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Lagrangians with linear velocities within Riemann-Liouville fractional derivatives

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Dumitru Baleanu at Institute of Space Science - INFLPR Subsidiary
Dumitru Baleanu
T. Avkar
T. Avkar
T. Avkar
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Abstract

Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained and the recovery of the classical results was discussed. Comment: 7 pages, LATEX,accepted for publication in Il Nuovo Cimento B
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arXiv:math-ph/0405012v1 4 May 2004
Lagrangians with linear velocities within Riemann-Liouville
fractional derivatives
Dumitru Baleanu
1
, Tansel Avkar
2
Department of Mathematics and Computer Science, Faculty of Arts and
Sciences, C¸ ankaya University- 06530, Ankara, Turkey
Abstract
Lagrangians linear in velocities were analyzed using the fractional
calculus and the Euler-Lagrange equations were derived. Two examples
were investigated in details, the explicit solutions of Euler-Lagrange equa-
tions were obtained and the recovery of the classical results was discussed.
PACS: 11.10.Ef. Lagrangian and Hamiltonian approa ch
1 Introduction
The mathematical idea of fractional derivatives, which goes back to the seven-
teenth century, has represented the subject of int erest for various mathematicians
[1, 2 , 3]. The fractional calculus, which means the calculus of derivatives and inte-
grals of any arbitrary real or complex order is gaining a considerable importance in
various branches of science, engineering and finance [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].
As it was observed during the past three decades, the fractional differential
calculus describes more accurately the physical systems [4, 5, 6, 12].Many applica-
tions of fractional calculus amount to replacing the time derivative in an evolution
equation with a derivative of fractional order. This is not merely a phenomeno-
logical procedure providing an additional fit parameter. One of the pro blems
encountered in this field is what kind of fr actional derivatives will replace the
integer derivative for a given problem [1, 2, 3, 13]. Depending on the specified
physical situation different authors have applied different derivatives [6, 9].
Nonconservative Lagrangia n and Hamiltonian mechanics were investigated by
Riewe within fractional calculus [14, 15]. Besides, Lagrangian and Hamiltonian
fractional sequential mechanics, the models with symmetric fractional derivative
were studied in [16, 17] and the properties of fractional differential forms were
introduced [18, 19].
Recently, an extension of the simplest f ractional variational pro blem and the
fractional variational problem of Lagrange was obtained [20]. A natural and
interesting generalization of Agrawal’s approach [20] is to apply the fractional
1
On leave of absence from Institute of Space Sciences, P.O BOX, MG-23, R 76900 Magurele-
Bucharest, Romania, E-mails: dumitru@cankaya.edu.tr, baleanu@venus.nipne.ro
2
E-Mail: avkar@cankaya.edu.tr
1
calculus to the constrained systems [21, 22]. In the present work, we apply the
concept of f ractional calculus to the Lagrangian with linear velocities.
The plan of this paper is as follows:
In Sec. 2 the Riemann-Liouville (RL) fractional derivatives were briefly presented.
In Sec. 3 the fractional Lagra ngia ns with linear velocities were constructed and
their corresponded Euler-Lagrange were a nalyzed. Sec. 4 was devoted to our
conclusions.
2 Riemann - Liouville fractional derivatives
In this section the definitions of the right and left RL derivatives as well as t heir
basic properties are briefly presented.
RL fractional derivatives [3 , 20] are defined as follows
a
D
α
t
f(t) =
1
Γ(n − α)
d
dt
!
n
t
Z
a
(t − τ)
n−α−1
f(τ)dτ , (1)
t
D
α
b
f(t) =
1
Γ(n − α)
−
d
dt
!
n
b
Z
t
(τ − t)
n−α−1
f(τ)dτ, (2)
where the order α fulfills n − 1 ≤ α < n and Γ represents the Euler’s gamma
function. The first derivative is called the RL left fractional derivative and in (2)
the expression of the RL rig ht fractional is present ed. If α is any integer, the
relations to the usual derivative are obtained as follows
a
D
α
t
f(t) =
d
dt
!
α
,
t
D
α
b
f(t) =
−
d
dt
!
