Hamilton-Jacobi Method Research Articles

Cosmological perturbations in FRW model with scalar field within Hamilton-Jacobi formalism and symplectic projector method

Abstract The Hamilton-Jacobi analysis is applied to the dynamics of the scalar fluctuations about the Friedmann-Robertson-Walker (FRW) metric. The gauge conditions are determined from the consistency conditions. The physical degrees of freedom of the model are obtained by the symplectic projector method. The role of the linearly dependent Hamiltonians and the gauge variables in the Hamilton-Jacobi formalism is discussed.

Tachyon matter in loop quantum cosmology

An analytical approach for studying the cosmological scenario with a homogeneous tachyon field within the framework of loop quantum gravity is developed. Our study is based on the semi-classical regime where space time can be approximated as a continuous manifold, but matter Hamiltonian gets non-perturbative quantum corrections. A formal correspondence between classical and loop quantum cosmology is also established. The Hamilton-Jacobi method for getting exact solutions is constructed and some exact power-law as well as bouncing solutions are presented.

Hamilton–Jacobi method for solving ordinary differential equations

The Hamilton?Jacobi method for solving ordinary differential equations is presented in this paper. A system of ordinary differential equations of first order or second order can be expressed as a Hamilton system under certain conditions. Then the Hamilton?Jacobi method is used in the integration of the Hamilton system and the solution of the original ordinary differential equations can be found. Finally, an example is given to illustrate the application of the result.

Two-dimensional supersymmetry: From SUSY quantum mechanics to integrable classical models

Two known two-dimensional SUSY quantum mechanical constructions—the direct generalization of SUSY with first-order supercharges and higher-order SUSY with second-order supercharges—are combined for a class of 2-dim quantum models, which are not amenable to separation of variables. The appropriate classical limit of quantum systems allows us to construct SUSY-extensions of original classical scalar Hamiltonians. Special emphasis is placed on the symmetry properties of the models thus obtained—the explicit expressions of quantum symmetry operators and of classical integrals of motion are given for all (scalar and matrix) components of SUSY-extensions. Using Grassmanian variables, the symmetry operators and classical integrals of motion are written in a unique form for the whole Superhamiltonian. The links of the approach to the classical Hamilton–Jacobi method for related “flipped” potentials are established.

Correcting Curvature-Density Effects in the Hamilton–Jacobi Skeleton

The Hamilton-Jacobi approach has proven to be a powerful and elegant method for extracting the skeleton of two-dimensional (2-D) shapes. The approach is based on the observation that the normalized flux associated with the inward evolution of the object boundary at nonskeletal points tends to zero as the size of the integration area tends to zero, while the flux is negative at the locations of skeletal points. Nonetheless, the error in calculating the flux on the image lattice is both limited by the pixel resolution and also proportional to the curvature of the boundary evolution front and, hence, unbounded near endpoints. This makes the exact location of endpoints difficult and renders the performance of the skeleton extraction algorithm dependent on a threshold parameter. This problem can be overcome by using interpolation techniques to calculate the flux with subpixel precision. However, here, we develop a method for 2-D skeleton extraction that circumvents the problem by eliminating the curvature contribution to the error. This is done by taking into account variations of density due to boundary curvature. This yields a skeletonization algorithm that gives both better localization and less susceptibility to boundary noise and parameter choice than the Hamilton-Jacobi method.

Integrating the geodesic equations in the Schwarzschild and Kerr space-times using Beltrami's “geometrical” method

We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.

Boundary counterterms and the thermodynamics of 2-D black holes

We utilize a novel method to study the thermodynamics of two dimensional type 0A black holes with constant RR flux. Our approach is based on the Hamilton-Jacobi method of deriving boundary counterterms. We demonstrate this approach by recovering the standard results for a well understood example, Witten's black hole. Between this example and the 0A black hole we find universal expressions for the entropy and black hole mass, as well as the infra-red divergence of the partition function. As a non-trivial check of our results we verify the first law of thermodynamics for these systems. Our results for the mass disagree with the predictions of a proposed matrix model dual of the 0A black hole.

Hamilton–Jacobi Quantization of Singular Lagrangians with Linear Velocities

In this paper, constrained Hamiltonian systems with linear velocities are investigated by using the Hamilton–Jacobi method. The integrablity conditions are considered on the equations of motion and the action function as well in order to obtain the path integral quantization of singular Lagrangians with linear velocities.

Quantization of constrained systems: A comparative look at three methods

Constrained Hamiltonian systems are investigated using three different methods: Hamilton-Jacobi, Batalin-Fradkin-Tyutin and gauge unfixing methods. The abelian Proca model is analyzed and the involutive Hamiltonian is obtained by the three methods.

