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.
Issue Section:
Mathematical Theory
1.
Podlubny
, I.
, 1999, Fractional Differential Equations
, Academic
, San Diego, CA
.2.
Zaslavsky
, G. M.
, 2005, Hamiltonian Chaos and Fractional Dynamics
, Oxford University Press
, Oxford
.3.
Kilbas
, A. A.
, Srivastava
, H. H.
, and Trujillo
, J. J.
, 2006, Theory and Applications of Fractional Differential Equations
, Elsevier
, New York
.4.
Magin
, R. L.
, 2006, Fractional Calculus in Bioengineering
, Begell House
, Redding, CT
.5.
Oldham
, K. B.
, and Spanier
, J.
, 1974, The Fractional Calculus
, Academic
, New York
.6.
Miller
, K. S.
, and Ross
, B.
, 1993, An Introduction to the Fractional Integrals and Derivatives: Theory and Applications
, Wiley
, New York
.7.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993, Fractional Integrals and Derivatives: Theory and Applications
, Gordon and Breach
, Linghorne, PA
.8.
Hilfer
, R.
, 2000, Applications of Fractional Calculus in Physics
, World Scientific
, Singapore
.9.
F. Mainardi
, A.
, 1997, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
,” Fractals and Fractional Calculus in Continuum Mechanics
, A.
Carpinteri
and F.
Mainardi
, eds., Springer-Verlag
, New York
.10.
Ortigueira
, M. D.
, 2003, “On the Initial Conditions in Continuous-Time Fractional Linear Systems
,” Signal Process.
0165-1684, 83
(11
), pp. 2301
–2309
.11.
Tenreiro Machado
, J. A.
, 1997, “Analysis and Design of Fractional Order Digital Control Systems
,” Syst. Anal. Model. Simul.
0232-9298, 27
, pp. 107
–122
.12.
Trujillo
, J. J.
, 1999, “On a Riemann–Liouville Generalized Taylor’S Formula
,” J. Math. Anal. Appl.
0022-247X, 231
, pp. 255
–265
.13.
Tofighi
, A.
, 2006, “The Intrisic Damping of the Fractional Oscillator
,” Physica A
0378-4371, 329
, pp. 29
–34
.14.
Chen
, W.
, and Holm
, S.
, 2004, “Fractional Laplacian Time-Space Models for Linear and Nonlinear Lossy Media Exhibiting Arbitrary Frequency Power-Law Dependency
,” J. Acoust. Soc. Am.
0001-4966, 115
(4
), pp. 1424
–1430
.15.
Goldfain
, E.
, 2004, “Fractional Dynamics, Cantorian Space-Time and the Gauge Hierarchy Problem
,” Chaos, Solitons Fractals
0960-0779, 22
(3
), pp. 513
–520
.16.
da Graa Marcos
, M.
, Duarte
, F. B. M.
, and Tenreiro Machado
, J. A.
, 2007, “Fractional Dynamics in the Trajectory Control of Redundant Manipulators
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, in press; doi: 10.1016/j.cnsns.2007.03.027.17.
Agrawal
, O. P.
, 2006, “Formulation and a Numerical Scheme for Fractional Optimal Control Problems
,” Proceedings of the IFAC∕FDA06
, Porto, Portugal
, Jul. 19–21.18.
Agrawal
, O. P.
, and Baleanu
, D.
, 2007, “A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems
,” J. Vib. Control
1077-5463, 13
(9-10
), pp. 1239
–1247
.19.
Tarasov
, V. E.
, and Zaslavsky
, G. M.
, 2006, “Fractional Dynamics of Systems With Long-Range Interaction
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 11
(8
), pp. 885
–898
.20.
Korabel
, N.
, Zaslavsky
, G. M.
, and Tarasov
, V. E.
, 2007, “Coupled Oscillators With Power-Law Interaction and Their Fractional Dynamics Analogues
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, 12
(8
), pp. 1405
–1417
.21.
Momani
, S.
, 2006, “A Numerical Scheme for the Solution of Multi-Order Fractional Differential Equations
,” Appl. Math. Comput.
0096-3003, 182
, pp. 761
–770
.22.
El-Wakil
, S. A.
, Elhanbaly
, A.
, and Abdou
, M. A.
, 2006, “A Domian Decomposition Method for Solving Fractional Nonlinear Differential Equations
,” Appl. Math. Comput.
0096-3003, 182
, pp. 313
–324
.23.
Riewe
, F.
, 1996, “Nonconservative Lagrangian and Hamiltonian Mechanics
,” Phys. Rev. E
1063-651X, 55
, pp. 1890
–1899
.24.
Riewe
, F.
, 1997, “Mechanics With Fractional Derivatives
,” Phys. Rev. E
1063-651X, 55
, pp. 3581
–3592
.25.
Klimek
, M.