α
. (3)
Under the assumptions that f (t) is continuous and p ≥ q ≥ 0 , the most
general pro perty of RL fractional derivatives can be written as
a
D
p
t
a
D
−q
t
f(t)
=
a
D
p−q
t
f(t). (4)
For p > 0 and t > a we obtain
a
D
p
t
a
D
−p
t
f(t)
= f(t), (5)
which means that the RL fractional differentiation operator is a left inverse to
the RL fractional integration operator of the same order. The relation (5) is
called the fundamental property of the RL fractional derivative. In addition, the
fractional derivative of a constant is not zero and the RL fractional derivative of
the power function (t − a)
ν
is given by
a
D
p
t
(t − a)
ν
=
Γ(ν + 1)
Γ(−p + ν + 1)
(t − a)
ν−p
, (6)
2
where ν > −1. The normal derivatives
d
n
dt
n
and
a
D
p
t
commute only if f
(j)
(a) =
0, j = 0, 1, . . . , n − 1 is fulfilled and two RL fractional derivative op erators
a
D
p
t
and
a
D
q
t
commute only if
h
a
D
p−j
t
f(t)
i
t=a
= 0 , j = 1, · · · , m (7)
and
h
a
D
q−j
t
f(t)
i
t=a
= 0 , j = 1, · · · , n . (8)
The above properties of RL fractional derivatives lead us to the conclusion that
there are many substantial differences from the usual derivatives a nd therefor e the
solutions of the differential fractional equations contain more information than
the classical ones.
3 Fractional Euler-Lagrange equations for
Lagrangians with linear velocities
Let J[q
1
, · · · , q
n
] be a functional of the form
b
Z
a
L
t, q
1
, · · · , q
n
,
a
D
α
t
q
1
, · · · ,
a
D
α
t
q
n
,
t
D
β
b
q
1
, · · · ,
t
D
β
b
q
n
dt (9)
defined on the set of f unctions q
i
(t), i = 1, · · · , n which have continuous left
RL fractional derivative of order α and right RL fractional derivative of order
β in [a, b] and satisfy the boundary conditions q
i
(a) = q
i
a
and q
i
(b) = q
i
b
. A
necessary condition for J[q
1
, · · · , q
n
] to admit a n extremum for given functions
q
i
(t), i = 1, · · · , n is that q
i
(t) satisfy Euler-Lagrange equations [20]
∂L
∂q
j
+
t
D
α
b
∂L
∂
a
D
α
t
q
j
+
a
D
β
t
∂L
∂
t
D
β
b
q
j
= 0 , j = 1, · · · , n . (10)
In the following we consider the Lagrangian with linear velocities
L = a
j
q
i
˙q
j
− V
q
i
, (11)
where a
j
(q
i
) and V (q
i
) are functions of their arguments.
The fist step is to construct the corresponding fractional generalization of
the Lagrangian given by (11). The fractional Lag rangian is not unique, in o ther
words there are several possibilities to replace the time derivative with fractional
ones. The requirement is to obtain the same Lagrangian expression if the order
α is 1. Having in mind the above considerations, for 0 < α ≤ 1, we propose two
fractional Lagrangians. The first one is as follows
3
L
′
= a
j
q
i
a
D
α
t
q
j
− V
q
i
. (12)
From (10) and (12), t he corresponding Euler-Lagrang e equations emerge as
∂a
j
(q
i
)
∂q
k
a
D
α
t
q
j
+
t
D
α
b
a
k
q
i
−
∂V (q
i
)
∂q
k
= 0 . (13)
The second Lagrangian is given by
L
′
= −a
j
q
i
t
D
α
b
q
j
− V
q
i
. (14)
Using (1 0) and (14) the corresponding Euler-Lagrange equations become
∂a
j
(q
i
)
∂q
k
t
D
α
b
q
j
+
a
D
α
t
a
k
q
i
+
∂V (q
i
)
∂q
k
= 0 . (15)
3.1 Examples
A. To illustrate our a pproa ch, let us consider the following Lagrangian
L = ˙q
1
q
2
− ˙q
2
q
1
− (q
1
− q
2
)q
3
(16)
which is a gauge invariant [23]. In this case we proposed the corresponding
fractional Lagrangian to be as
L
′
=
a
D
α
t
q
1
q
2
−
a
D
α
t
q
2
q
1
− (q
1
− q
2
)q
3
. (17)
Using (1 3), the Euler-Lagra nge equations corresponding to (17) become
q
1
= q
2
, −
a
D
α
t
q
2
− q
3
+
t
D
α
b
q
2
= 0 ,
a
D
α
t
q
1
+ q
3
−
t
D
α
b
q
1
= 0. (18)
The solution of (18) is given as follows
q
1
= q
2
, (19)
q
3
= (−
a
D
α
t
+
t
D
α
b
)q
1
. (20)
From (19) and (20) we conclude that the classical solution is obtained if α → 1.
