- Research Article
- Open Access
Fractional Odd-Dimensional Mechanics
© Ali Khalili Golmankhaneh et al. 2011
- Received: 28 August 2010
- Accepted: 26 October 2010
- Published: 27 October 2010
The classical Nambu mechanics is generalized to involve fractional derivatives using two different methods. The first method is based on the definition of fractional exterior derivative and the second one is based on extending the standard velocities to the fractional ones. Fractional Nambu mechanics may be used for nonintegrable systems with memory. Further, Lagrangian which is generate fractional Nambu equations is defined.
- Fractional Derivative
- Exterior Derivative
- Fractional Form
- Hamiltonian Mechanic
- Dimensional Phase Space
Derivatives and integrals of fractional-order have found many applications in recent studies in mechanics and physics, for example, in chaotic dynamics, quantum mechanics, plasma physics, anomalous diffusion, and so many fields of physics [1–12]. Fractional mechanics describes both conservative and nonconservative systems [13, 14]. In mechanics, Riewe has shown that Lagrangian involving fractional time derivatives leads to equation of motion with nonconservative classical derivatives such as friction [13, 14]. Motivated by this approach many researchers have explored this area giving new insight into this problem [15–37]. Agrawal has presented fractional Euler-Lagrangian equation involving Riemann-Liouville derivatives [16, 17]. Further fractional single and multi-time Hamiltonian formulation has been developed by Baleanu and coworkers .
In 1973, Nambu generalized Hamiltonian mechanics which is called now Nambu mechanics. This formalism is shown that provide a suitable framework for the odd dimensional phase space and nonintegrable systems [39–43]. By this motivation the authors have fractionalized this formalism .
In this work two methods are introduced for fractionalizing of Nambu mechanics. The first method is based on the definition of fractional exterior derivative and fractional forms. The second methods is based on fractionalizing of classical velocity. The resulted equations using these methods may use for complex memorial systems.
This paper is organized as follows.
Section 2 is devoted to a brief review of the fractional derivative definitions and fractional forms. Section 3 contains the classical Nambu mechanics. Section 4 deals with fractionalizing Hamiltonian mechanics using fractional differential forms. Using two different methods in Section 5 the Nambu mechanics has been fractionalized. In Section 6 is defined a Lagrangian which its variation gives the fractional Hamiltonian equations. In Section 7 we present our conclusions.
The following subsections contain all mathematical tools used in this manuscript.
2.1. Fractional Derivatives
2.2. Fractional Forms
In [45, 46] the author generalizes the definition of integer-order vector spaces form to fractional-order one, and denotes it by . In this notation is the order of differential form, the number of coordinate differential appearing in the basis elements, the number of coordinates. For instance (2.5) is an element of .
The Definition of Fractional Exterior Derivative
The Pfaffian equations is obtained, and then the Hamiltonian equations are resulted.
Now the Paffian equations. Equating them with zero we lead to Nambu mechanics equations.
By expanding the right hand and by comparing the coefficients of form we lead to fractional Nambu mechanic equations.
In this paper, we defined new equations corresponding to the complex systems described by the Nambu mechanics within the languages of the fractional differential forms. It is shown that variation of the corresponding new action using fractional Lagrangian gives fractional Nambu equations. The equivalent methods presented in this manuscript can be applied to investigate the dynamics of the complex nonintegrable systems with memory. The classical results are obtained in the limiting case .
