton–Jacobi formalism is elaborated.

THE HAMILTON–JACOBI FORMALISM FOR HIGHER-ORDER FIELD THEORIES

We extend the geometric Hamilton–Jacobi formalism for Hamiltonian mechanics to higher-order field theories with regular Lagrangian density. We also investigate the dependence of the formalism on the Lagrangian density in the class of those yielding the same Euler–Lagrange equations.

Quantum interference within the complex quantum Hamilton–Jacobi formalism

Quantum interference is investigated within the complex quantum Hamilton–Jacobi formalism. As shown in a previous work [Phys. Rev. Lett. 102 (2009) 250401], complex quantum trajectories display helical wrapping around stagnation tubes and hyperbolic deflection near vortical tubes, these structures being prominent features of quantum caves in space–time Argand plots. Here, we further analyze the divergence and vorticity of the quantum momentum function along streamlines near poles, showing the intricacy of the complex dynamics. Nevertheless, despite this behavior, we show that the appearance of the well-known interference features (on the real axis) can be easily understood in terms of the rotation of the nodal line in the complex plane. This offers a unified description of interference as well as an elegant and practical method to compute the lifetime for interference features, defined in terms of the average wrapping time, i.e., considering such features as a resonant process.

Geometric phase within the complex quantum Hamilton–Jacobi formalism

We derive a geometric phase using the quantum kinematic approach within the complex quantum Hamilton–Jacobi formalism. The single valuedness of the wave function implies that the geometric phase along an arbitrary path in the complex plane must be equal to an integer multiple of 2 π. The nonzero geometric phase indicates that we travel along the path through the branch cut of the phase function from one Riemann sheet to another.

First order description of [formula omitted] static black holes and the Hamilton–Jacobi equation

In this Note we discuss the application of the Hamilton–Jacobi formalism to the first order description of four-dimensional spherically symmetric and static black holes. In particular we show that the prepotential characterizing the flow coincides with the Hamilton principal function associated with the one-dimensional effective Lagrangian. This implies that the prepotential can always be defined, at least locally in the radial variable and in the moduli space, both in the extremal and non-extremal cases and allows us to conclude that it is duality invariant. We also give, in this framework, a general definition of the “Weinhold metric” in terms of which a necessary condition for the existence of multiple attractors is given. The Hamilton–Jacobi formalism can be applied both to the restricted phase space where the electromagnetic potentials have been integrated out as well as in the case where the electromagnetic potentials are dualized to scalar fields using the so-called three-dimensional Euclidean approach. We give some examples of application of the formalism, both for the BPS and the non-BPS black holes.

Hamilton–Jacobi and quantum theory formulations of thermal-wave propagation under the dual-phase lagging model of heat conduction

Dual-phase lagging model is one of the most promising approaches to generalize the Fourier heat conduction equation, and it can be reduced in the appropriate limits to the hyperbolic Cattaneo–Vernotte and to the parabolic equations. In this paper it is shown that the Hamilton–Jacobi and quantum theory formulations that have been developed to study the thermal-wave propagation in the Fourier framework can be extended to include the more general approach based on dual-phase lagging. It is shown that the problem of solving the heat conduction equation can be treated as a thermal harmonic oscillator. In the classical approach a formulation in canonical variables is presented. This formalism is used to introduce a quantum mechanical approach from which the expectation values of observables such as the temperature and heat flux are obtained. These formalisms permit to use a methodology that could provide a deeper insight into the phenomena of heat transport at different time scales in media with inhomogeneous thermophysical properties.

Left–right asymmetric holographic RG flow with gravitational Chern–Simons term

We consider the holographic renormalization group (RG) flow in three-dimensional gravity with the gravitational Chern–Simons term coupled to some scalar fields. We apply the canonical approach to this higher derivative case and employ the Hamilton–Jacobi formalism to analyze the flow equations of two-dimensional field theory. Especially we obtain flow equations of Weyl and gravitational anomalies, and derive c-functions for left and right moving modes. Both of them are monotonically non-increasing along the flow, and the difference between them is determined by the coefficient of the gravitational Chern–Simons term. This is completely consistent with the Zamolodchikov's c-theorem for parity-violating two-dimensional quantum field theories.

DISCUSSION ON SOME CHARACTERISTICS OF CHARGED BRANE-WORLD BLACK HOLES

Several physical natures of charged brane-world black holes are investigated. Firstly, the timelike and null geodesics of the charged brane-world black holes are presented. We also analyze all the possible motions by plotting the effective potentials for various parameters for circular and radial geodesics. Secondly, we investigate the motion of test particles in the gravitational field of the charged brane-world black holes using the Hamilton–Jacobi formalism. We consider charged and uncharged test particles and examine their behavior in both static and nonstatic cases. Thirdly, the thermodynamics of the charged brane-world black holes are studied. Finally, it is shown that there is no phenomenon of superradiance for an incident massless scalar field for such a black hole.

