J. Comp. & Math. Sci. Vol.3 (3), 338-342 (2012)
Applications of Fractional Hamilton Equations within Caputo Derivatives HARISH NAGAR1 and ANIL KUMAR MENARIA2 1
Head, Department of Mathematics, Mewar University, Gangrar, INDIA. 2 Research Scholar, Mewar University, Gangrar, INDIA (Received on: May 29, 2012) ABSTRACT The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the equivalent Lagrangians play an important role because they admit the same Euler-Lagrange equations. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann-Liouville fractional derivatives. As a consequence of applying this procedure the classical results are re-obtained as a special case. The fractional Euler-Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated and two examples are analyzed in details. In this paper, we review some new trends in this field and we discuss some of their potential applications. Keywords: Fractional calculus, Fractional variational principles, Fractional Euler-Lagrange equations, Fractional Hamilton equations. MSC2010: 33C90, 33E99.
1. INTRODUCTION Various applications of fractional calculus are based on replacing the time derivative in an evolution equation with a derivative of fractional order. The results of
several recent researchers confirm that fractional derivatives seem to arise for important mathematical reasons. During the last decades the fractional calculus1-3 started to be used in various fields, e.g. engineering, physics, biology and many important results
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Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (3), 338-342 (2012)
were reported4-19. The direct method is to replace the classical derivatives with the fractional ones and after that to apply the generalization of the classical techniques to find the corresponding fractional Euler– Lagrange equations. As a result several formulations of the fractional Euler– Lagrange equations have been reported in the literature and applied to several important dynamical systems21,22.
However, the RL derivative of a power has the form
The paper is organized as follows: in section 2, some basic formulas of the fractional calculus are briefly reviewed. In Section 3, we present the fractional Lagrangian and Hamiltonian analysis of discrete systems. Section 4 is dedicated to the fractional Hamilton equations within Caputo derivatives. Finally, section 5 is devoted to our conclusions.
ŕł&#x; ŕł&#x; ŕł&#x;
(4)
For 1, 0. Composite of fractional derivatives is given by the following formula
2. PRELIMINARIES
1 Γ
ŕ´€
|
– Γ 1
(5) Here 0 1 ; 0 and k is an integer number. The fractional product rule is given below
∑
(6)
(1)
And the right RL fractional derivative becomes ŕł&#x;
ഀజ഑
ŕ´‘
In the following we define the left and the right Caputo derivatives. Namely, the left Caputo fractional derivative has the form
The left RL derivative has the form
ŕ´€
1 Γ
(7) and the right Caputo fractional derivative is given by
(2)
(8)
Where 1 As it can be seen from (1), the RL derivative of a constant is not zero and its expression is given by
(3)
where 1 . In this case the derivative of a constant is zero and we can define properly the initial conditions for the fractional differential equations which can be handle by using an analogy with the classical case.
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Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (3), 338-342 (2012)
3. FRACTIONAL LAGRANGIAN AND HAMILTONIAN ANALYSIS OF DISCRETE SYSTEMS: Let us consider Lagrangian given below:
the
L f (q p (t ), a Dta q p (t ), t Dbb q p (t )),
fractional
p = 1, 2,..., N .
(9) The corresponding Euler–Lagrange equations become
0
Since the time it was proposed the fractional Lagrangians involving both the left and the right derivatives have been without a clear application meaning. However, we point out here that we can find the meaning of this type of Lagrangian when we try to analyze the fractional generalization of the classical Lagrangians involving higher derivatives. However, the fractional generalization is not unique and it depends on the fractional derivative we have in hand. The use of one or another fractional derivative depends on the analyzed problem.
1. (10) 4. FRACTIONAL HAMILTON EQUATIONS WITHIN CAPUTO DERIVATIVES for 0 < a , b < 1. The next step is to A Hamiltonian dynamics in terms of consider the following form of the fractional Caputo derivatives was developed (see e.g. Lagrangian: [48, 55]). Let us consider for simplicity the fractional Lagrangian
0 (11)
L(q p (t ), ac Dta q p (t ), ct Dba q p (t )),
for 0 < a < 1, p = 1, 2,..., N . By using (11) we define the generalized momenta as
, 1, 2, â&#x20AC;Ś ,
(12)
Taking into account (11) and (12), a fractional Hamiltonian function is defined as
0 < a , b < 1. (15)
The corresponding canonical momenta have the forms
, ŕ°&#x2030;
(16)
#
Defining the fractional Hamiltonian as
H=
pa p a Dta q p (t ) -
L' f .
(13)
Therefore, the canonical equations corresponding to (13) are given below: థ௠థ௧
,
! " ,
#
for 0 < a < 1, p = 1, 2,..., N .
" .
(14)
H = pa ( ac Dta q ) + pb ( ct Dba q ) - L, we obtain the fractional equations as given below: థ௠థ௧
,
$ !
" ,
(17)
Hamiltonâ&#x20AC;&#x2122;s
' & = &% , % &.
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(18)
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Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (3), 338-342 (2012)
The treatment of the constraint should be investigated with special care (see [33â&#x20AC;&#x201C;35, 51] and references therein). The main problem to generalize the classical procedure for fractional Lagrangians appears when we perform the fractional derivation of the primary constraint and it is related to the fact that the fractional Hamiltonian is not a conservative Hamiltonian [33â&#x20AC;&#x201C;35, 51, 54]. 5. CONCLUSION In this paper, some of the new trends in the field of fractional variational area were summarized. Finding the fractional Lagrangian corresponding to an equation involving both the left and the right derivatives is not an easy task. However, as it was shown in this paper, for a special type of fractional differential equation, it is always possible to find a corresponding fractional Lagrangian. REFERENCES 1. Agrawal, O. P.: Application of fractional derivatives in thermal analysis of disk brakes. Nonlinear Dynamics. 38, 191206 (2004). 2. Agrawal, O. P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics. 38, 323-337 (2004). 3. Agrawal, O. P.: Formulation of EulerLagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368-379 (2002). 4. Baleanu, D., Muslih, S.: Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Physica Scripta. 72(2-3), 119-121 (2005).
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Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)