Eqab M. Rabei

POTENTIALS OF ARBITRARY FORCES WITH FRACTIONAL DERIVATIVES

The Laplace transform of fractional integrals and fractional derivatives is used to develop a general formula for determining the potentials of arbitrary forces: conservative and nonconservative in order to introduce dissipative effects (such as friction) into Lagrangian and Hamiltonian mechanics. The results are found to be in exact agreement with Riewes results of special cases. Illustrative examples are given.

Hamilton–Jacobi formulation of systems within Caputo's fractional derivative

A new fractional Hamilton–Jacobi formulation for discrete systems in terms of fractional Caputo derivatives was developed. The fractional action function is obtained and the solutions of the equations of motion are recovered. Two examples are studied in detail.

Hamilton–Jacobi fractional mechanics

Abstract As a continuation of Rabei et al. work [Eqab M. Rabei, Khaled I. Nawafleh, Raed S. Hijjawi, Sami I. Muslih, Dumitru Baleanu, The Hamilton formalism with fractional derivatives, J. Math. Anal. Appl. 327 (2007) 891–897], the Hamilton–Jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives. The Hamilton–Jacobi function in configuration space is obtained in a similar manner to the usual mechanics. Two problems are considered to demonstrate the application of the formalism. The result is found to be in exact agreement with Agrawals formalism.

Heisenberg's Equations of Motion with Fractional Derivatives

Fractional variational principles is a new topic in the field of fractional calculus and it has been subject to intense debate during the last few years. One of the important applications of fractional variational principles is fractional quantization. In this present study, fractional calculus is applied to obtain the Hamiltonian formalism of non-conservative systems. The definition of Poisson bracket is used to obtain the equations of motion in terms of these brackets. The commutation relations and the Heisenberg equations of motion are also obtained. The proposed approach was tested on two examples and good agreements with the classical fractional are reported.

Quantization of Higher-Order Constrained Lagrangian Systems Using the WKB Approximation

A general theory is given for solving the Hamilton–Jacobi partial differential equations (HJPDEs) for both constrained and unconstrained systems with arbitrarily higher-order Lagrangians. The Hamilton–Jacobi function is obtained for both types of systems by solving the appropriate set of HJPDEs. This is used to determine the solutions of the equations of motion. The quantization of both systems is then achieved using the WKB approximation. In constrained systems, the constraints become conditions on the wave function to be satisfied in the semiclassical limit.

Bargmann invariants and geometric phases: A generalized connection

We develop the broadest possible generalization of the well known connection between quantum-mechanical Bargmann invariants and geometric phases. The key concept is that of null phase curves in quantum-mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalization is shown to be essential for properly understanding geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.

Fractional Hamilton’s equations of motion in fractional time

The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.

HAMILTON JACOBI TREATMENT OF CONSTRAINED SYSTEMS

One approach for solving mechanical problems of constrained systems using the Hamilton–Jacobi formulation is examined. The Hamilton–Jacobi function is obtained in the same manner as for regular systems. This is used to determine the solutions of the equations of motion for constrained systems.

Gravitational potential in fractional space

AbstractIn this paper the gravitational potential with β-th order fractional mass distribution was obtained in α dimensionally fractional space. We show that the fractional gravitational universal constant Gα is given by