B. Let us consider the second Lagrangian given by
L = ˙q
1
q
2
+ ˙q
3
q
4
− V (q
2
, q
3
, q
4
), (21)
where V (q
2
, q
3
, q
4
) =
−1
2
[(q
4
)
2
− 2q
2
q
3
]. We observe that (21) is a second class
constrained system in Dirac’s classification [21].
4
We propose the fractional generalization of (21) t o be as follows
L
′
= −[(
t
D
α
b
q
1
)q
2
+ (
t
D
α
b
q
3
)q
4
+ V (q
2
, q
3
, q
4
)] . (22)
From (15) and (22) the Euler-Lagrange equations are given by
a
D
α
t
q
2
= 0 , (23)
t
D
α
b
q
1
+ q
3
= 0 , (24)
a
D
α
t
q
4
+ q
2
= 0 , (25)
t
D
α
b
q
3
− q
4
= 0 . (26)
From (23), we conclude that the solution for q
2
(t) has t he form
q
2
(t) = C
1
(t − a)
α−1
. (27)
From (25) and (27) an equation for q
4
(t) is o bta in as follows
a
D
α
t
q
4
= −C
1
(t − a)
α−1
. (28)
The solution of (28) has the form
q
4
(t) = C
2
(t − a)
α−1
−
C
1
Γ(α)
t
Z
a
(−τ + t)
α−1
(τ − a)
α−1
dτ . (29)
Using (2 6) and (29), the solution of q
3
(t) becomes
q
3
(t) = C
3
(b − t)
α−1
+
C
2
Γ(α)
b
Z
t
(−t + τ)
α−1
(−a + τ)
α−1
dτ
−
C
1
Γ(α)
2
b
Z
t
(−t + τ)
α−1
τ
Z
a
(τ − η)
α−1
(η − a)
α−1
dηdτ . (30)
Finally, the equation (24) together with (30) give the solution for q
1
(t) as
follows
q
1
(t) = C
4
(b − t)
α−1
−
C
3
Γ(α)
b
Z
t
(−t + τ)
α−1
(b − τ)
α−1
dτ
−
C
2
Γ(α)
2
b
Z
t
(−t + τ)
α−1
b
Z
τ
(−τ + η)
α−1
(−a + η)
α−1
dηdτ
+
C
1
Γ(α)
3
b
Z
t
(σ − t)
α−1
b
Z
σ
(σ − η)
α−1
η
Z
a
(τ − η)
α−1
(η − a)
α−1
dτdηdσ .(31)
5
Here C
1
, C
2
, C
3
and C
4
are constants. If α → 1, a → 0, b → 1, then the standard
solutions
q
1
(t) =
C
′
4
t
3
6
−
C
′
3
t
2
2
+C
′
2
t+C
′
1
, q
2
(t) = C
′
4
, q
3
(t) =
C
′
4
2
t
2
−C
′
3
t+C
′
2
, q
4
(t) = −C
′
4
t+C
′
3
(32)
are recovered if we redefine the constants fro m (27), (29), (30) and (31 ) as
C
′
1
= C
4
− C
3
−
C
2
2
+
C
1
3
, C
′
2
=
−C
1
2
+ C
2
+ C
3
, C
′
3
= C
2
, C
′
4
= C
1
.
4 Conclusion
The Lagrangians with linear velocities were investigated using left and right R L
fractional derivatives. The corresponding fractional Lagrangians were proposed
and the fractional Euler-Lagrange equations were obtained. Although the frac-
tional Lagrangia ns contain only left or right derivatives, both derivatives are in-
volved in Euler- Lagrange equations and both played an important role in finding
the solutions. The exact solutions of the Euler-Lagrange equations were obtained
for two examples corresponding to first and second-class constrained systems.
A “gauge invariance” was reported for the first example. The solutions of the
investigated examples depend on the limits a and b and t he limiting procedure
recovered the standard results.