- Sabatier J, Agrawal OP, Tenreiro Machado JA: Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht, The Netherlands; 2007:xiv+552.View ArticleMATHGoogle Scholar
- Baleanu D, Guvenc Ziya B, Tenreiro MJA: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Dordrecht, The Netherlands; 2009.MATHGoogle Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000:viii+463.View ArticleMATHGoogle Scholar
- Gorenflo R, Mainardi F: Fractional Calculus: Integral and Differential Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York, NY, USA; 1997.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier, Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York, NY, USA; 1974:xiii+234.MATHGoogle Scholar
- Miller KS, Ross B: An Introduction to the Fractional Integrals and Derivatives-Theory and Application, A Wiley-Interscience Publication. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar
- Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
- Magin RL: Fractional Calculus in Bioengineering. Begell House, Connecticut, Mass, USA; 2006.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach, New York, NY, USA; 1993:xxxvi+976.MATHGoogle Scholar
- West BJ, Bologna M, Grigolini P: Physics of Fractal Operators, Institute for Nonlinear Science. Springer, New York, NY, USA; 2003:x+354.View ArticleGoogle Scholar
- Zaslavsky GM: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford, UK; 2005:xiv+421.MATHGoogle Scholar
- Riewe F: Nonconservative Lagrangian and Hamiltonian mechanics. Physical Review E 1996,53(2):1890-1899. 10.1103/PhysRevE.53.1890MathSciNetView ArticleGoogle Scholar
- Riewe F: Mechanics with fractional derivatives. Physical Review E 1997,55(3):3581-3592. 10.1103/PhysRevE.55.3581MathSciNetView ArticleGoogle Scholar
- Golmankhaneh AK, Golmankhaneh AK, Baleanu D, Baleanu MC: Hamiltonian structure of fractional first order Lagrangian. International Journal of Theoretical Physics 2010,49(2):365-375. 10.1007/s10773-009-0209-5MathSciNetView ArticleMATHGoogle Scholar
- Agrawal OP: Formulation of Euler-Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications 2002,272(1):368-379. 10.1016/S0022-247X(02)00180-4MathSciNetView ArticleMATHGoogle Scholar
- Agrawal OP: Fractional variational calculus and the transversality conditions. Journal of Physics A 2006,39(33):10375-10384. 10.1088/0305-4470/39/33/008MathSciNetView ArticleMATHGoogle Scholar
- Baleanu D, Muslih SI: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Physica Scripta 2005,72(2-3):119-121. 10.1238/Physica.Regular.072a00119MathSciNetView ArticleMATHGoogle Scholar
- Baleanu D, Agrawal OP: Fractional Hamilton formalism within Caputo's derivative. Czechoslovak Journal of Physics 2006,56(10-11):1087-1092. 10.1007/s10582-006-0406-xMathSciNetView ArticleMATHGoogle Scholar
- Baleanu D, Muslih SI, Taş K: Fractional Hamiltonian analysis of higher order derivatives systems. Journal of Mathematical Physics 2006,47(10):8.View ArticleMathSciNetMATHGoogle Scholar
- Baleanu D, Golmankhaneh AK, Golmankhaneh AK: Fractional Nambu mechanics. International Journal of Theoretical Physics 2009,48(4):1044-1052. 10.1007/s10773-008-9877-9MathSciNetView ArticleMATHGoogle Scholar
- Frederico GSF, Torres DFM: A formulation of Noether's theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications 2007,334(2):834-846. 10.1016/j.jmaa.2007.01.013MathSciNetView ArticleMATHGoogle Scholar
- Golmankhaneh AK: Fractional poisson bracket. Turkish Journal of Physics 2008,32(5):241-250.Google Scholar
- Klimek M: Fractional sequential mechanics—models with symmetric fractional derivative. Czechoslovak Journal of Physics 2001,51(12):1348-1354. 10.1023/A:1013378221617MathSciNetView ArticleMATHGoogle Scholar