A variational approach to multi-phase motion of gas, liquid and solid based on the level set method

We propose a simple and robust numerical algorithm to deal with multi-phase motion of gas, liquid and solid based on the level set method [S. Osher, J.A. Sethian, Front propagating with curvature-dependent speed: Algorithms based on Hamilton–Jacobi formulation, J. Comput. Phys. 79 (1988) 12; M. Sussman, P. Smereka, S. Osher, A level set approach for capturing solution to incompressible two-phase flow, J. Comput. Phys. 114 (1994) 146; J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999; S. Osher, R. Fedkiw, Level Set Methods and Dynamics Implicit Surface, Applied Mathematical Sciences, vol. 153, Springer, 2003]. In Eulerian framework, to simulate interaction between a moving solid object and an interfacial flow, we need to define at least two functions (level set functions) to distinguish three materials. In such simulations, in general two functions overlap and/or disagree due to numerical errors such as numerical diffusion. In this paper, we resolved the problem using the idea of the active contour model [M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision 1 (1988) 321; V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, International Journal of Computer Vision 22 (1997) 61; G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001; R. Kimmel, Numerical Geometry of Images: Theory, Algorithms, and Applications, Springer-Verlag, 2003] introduced in the field of image processing.

ON THE CLASSICAL CONFINEMENT OF TEST PARTICLES TO A THIN 3-BRANE IN THE ABSENCE OF NON-GRAVITATIONAL FORCES

The classical confinement condition of test particles to a brane universe in the absence of non-gravitational forces is transformed using the Hamilton–Jacobi formalism. The transformed condition provides a direct criterion for selecting in a cosmological scenario 5D bulk manifolds wherein it is possible to obtain confinement of trajectories to 4D hypersurfaces purely due to classical gravitational effects.

Quantum streamlines within the complex quantum Hamilton–Jacobi formalism

Quantum streamlines are investigated in the framework of the quantum Hamilton-Jacobi formalism. The local structures of the quantum momentum function (QMF) and the Polya vector field near a stagnation point or a pole are analyzed. Streamlines near a stagnation point of the QMF may spiral into or away from it, or they may become circles centered on this point or straight lines. Additionally, streamlines near a pole display east-west and north-south opening hyperbolic structure. On the other hand, streamlines near a stagnation point of the Polya vector field for the QMF display general hyperbolic structure, and streamlines near a pole become circles enclosing the pole. Furthermore, the local structures of the QMF and the Polya vector field around a stagnation point are related to the first derivative of the QMF; however, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Two nonstationary states constructed from the eigenstates of the harmonic oscillator are used to illustrate the local structures of these two fields and the dynamics of the streamlines near a stagnation point or a pole. This study presents the abundant dynamics of the streamlines in the complex space for one-dimensional time-dependent problems.

Non-involutive constrained systems and Hamilton–Jacobi formalism

In this work we discuss the natural appearance of the Generalized Brackets in systems with non-involutive (equivalent to second class) constraints in the Hamilton–Jacobi formalism. We show how a consistent geometric interpretation of the integrability conditions leads to the reduction of degrees of freedom of these systems and, as consequence, naturally defines a dynamics in a reduced phase space.

Solution for a Free Particle in a Circular Sector Billiard

We study a particle moving inside a circular sector billiard. This system has two integrals of motion: the kinetic energy and the magnitude of the angular momentum, therefore it is an integrable system. We show the foliation of the phase space and we use the Hamilton–Jacobi formalism to find any periodic or quasiperiodic orbit with a given winding number. The periodic orbits are also classified according to the properties of an elementary sector and whether they are in one direction or in two directions retracing back themselves

Quantum vortices within the complex quantum Hamilton–Jacobi formalism

Quantum vortices are investigated in the framework of the quantum Hamilton-Jacobi formalism. A quantum vortex forms around a node in the wave function in the complex space, and the quantized circulation integral originates from the discontinuity in the real part of the complex action. Although the quantum momentum field displays hyperbolic flow around a node, the corresponding Polya vector field displays circular flow. It is shown that the Polya vector field of the quantum momentum function is parallel to contours of the probability density. A nonstationary state constructed from eigenstates of the harmonic oscillator is used to illustrate the formation of a transient excited state quantum vortex, and the coupled harmonic oscillator is used to illustrate quantization of the circulation integral in the multidimensional complex space. This study not only analyzes the formation of quantum vortices but also demonstrates the local structures for the quantum momentum field and for the Polya vector field near a node of the wave function.

Interplay of causticity and vorticality within the complex quantum Hamilton–Jacobi formalism

Interference dynamics is analyzed in the light of the complex quantum Hamilton–Jacobi formalism, using as a working model the collision of two Gaussian wave packets. Though simple, this model nicely shows that interference in quantum scattering processes gives rise to rich dynamics and trajectory topologies in the complex plane, both ruled by two types of singularities: caustics and vortices, where the former are associated with the regime of free wave-packet propagation, and the latter with the collision (interference) process. Furthermore, an unambiguous picture connecting the complex and real frameworks is also provided and discussed.