5 Acknowledgments
This work is partially supported by the Scientific and Technical Research Council
of Turkey. One o f the authors (D. B.) would like to tha nk R. Hilfer, O. Agrawal
and N. Engheta for providing him impo r tant references and M. Naber and M.
Henneaux for interesting discussions.
References
[1] K. B. OLDHAM K. B. and SPANIER J., The Fractional Calculus, (Academic
Press, New Yor k)1974.
[2] MILLER K.S.and ROSS B., An introduction to the Fractional Integrals and
Derivatives-Theory and Applications, (Gordon and Breach, Longhorne, PA)
1993.
[3] PODLUBNY I., Fractional D ifferential Equations, (Academic Press) 1999.
[4] MAINARDI F., Fractional calculus:Some basic problems in continuum and
statistical mechanics, in A.Carpinteri, F. Mainardi (Eds.), Fractals and Frac-
tional Calculus in Continuum Mechanics, (Springer-Verlag, New York) 1997.
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7
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The fractional calculus of variations is considered today as an important in applied mathematics. It consists of minimizing or maximizing functionals that depend on different types of fractional derivatives and integral operators. This mathematical subject has proved to be relevant because of its motivating implications in describing dissipative and nonconservative physical systems ranging from classical to quantum mechanics and field theories. However, in some complex dynamical systems, the analytic solutions of the consequential fractional Euler–Lagrange equation are tricky to obtain, and, besides, the generalized fractional variational problems require tricky boundary conditions and further properties in order to obtain the necessary optimality conditions of the Euler–Lagrange type for the given problem. Advanced methods in the fractional calculus of variations have been found to be very practical in studying a large number of nonlinear dynamical systems; however, describing some nonlinear wave phenomena, which are of particular relevance in various fields of science, is still not extensively elaborated in literature. In this study, a new action functional based on Kiryakova fractional operators involving the Meijer-G functions is introduced and discussed. This approach was found to be practical to study periodic dampened oscillators exhibiting traveling soliton-like solutions and a family of modified Korteweg-de Vries equations and lumps obtained within the framework of weakly nonlinear dispersive differential equations. We extended our approach to the case of time-fractal Kiryakova fractional-action integral, where a set of time-fractal Korteweg-de Vries equations exhibiting lump-like evanescent wave solutions is obtained. Our approach may be applied to other nonlinear dynamical systems.
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A reliable numerical algorithm mixed with hypergeometric function for analyzing fractional variational problems
Article
Full-text available
  • Jun 2024
  • NUMER ALGORITHMS
The present study aims to introduce a numerical approach based on the hybrid of block-pulse functions (BPFs), Bernoulli polynomials (BPs), and hypergeometric function for analyzing a class of fractional variational problems (FVPs). The FVPs are made by the Caputo derivative sense. To analyze this problem, first, we create an approximate for the Riemann-Liouville fractional integral operator for BPFs and BPs of the fractional order. In this framework and using the Gauss-Legendre points, the main problem is converted into a system of algebraic equations. In the follow-up, an accurate upper bound is obtained and some theorems are established on the convergence analysis. Moreover, the computational order of convergence and solvability of the proposed approach are displayed and approximated theoretically and numerically. Meanwhile, the thrust of the proposed scheme is compared with other sophisticated examples in the literature, demonstrating that the process is accurate and efficient.
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Fractional Spectral and Fractional Finite Element Methods: A Comprehensive Review and Future Prospects
Article
  • Mar 2024
In this article, we will discuss the applications of the Spectral element method (SEM) and Finite element Method (FEM) for fractional calculusThe so-called fractional Spectral element method (f-SEM) and fractional Finite element method (f-FEM) are crucial in various branches of science and play a significant role. In this review, we discuss the advantages and adaptability of FEM and SEM, which provide the simulations of fractional derivatives and integrals and are, therefore, appropriate for a broad range of applications in engineering, biology, and physics. We emphasize that they can be used to simulate a wide range of real-world phenomena because they can handle fractional differential equations that are both linear and nonlinear. Although many researchers have already discussed applications of FEM in a variety of fractional differential equations (FDEs) and delivered very significant results, in this review article, we aspire to enclose fundamental to advanced articles in this field which will guide the researchers through recent achievements and advancements for the further studies.
