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.

HAMILTON–JACOBI QUANTIZATION OF TWO-DIMENSIONAL GRAVITY WITH TORSION

2004 ◽  
Vol 19 (02) ◽  
pp. 151-157 ◽  
Author(s):  
SAMI I. MUSLIH
Keyword(s):  
Phase Space ◽  
Differential Equations ◽  
Path Integral ◽  
Equations Of Motion ◽  
Two Dimensional ◽  
Path Integral Quantization ◽  
Action Integral ◽  
Total Differential ◽  
Jacobi Formalism ◽  
Dimensional Gravity

Two-dimensional gravity with torsion is investigated using the Hamilton–Jacobi formalism. The equations of motion and the action integral are obtained as total differential equations in many variables. The integrabilty conditions, lead us to obtain the path integral quantization as an integration over the canonical phase-space coordinates.

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.

THE CANONICAL PATH INTEGRAL QUANTIZATION OF CHIRAL SCHWINGER MODEL

2003 ◽  
Vol 18 (17) ◽  
pp. 1187-1196 ◽  
Author(s):  
S. I. MUSLIH
Keyword(s):  
Differential Equations ◽  
Path Integral ◽  
Integral Method ◽  
Equation Of Motion ◽  
Action Function ◽  
Schwinger Model ◽  
Integrability Conditions ◽  
Total Differential ◽  
Jacobi Formalism ◽  
Path Integral Method

We quantize the chiral Schwinger model by using the Hamilton–Jacobi formalism. 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.

FRACTIONAL TIME ACTION AND PERTURBED GRAVITY

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 243-247 ◽  
Author(s):  
MADHAT SADALLAH ◽  
SAMI I. MUSLIH ◽  
DUMITRU BALEANU ◽  
EQAB RABEI
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variational Problems ◽  
Integral Function ◽  
Lagrange Equations ◽  
Action Integral

In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < α ≤ 1. Two examples are investigated in detail.

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.

Dynamics of Rolling-Element Bearings—Part I: Cylindrical Roller Bearing Analysis

1979 ◽  
Vol 101 (3) ◽  
pp. 293-302 ◽  
Author(s):  
P. K. Gupta
Keyword(s):  
Differential Equations ◽  
Normal Force ◽  
Equations Of Motion ◽  
Roller Bearing ◽  
Rolling Element Bearings ◽  
Analytical Formulation ◽  
Rolling Element ◽  
Cylindrical Roller Bearing ◽  
Cylindrical Roller ◽  
Bearing Analysis

An analytical formulation for the roller motion in a cylindrical roller bearing is presented in terms of the classical differential equations of motion. Roller-race interaction is analyzed in detail and the resulting normal force and moment vectors are determined. Elastohydrodynamic traction models are considered in determining the roller-race tractive forces and moments. Formulation for the roller end and race flange interaction during skewing of the roller is also considered. Roller-cage interactions are assumed to be either hydrodynamic or fully metallic. Simple relationships are used to determine the churning and drag losses.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

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