Hamilton-Jacobi Method Research Articles

On the Hamilton–Jacobi Treatment of Constrained Systems

Systems with singular-higher order Lagrangians are investigated by two methods: Dirac method and Hamilton–Jacobi method. An example is studied and it is shown that the Hamilton–Jacobi method gives the correct canonical generalized equations of motion, contrary to Dirac method, where Dirac conjecture is invalid.

Hamilton-Jacobi method and D-brane actions in supergravity

Hamilton-Jacobi method and D-brane actions in supergravity

Investigation of Constrained Systems with Singular Higher-Order Lagrangians

Constrained Hamiltonian systems with singular higher-order Lagrangians are investigated by using two methods: the Dirac's and the Hamilton-Jacobi methods. Three examples are studied and it is shown that the equations of motion which are obtained by these two methods are in exact agreement.

On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems

Introduction and some salient features of the model Hamiltonian Symplectic geometry and the splitting of invariant manifolds Estimating the splitting matrix using normal forms The Hamilton-Jacobi method for a simple resonance Appendix. Invariant tori with vanishing or zero torsion Bibliography.

Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

A systematic construction of Stäckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton–Jacobi method, is derived. One-Casimir bi-Hamiltonian case is studied in details and in this case, a systematic construction of related hydrodynamic systems in arbitrary coordinates is presented, using the cofactor method and soliton symmetry constraints.

The correlation time of mesoscopic chemical clocks

A formula is proved for the correlation time of nonequilibrium chemical clocks in the presence of molecular noise. The correlation time is defined as the inverse of the damping rate of the autocorrelation functions of the chemical concentrations. Using the Hamilton–Jacobi method for stochastic systems as well as a Legendre transform from the Onsager–Machlup action to a reduced action depending only on the Hamilton–Jacobi pseudoenergy, the correlation time is given in the weak-noise limit in terms of the extensivity parameter, the period of oscillations, as well as the derivative of the period with respect to the pseudoenergy. Using this result, an estimation is obtained for the minimum number of molecules required for the oscillations of the chemical concentrations to remain correlated in time. This estimation puts a fundamental lower limit on the size of chemical clocks. For typical oscillators, the minimum number of molecules is estimated between ten and one hundred, which essentially corresponds to nanometric systems.

Long-wavelength cosmological perturbation in the Universe with multiple perfect fluids

We investigate large-scale cosmological perturbation in the Universe with multiple perfect fluids. Using the long-wavelength approximation with the Hamilton–Jacobi method, we derive a formula for the gauge-invariant comoving curvature perturbation. As an application of our approach, we examine large-scale perturbation in a brane cosmology.

Quantization of the O(N) Nonlinear Sigma Model

The Hamilton–Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, one can obtain the canonical phase space coordinates and set of canonical Hamilton–Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the O(2) nonlinear sigma model using two different approaches: the functional Schrödinger method to obtain the wave functionals for the ground and the exited state and then we quantize the same model using the canonical path integral quantization as an integration over the canonical phase-space coordinates.

The Hamilton-Jacobi method and Hamiltonian maps

A method for constructing time-step-based symplectic maps for a generic Hamiltonian system subjected to perturbation is developed. Using the Hamilton-Jacobi method and Jacobi's theorem in finite periodic time intervals, the general form of the symplectic maps is established. The generating function of the map is found by the perturbation method in the finite time intervals. The accuracy of the maps is studied for fully integrable and partially chaotic Hamiltonian systems and compared to that of the symplectic integration method.