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Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain
Article
  • Jul 2002
A general solution is given for a fractional diffusion-wave equation defined in a bounded space domain. The fractional time derivative is described in the Caputo sense. The finite sine transform technique is used to convert a fractional differential equation from a space domain to a wavenumber domain. Laplace transform is used to reduce the resulting equation to an ordinary algebraic equation. Inverse Laplace and inverse finite sine transforms are used to obtain the desired solutions. The response expressions are written in terms of the Mittag–Leffler functions. For the first and the second derivative terms, these expressions reduce to the ordinary diffusion and wave solutions. Two examples are presented to show the application of the present technique. Results show that for fractional time derivatives of order 1/2 and 3/2, the system exhibits, respectively, slow diffusion and mixed diffusion-wave behaviors.
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Fractional calculus and generalized Rodrigues formula
Article
  • Jan 2004
  • APPL MATH COMPUT
The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers. The two generalizations of the Rodrigues formula of the Laguerre polynomials Lαβ(x)=(x−β/n!)exDαe−xxn+β, and Lαβ(x)=((x−βex)/Γ(1+α))Dαe−xxα+β, are defined in [Math. Sci. Res. Hot-line 1 (10) (1997) 7; Appl. Math. Comput. 106 (1) (1999) 51] and some of their properties are proved.Here we define the new special function Lαβ(γ,a;x) based on a generalization of the Rodrigues formula, then we study some of its properties, some recurrence relations and prove that the set of functions {Lαβ(γ,a;x),α∈R} is continuous as a function of α∈R. The continuation as α,γ→n and a=1 to the Rodrigues formula of the Laguerre polynomials Lnβ(x) are proved. Also the confluent hypergeometric representation will be given.
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Experimental evidence for fractional time evolution in glass materials
Article
  • Nov 2002
  • CHEM PHYS
The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.
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Mechanics with fractional derivative
Article
  • Mar 1997
Lagrangian and Hamiltonian mechanics can be formulated to include derivatives of fractional order [F. Riewe, Phys. Rev. 53, 1890 (1996)]. Lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction. The present work continues the development of fractional-derivative mechanics by deriving a modified Hamilton's principle, introducing two types of canonical transformations, and deriving the Hamilton-Jacobi equation using generalized mechanics with fractional and higher-order derivatives. The method is illustrated with a frictional force proportional to velocity. In contrast to conventional mechanics with integer-order derivatives, quantization of a fractional-derivative Hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives. Instead, a quantum-mechanical wave equation is proposed that follows from the Hamilton-Jacobi equation by application of the correspondence principle.
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The Fractional Calculus
Book
  • Jan 1974
Here, we should mention the most important function used in fractional calculus — Euler’s Gamma function, which is defined as Γ(n)=∫0∞tn−1e−tdt. \Gamma (n) = \int_0^\infty {{t^{n - 1}}{e^{ - t}}dt.} (2.1) This function is generalization of a factorial in the following form: Gamma(n)=(n−1)! Gamma (n) = (n - 1)! (2.2)
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Fractional Sequential Mechanics - Models with Symmetric Fractional Derivative
Article
  • Dec 2001
The symmetric fractional derivative is introduced and its properties are studied. The Euler-Lagrange equations for models depending on sequential derivatives of type are derived using minimal action principle. The Hamiltonian for such systems is introduced following methods of classical generalized mechanics and the Hamilton’s equations are obtained. It is explicitly shown that models of fractional sequential mechanics are non-conservative. The limiting procedure recovers classical generalized mechanics of systems depending on higher order derivatives. The method is applied to fractional deformation of harmonic oscillator and to the case of classical frictional force proportional to velocity.
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Lagrangean and Hamiltonian fractional sequential mechanics
Article
  • Jan 2002
The models described by fractional order derivatives of Riemann-Liouville type in sequential form are discussed in Lagrangean and Hamiltonian formalism. The Euler-Lagrange equations are derived using the minimum action principle. Then the methods of generalized mechanics are applied to obtain the Hamilton’s equations. As an example free motion in fractional picture is studied. The respective fractional differential equations are explicitly solved and it is shown that the limitα→1+ recovers classical model with linear trajectories and constant velocity.
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Formulation of Euler-Lagrange equations for fractional variational problems
Article
  • Aug 2002
This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann–Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler–Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.
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