Quantization of fractional singular Lagrangian systems using WKB approximation

2018 ◽  
Vol 33 (36) ◽  
pp. 1850222 ◽  
Author(s):  
Eqab M. Rabei ◽  
Mohammed Al Horani
Keyword(s):  
Wave Function ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Singular Systems ◽  
Equations Of Motion ◽  
Singular System ◽  
Wkb Approximation ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Action Integral

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

2010 ◽  
Vol 25 (02) ◽  
pp. 135-141
Author(s):  
H. A. ELEGLA ◽  
N. I. FARAHAT
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Path Integral ◽  
Equations Of Motion ◽  
Singular System ◽  
Constrained Systems ◽  
Classical Structure ◽  
Path Integral Quantization ◽  
Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

INTEGRABILITY AND ACTION FUNCTION IN MULTI-HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (11) ◽  
pp. 863-870 ◽  
Author(s):  
S. I. MUSLIH
Keyword(s):  
Differential Equations ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Integral Function ◽  
Jacobi Method ◽  
Action Function ◽  
Action Integral ◽  
Method Integration ◽  
Total Differential

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.

QUANTIZATION OF SECOND-ORDER CONSTRAINED LAGRANGIAN SYSTEMS USING THE WKB APPROXIMATION

2005 ◽  
Vol 02 (03) ◽  
pp. 485-504 ◽  
Author(s):  
EQAB M. RABEI ◽  
EYAD H. HASSAN ◽  
HUMAM B. GHASSIB ◽  
S. MUSLIH
Keyword(s):  
Wave Function ◽  
General Theory ◽  
Semiclassical Limit ◽  
Second Order ◽  
Constrained Systems ◽  
Wkb Approximation ◽  
Lagrangian Systems

A general theory is given for quantizing both constrained and unconstrained systems with second-order Lagrangian, using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit. This is illustrated with two examples.

scholarly journals Vibration Suppression Control of a Flexible Gantry Crane System with Varying Rope Length

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Tung Lam Nguyen ◽  
Trong Hieu Do ◽  
Hong Quang Nguyen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Vibration Suppression ◽  
Direct Method ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Partial Differential ◽  
Control Approach ◽  
Lyapunov’S Direct Method

The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

XIV.—On General Dynamics:—I. Hamilton's Partial Differential Equations and the Determination of their Complete Integrals

1913 ◽  
Vol 32 ◽  
pp. 164-174
Author(s):  
A. Gray
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Equations Of Motion ◽  
Partial Differential ◽  
General Dynamics ◽  
Elementary Manner ◽  
Derivatives Of

The present paper contains the first part of a series of notes on general dynamics which, if it is found worth while, may be continued. In § 1 I have shown how the first Hamiltonian differential equation is led up to in a natural and elementary manner from the canonical equations of motion for the most general case, that in which the time t appears explicitly in the function usually denoted by H. The condition of constancy of energy is therefore not assumed. In § 2 it is proved that the partial derivatives of the complete integral of Hamilton's equation with respect to the constants which enter into the specification of that integral do not vary with the time, so that these derivatives equated to constants are the integral equations of motion of the system.*

Hamilton–Jacobi Treatment of Lagrangians with Linear Velocities

2003 ◽  
Vol 18 (23) ◽  
pp. 1591-1596 ◽  
Author(s):  
Eqab M. Rabei ◽  
Khaled I. Nawafleh ◽  
Yacoub S. Abdelrahman ◽  
H. Y. Rashed Omari
Keyword(s):  
Equations Of Motion ◽  
Lagrangian Systems ◽  
New Approach ◽  
Consistency Conditions ◽  
Action Integral ◽  
Jacobi Formulation

A new approach for solving mechanical problems of Linear Lagrangian systems using the Hamilton–Jacobi formulation is proposed. The equations of motion are recovered from the action integral. It has been proved that there is no need to follow the consistency conditions of the Dirac approach.

scholarly journals ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS

1995 ◽  
Vol 10 (04) ◽  
pp. 579-610 ◽  
Author(s):  
V. MUKHANOV ◽  
A. WIPF
Keyword(s):  
String Theory ◽  
Hamiltonian Systems ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Linear Constraints ◽  
Lagrangian System ◽  
Lagrangian Systems ◽  
Diffeomorphism Invariance ◽  
Symmetry Transformations ◽  
Local Symmetries

In this paper we show how the well-known local symmetries of Lagrangian systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangian system. The non-linear constraints (which we have, for instance, in gravity, supergravity and string theory) generate the dynamics of the corresponding Lagrangian system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We show the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems, in particular those which are diffeomorphism-invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian and Lagrangian formalisms is found. The possible applications of our results are discussed.

On the Parametrically Excited Response of the Quadratically Nonlinear Model of a Rotating Ring on an Elastic Hub

1993 ◽  
Author(s):  
Rick I. Zadoks ◽  
Charles M. Krousgrill
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Normal Modes ◽  
Point Load ◽  
Tangential Displacement ◽  
Partial Differential

Abstract As a first approximation, a steel-belted radial tire can be modeled as a one dimensional rotating ring connected elastically to a moving hub. This ring can be modeled mathematically using a set of three nonlinear partial differential equations, where the three degrees of freedom are a radial displacement, a tangential displacement and a section rotation. In this study, only quadratic geometric nonlinearities are considered. The system is excited by a temporally harmonic point load f^(t) and a temporally harmonic hub motion z^(t) that have the same harmonic frequency. The point load f^(t) appears in the equations of motion as a single in-homogeneous term, while the hub motion z^(t) appears in inhomogeneous and parametric excitation terms. To simplifying the ensuing analysis, the rotation rate of the hub is assumed to be constant. The partial differential equations of motion are reduced to a set of four second-order ordinary differential equations by using two linear normal modes to approximate the spatial distribution of the displacements. A region of the parameter space, as defined by ranges of values of the excitation amplitude z and the excitation frequency ω (or detuning parameter σ), is identified, from a Strutt diagram, where the parametric excitation is expected to be dominant. In this region σ is varied to locate a secondary Hopf bifurcation that leads to a set of complex steady-state quasi-periodic solutions. These solutions contain two families of frequency components where the fundamental frequencies of these families are non-commensurate, and they are characterized by Poincaré sections with closed or nearly closed “orbits” as opposed to the distinct points displayed by periodic responses and the strange attractor sections displayed by chaotic solutions.

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