Control Systems Theory with Engineering Applications

3R22. Control Systems Theory with Engineering Applications. - SE Lyshevski (Dept of Elec and Comput Eng, Purdue Univ, Indianapolis IN 46202-5132). Birkhauser Boston, Cambridge MA. 2001. 416 pp. CD-ROM included. ISBN 0-8176-4203-X. $79.95.Reviewed by PJ Eagle (Exp and Comput Mech, DaimlerChrysler Corp, 800 Chrysler Dr, Auburn Hill MI 48326-2757).This book is a monograph devoted to methods for analyzing linear and nonlinear multivariate control system problems. The stated aim of the text is to be “used to teach undergraduate and graduate classes in automatic control at electrical, mechanical and aerospace engineering departments.” While this text has several problems achieving this aim (as described below), it is a unique resource that should not be overlooked as an addition to a library of books on control systems. The book is geared toward using the popular MATLAB® and SIMULINK™ software as tools for analysis and data presentation. The book is not at a very fundamental level and does not contain practice problems. As such, it would not be a good choice as an undergraduate text. However, it is a resource of a large quantity of solved example problems (including a CD-ROM of MATLAB/SIMULINK source code) related to multivariate control and nonlinear systems that is not found in many similar texts. For this reason, this book would be a valuable resource for upper-level graduate students pursuing research topics in control systems. The book consists of five main sections divided into numerous subsections. There is no list of nomenclature and symbols, but the notation is consistent with those used in standard control system practice. There is a brief index as well as a list of references which is simply a list of publications that are not necessarily related to the content of the book. The reference list is divided into topical sections. For reasons that are not clear, the list of “Papers in Control” is largely populated by the author’s publications. The book was well reviewed for typesetting irregularities and typographic errors. There is an abundance of figures, most of which are graphs and other screen output from MATLAB. The line art does not appear to be professionally executed, but it is clear and consistent.The book contains an introduction to MATLAB that begins as being overly simple, but covers a wide range of the fundamentals necessary to apply this tool to solving controls problems. The level of detail is useful for readers who are teaching themselves MATLAB for use in these types of applications. The coverage of MATLAB examples is so extensive that some diagrams and output figures are too complex for a book of this type. Nonetheless, the author has taken great care to painstakingly document the examples in text and figure. Some editorial oversight is the coverage of state space theory after MATLAB has already been applied to some state space analysis. One of the best features of this book is its clear presentation of Hamilton-Jacobi and Lyapunov methods coupled with numerous examples, complete with simulation data and MATLAB files on CD-ROM. Graduate students and researchers in optimal control will benefit from this level of detail. It should be noted that while these examples are detailed enough to run simulations and verify output, they expect the reader to be familiar with a fairly high level of domain-specific knowledge (such as aircraft dynamics). Some examples are flush with irrelevant details like how long the simulation took to run using a particular computer. Other examples suffer from poor practices in significant figures and the lack of appropriate diagrams (a complex dynamic model of a robot is presented with no supporting diagram). In contrast, there is a well-documented and laboriously complete example on induction motor control that would be invaluable to students struggling with the practices in nonlinear control. Even with the above criticisms, this book would be an excellent starting point for a graduate student pursuing problems in discrete-time systems with optimal control. In summary, this reviewer would definitely recommend Control Systems Theory with Engineering Applications to students or libraries seeking control system and MATLAB/SIMULINK reference texts. This is a very complete and practical resource. It is not a teaching textbook, but it would be an outstanding supplement to an advanced graduate course in nonlinear control.

Holographic cosmic quintessence on the dilatonic brane world

Recently quintessence has been proposed to explain the observation data of supernovae indicating a time-varying cosmological constant and accelerating universe. Inspired by this and its mysterious origin, we look for the possibility that quintessence is the holographic dark matter dominating in the late time in the brane world scenarios. We consider both the cases of a static and moving brane in a dilaton gravity background. For the static brane we use the Hamilton-Jacobi method motivated by holographic renormalization group to study the intrinsic FRW cosmology on the brane and find the constraint on the bulk potential for quintessence. This constraint requires a negative slowly varying bulk potential which implies an anti--de Sitter--like bulk geometry and could be possibly realized from higher dimensional supergravities or string theory. We find a similar constraint for the moving brane cases and that the quintessence on it has the effect of a mildly time-varying Newton constant.

The Canonical Path Integral Quantization of CP1 Model

The Hamilton-Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, then one can obtain the canonical phase space coordinates and the set of the canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the CP1 model using the canonical path integral quantization formalism to obtain the path integral as an integration over the canonical phase-space coordinates.

A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

We propose a geometrical approach to the problem of integrability of Hamiltonian systems of low dimensions using the Hamilton–Jacobi method of separation of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems defined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found.

HAMILTON–JACOBI SOLUTION TO SOLITON PATHS AND TRIANGULAR MASS RELATION IN TWO-DIMENSIONAL EXTENDED SUPERSYMMETRIC THEORY

D=2, N=2 generalized Wess–Zumino theory is investigated by the dimensional reduction from D=4, N=1 theory. For each solitonic configuration (i,j), the classical static solution is solved by the Hamilton–Jacobi method of equivalent one-dimensional classical mechanics. It is easily shown that the Bogomol'nyi mass bound is saturated by these solutions and triangular mass inequality [Formula: see text] is automatically satisfied.