- Klimek M: Lagrangean and Hamiltonian fractional sequential mechanics. Czechoslovak Journal of Physics 2002,52(11):1247-1253. 10.1023/A:1021389004982MathSciNetView ArticleMATHGoogle Scholar
- Laskin N: Fractional quantum mechanics. Physical Review E 2000,62(3 A):3135-3145.MathSciNetView ArticleMATHGoogle Scholar
- Laskin N: Fractional Schrödinger equation. Physical Review E 2002,66(5):-7.Google Scholar
- Muslih SI, Baleanu D: Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives. Journal of Mathematical Analysis and Applications 2005,304(2):599-606. 10.1016/j.jmaa.2004.09.043MathSciNetView ArticleMATHGoogle Scholar
- Rabei EM, Nawafleh KI, Hijjawi RS, Muslih SI, Baleanu D: The Hamilton formalism with fractional derivatives. Journal of Mathematical Analysis and Applications 2007,327(2):891-897. 10.1016/j.jmaa.2006.04.076MathSciNetView ArticleMATHGoogle Scholar
- Rabei EM, Tarawneh DM, Muslih SI, Baleanu D: Heisenberg's equations of motion with fractional derivatives. Journal of Vibration and Control 2007,13(9-10):1239-1247. 10.1177/1077546307077469MathSciNetView ArticleMATHGoogle Scholar
- Rabei EM, Almayteh I, Muslih SI, Baleanu D: Hamilton-Jacobi formulation of systems within Caputo's fractional derivative. Physica Scripta 2008.,77(1):Google Scholar
- Tarasov VE: Fractional variations for dynamical systems: Hamilton and Lagrange approaches. Journal of Physics A 2006,39(26):8409-8425. 10.1088/0305-4470/39/26/009MathSciNetView ArticleMATHGoogle Scholar
- Tarasov VE: Fractional generalization of gradient and Hamiltonian systems. Journal of Physics A 2005,38(26):5929-5943. 10.1088/0305-4470/38/26/007MathSciNetView ArticleMATHGoogle Scholar
- Tarasov VE: Fractional vector calculus and fractional Maxwell's equations. Annals of Physics 2008,323(11):2756-2778. 10.1016/j.aop.2008.04.005MathSciNetView ArticleMATHGoogle Scholar
- Zaslavsky GM: Chaos, fractional kinetics, and anomalous transport. Physics Reports A 2002,371(6):461-580. 10.1016/S0370-1573(02)00331-9MathSciNetView ArticleMATHGoogle Scholar
- Baleanu D, Golmankhaneh AK, Nigmatullin R, Golmankhaneh AK: Fractional Newtonian mechanics. Central European Journal of Physics 2010,8(1):120-125. 10.2478/s11534-009-0085-xMATHGoogle Scholar
- Golmankahneh AK, Golmankahneh AK, Baleanu D: On nonlinear fractional Klein Gordab equation. Signal Processing 2010,91(3):466-451.Google Scholar
- Baleanu D, Golmankhaneh AK, Golmankhaneh AK: The dual action of fractional multi time Hamilton equations. International Journal of Theoretical Physics 2009,48(9):2558-2569. 10.1007/s10773-009-0042-xView ArticleMATHGoogle Scholar
- Nambu Y: Generalized Hamiltonian dynamics. Physical Review D 1973, 7: 2405-2412. 10.1103/PhysRevD.7.2405MathSciNetView ArticleMATHGoogle Scholar
- Mukunda N, Sudarshan ECG: Relation between Nambu and Hamiltonian mechanics. Physical Review D 1976,13(10):2846-2850. 10.1103/PhysRevD.13.2846MathSciNetView ArticleGoogle Scholar
- Takhtajan L: On foundation of the generalized Nambu mechanics. Communications in Mathematical Physics 1994,160(2):295-315. 10.1007/BF02103278MathSciNetView ArticleMATHGoogle Scholar
- Fecko M: On geometrical formulation of the Nambu dynamics. Journal of Mathematical Physics 1992,33(3):926-929. 10.1063/1.529744MathSciNetView ArticleMATHGoogle Scholar
- Cayley A: Collected Mathematical Papers. Cambridge University Press, Cambridge, UK; 1890.MATHGoogle Scholar
- Flanders H: Differential Forms with Applicatons to the Physics Sciences, Dover Books on Advanced Mathematics. 2nd edition. Dover, New York, NY, USA; 1989:xvi+205.Google Scholar
- Shepherd C, Naber M: Fractional differential forms II. http://arxiv.org/abs/math-ph/0301016
- Cottrill-Shepherd K, Naber M: Fractional differential forms. Journal of Mathematical Physics 2001,42(5):2203-2212. 10.1063/1.1364688MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.