- 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
where the order satisfies . The Riemann-Liouville derivative of constant isn't zero, but the Caputo derivative of a constant is zero.
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
Note that in the following equations, the sign is omitted between the differential forms.
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 .
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