Fractional Constrained Systems and Caputo Derivatives

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Dumitru Baleanu
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principles ◽  
Constrained Systems ◽  
Control Problems ◽  
Fractional Dynamics ◽  
Constrained System ◽  
Caputo Derivatives ◽  
Fractional Dynamical System ◽  
Made In

During the last few years, remarkable developments have been made in the theory of the fractional variational principles and their applications to control problems and fractional quantization issue. The variational principles have been used in physics to construct the phase space of a fractional dynamical system. Based on the Caputo derivatives, the fractional dynamics of discrete constrained systems is presented and the notion of the reduced phase space is discussed. Two examples of discrete constrained system are analyzed in detail.

On Constrained Systems Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principles ◽  
Constrained Systems ◽  
Control Problems ◽  
Fractional Dynamics ◽  
Caputo Derivatives ◽  
Fractional Dynamical System

The constraints systems play a very important role in physics and engineering. The fractional variational principles were successfully applied to control problems as well as to construct the phase space of a fractional dynamical system. In this paper the fractional dynamics of discrete constrained systems is presented and the notion of the reduced phase-space is analyzed. One system possessing two primary first class constraints is analyzed in detail.

Fractional Mechanics on the Extended Phase Space

2009 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Alireza K. Golmankhaneh ◽  
Ali K. Golmankhaneh ◽  
Eqab M. Rabei
Keyword(s):  
Phase Space ◽  
Variational Principles ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Extended Phase Space ◽  
Science And Engineering ◽  
Extended Phase ◽  
Potential Applications ◽  
Fractional Mechanics ◽  
Complex Phenomena

Fractional calculus has gained a lot of importance and potential applications in several areas of science and engineering. The fractional dynamics and the fractional variational principles started to be used intensively as an alternative tool in order to describe the physical complex phenomena. In this paper we have discussed the fractional extension of the classical dynamics. The fractional Hamiltonian is constructed and the fractional generalized Poisson’s brackets on the extended phase space is established.

scholarly journals POISSON GEOMETRY IN CONSTRAINED SYSTEMS

2003 ◽  
Vol 15 (07) ◽  
pp. 663-703 ◽  
Author(s):  
MARTIN BOJOWALD ◽  
THOMAS STROBL
Keyword(s):  
Phase Space ◽  
Symplectic Manifold ◽  
Sufficient Conditions ◽  
Constrained Systems ◽  
Poisson Manifolds ◽  
Poisson Geometry ◽  
Dirac Bracket ◽  
Semisimple Lie Algebra ◽  
Constrained System ◽  
Symplectic Embedding

Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P,Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M,ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.

Energy method for modelling delamination buckling in geometrically constrained systems

2003 ◽  
Vol 217 (9) ◽  
pp. 1015-1026 ◽  
Author(s):  
B J Hicks ◽  
G Mullineux ◽  
C Berry ◽  
C J McPherson ◽  
A J Medland
Keyword(s):  
Finite Element ◽  
Full Range ◽  
Constrained Systems ◽  
Third Party ◽  
Constrained System ◽  
Finite Element Approach ◽  
Delamination Buckling ◽  
Element Approach ◽  
Geometrically Constrained ◽  
Energy Formulation

Delamination buckling analysis of laminates is of considerable interest to the mechanical and materials engineering sectors, as well as having wider applications in geology and civil engineering. With advances in computing power, the ability to model ever increasingly complex problems at more detailed levels becomes more of a reality. However, many of the common finite element packages, with the exception of all but the most specialized, do not perform particularly well where complex non-linear problems are dealt with. In many cases, these packages can fail to determine the full range of solutions or accurately predict the properties and geometry of the final state. This is particularly the case where large deformations and buckling of laminates are considered. Because of this, many researchers prefer to use what they perceive to be more reliable techniques, such as the symbolic computation of the underlying differential equations, rather than finite element approaches. The use of finite element packages is further frustrated by the steep learning curve and implicit restrictions imposed by using third-party software. In this paper, a finite element approach and an energy formulation method are considered and used to model the delamination buckling in a geometrically constrained system. These methods are compared with experimental results and their relative merits are discussed. In particular, the accuracy and the ability to represent the geometry of the buckled system are discussed. Both the finite element approach and the energy formulation are described in detail and the numerical results are compared.

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.

Fractal–fractional dynamical system of Typhoid disease including protection from infection

2021 ◽  
Author(s):  
Qu Haidong ◽  
Mati ur Rahman ◽  
Muhammad Arfan ◽  
Mehdi Salimi ◽  
Soheil Salahshour ◽  
...  
Keyword(s):  
Dynamical System ◽  
Fractional Dynamical System

scholarly journals A Dynamical System with q-Deformed Phase Space Represented in Ordinary Variable Spaces

2010 ◽  
Vol 124 (6) ◽  
pp. 1019-1035
Author(s):  
S. Naka ◽  
H. Toyoda ◽  
T. Takanashi
Keyword(s):  
Dynamical System ◽  
Phase Space

scholarly journals A VARIATIONAL FORMULATION OF SYMPLECTIC NONCOMMUTATIVE MECHANICS

2007 ◽  
Vol 04 (05) ◽  
pp. 789-805 ◽  
Author(s):  
IGNACIO CORTESE ◽  
J. ANTONIO GARCÍA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Variational Principle ◽  
Configuration Space ◽  
Variational Formulation ◽  
Noncommutative Space ◽  
First Order ◽  
Dynamical Properties ◽  
Definition Of ◽  
Space Noncommutativity

The standard lore in noncommutative physics is the use of first order variational description of a dynamical system to probe the space noncommutativity and its consequences in the dynamics in phase space. As the ultimate goal is to understand the inherent space noncommutativity, we propose a variational principle for noncommutative dynamical systems in configuration space, based on results of our previous work [18]. We hope that this variational formulation in configuration space can be of help to elucidate the definition of some global and dynamical properties of classical and quantum noncommutative space.

Phase space geometry of constrained systems

1995 ◽  
Vol 105 (3) ◽  
pp. 1539-1545 ◽  
Author(s):  
V. P. Pavlov ◽  
A. O. Starinetz
Keyword(s):  
Phase Space ◽  
Constrained Systems ◽  
Space Geometry ◽  
Phase Space Geometry

On Fractional Hamilton Formulation Within Caputo Derivatives

2007 ◽  
Author(s):  
Dumitru Baleanu ◽  
Sami I. Muslih ◽  
Eqab M. Rabei
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivatives ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Fractional Integration ◽  
Fractional Dynamics ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Caputo Fractional Derivatives ◽  
Non Locality

The fractional Lagrangian and Hamiltonian dynamics is an important issue in fractional calculus area. The classical dynamics can be reformulated in terms of fractional derivatives. The fractional variational principles produce fractional Euler-Lagrange equations and fractional Hamiltonian equations. The fractional dynamics strongly depends of the fractional integration by parts as well as the non-locality of the fractional derivatives. In this paper we present the fractional Hamilton formulation based on Caputo fractional derivatives. One example is treated in details to show the characteristics of the fractional dynamics.

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