Front propagation in reaction-dispersal models with finite jump speed

The continuous-time random walk and the Hamilton-Jacobi method are used to reach analytical expressions for the speed of traveling fronts in reaction-dispersal models. In this work the waiting time and jump length are assumed to be coupled random variables. The jump length for any jump is selected according to the waiting time at the end of the previous jump, and in consequence jumps of finite speed are performed. We study the effect of finite jump speed of the particles on the speed of the traveling fronts and find that in the parabolic and hyperbolic limits it can exceed the jump speed of the particles. We report analytical expressions for different probability distribution functions. Finally, we introduce the possibility that several particle speeds are allowed, so different dispersal mechanisms can be considered simultaneously.

INTEGRABILITY AND ACTION FUNCTION IN MULTI-HAMILTONIAN SYSTEMS

Multi-Hamiltonian systems are investigated by using the Hamilton–Jacobi method. Integration of a set of total differential equations which includes the equations of motion and the action integral function is discussed. It is shown that this set is integrable if and only if the total variations of the Hamiltonians vanish. Two examples are studied.

The quantization of the Chiral Schwinger model based on the Hamilton-Jacobi formalism

Abstract We apply the Hamilton-Jacobi method to the Chiral Schwinger model in the case of regularization ambiguity a = 1. We show that one can obtain the integrable set of equation of motion and the action function by using the integrability conditions of total differential equations and without any need to introduce unphysical auxiliary fields. The path integral for this model is obtained by using the canonical path integral method.

Hamilton–Jacobi Quantization of Constrained Systems with Second-Order Lagrangians

We investigate the path-integral quantization of constrained systems with second-order Lagrangians using the Hamilton-Jacobi method. The path-integral quantization for two models is obtained using the canonical path-integral method.

Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach

In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton–Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalize the approach to the N-body case. We reexamine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N−1)-dimensional submanifold of the 12(N−1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed “by hand” to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry instilled into the equations of celestial mechanics. Existence of this internal symmetry has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system noncanonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a “generalized Lagrange gauge” wherein the Delaunay system is symplectic. This special gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity independent.

Canonical phase space formulation of quasi-local general relativity

We construct a Hamiltonian formulation of quasi-local general relativity using an extended phase space that includes boundary coordinates as configuration variables. This allows us to use Hamiltonian methods to derive an expression for the energy of a non-isolated region of spacetime that interacts with its neighbourhood. This expression is found to be very similar to the Brown–York quasi-local energy that was originally derived by Hamilton–Jacobi methods. We examine the connection between the two formalisms and find that when the boundary conditions for the two are harmonized, the resulting quasi-local energies are identical.

Hamilton–Jacobi method and effective actions of D-brane and M-brane in supergravity

We show that the effective actions of D-brane and M-brane are solutions to the Hamilton–Jacobi (H–J) equations in supergravities. This fact means that these effective actions are on-shell actions in supergravities. These solutions to the H–J equations reproduce the supergravity solutions that represent D-branes in a B 2 field, M2 branes and the M2–M5 bound states. The effective actions in these solutions are those of a probe D-brane and a probe M-brane. Our findings can be applied to the study of the gauge/gravity correspondence, especially the holographic renormalization group, and a search for new solutions of supergravity.

Hamilton-Jacobi formalism of interacting fields

Two singular field systems are investigated by the Hamilton-Jacobi method. The total differential equations are obtained for the Proca model. Electromagnetic interactions are studied by this method and it is verified thatinvestigation of interactions by constraints is consistent with the Feynman diagrams and S-matrix.

The Hamilton-Jacobi method for arbitrary rheo-linear dynamical systems

The Hamilton-Jacobi method is briefly summarized and then applied to arbitrary rheo-linear systems with a single degree of freedom. Various means of finding a complete solution of the Hamilton-Jacobi equation and applying the Jacobi theorem to solve the canonical differential equations are discussed. Basic to the procedure is the separation of variables in the Hamilton-Jacobi equation which leads to a Riccati equation which must be solved for the particular rheonomic differential equation. Five different cases of complete solution are illustrated by a simple rheonomic example. As a direct application of the method, a number of canonical systems corresponding to many “named” equations are solved. They are the: Airy, Bessel, Chebyshev, Error function, Euler, Gegenbauer, Hermite, Hypergeometric, Kelvin, Kummer, Laguerre, Legendre, Jacobi, Mathieu, Spherical Bessel, Weber-Hermite and Whittaker equations. Finally, the conservation law is given for a forced and damped oscillator.

Holographic renormalization and Ward identities with the Hamilton–Jacobi method

We illustrate how holographic renormalization is performed using the Hamilton–Jacobi approach.

Illustrating dynamical symmetries in classical mechanics: The Laplace–Runge–Lenz vector revisited

The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetries. We define a conserved dynamical variable α that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable β for the isotropic harmonic oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canonical one-parameter group of transformations generated by α(β). Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to use them as a coordinate-momentum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of dynamics in phase space. The procedure we use is in the spirit of the Hamilton–Jacobi method.