Bihamiltonian Geometry and Separation of Variables for Toda Lattices

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifold, we show that these systems can be explicitly integrated via the classical Hamilton–Jacobi method in the so-called Darboux–Nijenhuis coordinates.

Separability of the Toda Lattice

We study integrability of the nonperiodic, finite-dimensional Toda Lattice (TL) from the point of view of the Hamilton-Jacobi (HJ) theory of separation of variables. Known to be completely integrable in the sense of Arnol'd-Liouville, the system nonetheless cannot be integrated by the HJ method in the ‘original’ generalized physical position-momenta coordinates. The intrinsic (coordinate-free) characterization method by Benenti provides a tool to investigate the separability of the TL system independently of a local system of coordinates.

Hamilton–Jacobi method for classical mechanics in Grassmann algebra

Hamilton–Jacobi method for classical mechanics in Grassmann algebra

Phase space structure for matched intense charged-particle beams in periodic focusing transport systems

Test particle motion is analyzed analytically and numerically in the field configuration consisting of the equilibrium self-electric and self-magnetic fields of a well-matched, thin, continuous, intense charged-particle beam and an applied periodic focusing solenoidal magnetic field. The self fields are determined self-consistently, assuming the beam to have a uniform-density, rigid-rotor Vlasov equilibrium distribution. Using the Hamilton–Jacobi method, the betatron oscillations of test particles in the average self fields and applied focusing field are analyzed, and the nonlinear resonances induced by periodic modulations in the self fields and applied field are determined. The Poincaré surface-of-section method is used to analyze numerically the phase-space structure for test particle motion outside the outermost envelope of the beam over a wide range of system parameters. For vacuum phase advance σv=80°, it is found that the phase-space structure is almost entirely regular at low beam intensity (phase advance σ≳70°, say), whereas at moderate beam intensity (30°≲σ≲70°), nonlinear resonances appear, the most pronounced of which is the third-order primary nonlinear resonance. As the beam intensity is further increased (σ≲30°), the widths of the higher-order nonlinear resonances increase, and the chaotic region of phase space increases in size. Furthermore, the many chaotic layers associated with the separatrices of the primary and secondary nonlinear resonances are still divided by the remaining invariant Kolmogorov–Arnold–Moser surfaces, even at very high beam intensities. The implications of the rich nonlinear resonance structure and chaotic particle motion found in the present test-particle studies are discussed in the context of halo formation.

Canonical quasilocal energy and small spheres

Consider the definition E of quasilocal energy stemming from the Hamilton-Jacobi method as applied to the canonical form of the gravitational action. We examine E in the standard "small-sphere limit," first considered by Horowitz and Schmidt in their examination of Hawking's quasilocal mass. By the term "small sphere" we mean a cut S(r), level in an affine radius r, of the lightcone belonging to a generic spacetime point. As a power series in r, we compute the energy E of the gravitational and matter fields on a spacelike hypersurface spanning S(r). Much of our analysis concerns conceptual and technical issues associated with assigning the zero-point of the energy. For the small-sphere limit, we argue that the correct zero-point is obtained via a "lightcone reference," which stems from a certain isometric embedding of S(r) into a genuine lightcone of Minkowski spacetime. Choosing this zero-point, we find agreement with Hawking's quasilocal mass expression, up to and including the first non-trivial order in the affine radius. The vacuum limit relates the quasilocal energy directly to the Bel-Robinson tensor.

Moving observers, nonorthogonal boundaries, and quasilocal energy

The popular Hamilton-Jacobi method first proposed by Brown and York for defining quasilocal quantities such as energy for spatially bound regions assumes that the spatial boundary is orthogonal to the foliation of the spacetime. Such a restriction is undesirable for both theoretical and computational reasons. We remove the orthogonality assumption and show that it is more natural to focus on the foliation of the spatial boundary rather than the foliation of the entire spatially bound region. Reference spacetimes which define additional terms in the action are discussed in detail. To demonstrate this new formulation, we calculate the quasilocal energies seen by observers who are moving with respect to a Schwarzschild black hole.