, 2002, “Lagrangian and Hamiltonian Fractional Sequential Mechanics
,” Czech. J. Phys.
0011-4626, 52
, pp. 1247
–1254
.26.
Tarasov
, V. E.
, and Zaslavsky
, G. M.
, 2007, “Conservation Laws and Hamilton’S Equations for Systems With Long-Range Interaction and Memory
,” Commun. Nonlinear Sci. Numer. Simul.
1007-5704, in press; doi: 10.1016/j.cnsns.2007.05.017.27.
Tenreiro Machado
, J. A.
, 2001, “Discrete-Time Fractional-Order Controllers
,” Fractional Calculus Appl. Anal.
1311-0454, 4
(1
), pp. 47
–68
.28.
Agrawal
, O. P.
, 2002, “Formulation of Euler-Lagrange Equations for Fractional Variational Problems
,” J. Math. Anal. Appl.
0022-247X, 272
, pp. 368
–379
.29.
Naber
, M.
, 2004, “Time Fractional Schrödinger Equation
,” J. Math. Phys.
0022-2488, 45
, pp. 3339
–3352
.30.
Stanislavsky
, A. A.
, 2005, “Probability Interpretation of the Integral of Fractional Order
,” Theor. Math. Phys.
0040-5779, 138
, pp. 418
–431
.31.
Nigmatullin
, R. R.
, 1992, “The Fractional Integral and Its Physical Interpretation
,” Theor. Math. Phys.
0040-5779, 90
, pp. 242
–251
.32.
Tarasov
, V. E.
, 2006, “Electromagnetic Fields on Fractals
,” Mod. Phys. Lett. A
0217-7323, 12
, pp. 1587
–1600
.33.
El-Nabulusi
, R. A.
, 2007, “Differential Geometry and Modern Cosmology With Fractionaly Differentiated Lagrangian Function and Fractional Decaying Force Term
,” Rom. J. Phys.
1221-146X, 52
, pp. 441
–450
.34.
Agrawal
, O. P.
, 2006, “Fractional Variational Calculus and the Transversality Conditions
,” J. Phys. A
0305-4470, 39
, pp. 10375
–10384
.35.
Muslih
, S.
, and Baleanu
, D.
, 2005, “Hamiltonian Formulation of Systems With Linear Velocities Within Riemann–Liouville Fractional Derivatives
,” J. Math. Anal. Appl.
0022-247X, 304
(3
), pp. 599
–603
.36.
Baleanu
, D.
, 2006, “Fractional Hamiltoian Analysis of Irregular Systems
,” Signal Process.
0165-1684, 86
(10
), pp. 2632
–2636
.37.
Baleanu
, D.
, and Muslih
, S. I.
, 2005, “Formulation of Hamiltonian Equations for Fractional Variational Problems
,” Czech. J. Phys.
0011-4626, 55
(6
), pp. 633
–642
.38.
Baleanu
, D.
, and Muslih
, S. I.
, 2005, “Lagrangian Formulation of Classical Fields Within Riemann-Liouville Fractional Derivatives
,” Phys. Scr.
0031-8949, 72
(2–3
), pp. 119
–121
.39.
Baleanu
, D.
, and Avkar
, T.
, 2004, “Lagrangians With Linear Velocities Within Riemann–Liouville Fractional Derivatives
,” Nuovo Cimento B
0369-3554, 119
, pp. 73
–79
.40.
Baleanu
, D.
, Muslih
, S. I.
, and Tas
, K.
, 2006, “Fractional Hamiltonian Analysis of Higher Order Derivatives Systems
,” J. Math. Phys.
0022-2488, 47
(10
), pp. 1
–10
.41.
Rabei
, E. M.
, Nawafleh
, K. I.
, Hijjawi
, R. S.
, Muslih
, S. I.
, and Baleanu
, D.
, 2007, “The Hamilton Formalism With Fractional Derivatives
,” J. Math. Anal. Appl.
0022-247X, 327
(2
), pp. 891
–897
.42.
Baleanu
, D.
, and Agrawal
, O. P.
, 2006, “Fractional Hamilton Formalism Within Caputo’s Derivative
,” Czech. J. Phys.
0011-4626, 56
(10–11
), pp. 1087
–1092
.43.
Hermann
, R.
, 2007, “The Fractional Symmetric Rigid Rotor
,” J. Phys. G
0954-3899, 34
, pp. 607
–625
.44.
Henneaux
, M.
, and Teitelboim
, C.
, 1993, Quantization of Gauge Systems
, Princeton University
, Princeton, NJ
.45.
Jumarie
, G.
, 2007, “Lagrangian Mechanics of Fractional Order, Hamilton-Jacobi Fractional PDE and Taylor’s Series of Nondifferntiable Functions
,” Chaos, Solitons Fractals
0960-0779, 32
, pp. 969
–987
.Copyright © 2008
by American Society of Mechanical Engineers
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