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Communications in Applied Analysis 12 (2008), no. 4, 441­450

FRACTIONAL POWERS OF DERIVATIVES IN CLASSICAL MECHANICS
VASILY E. TARASOV Skobeltsyn Institute of Nuclear Physics, Moscow State University Moscow 119991, Russia tarasov@theory.sinp.msu.ru ABSTRACT. Fractional analysis is applied to describe classical dynamical systems. Fractional derivative can be defined as a fractional power of derivative. The infinitesimal generators {H, ·} and L = G(q , p)q + F (q , p)p , which are used in equations of motion, are derivative operators. We consider fractional derivatives on a set of classical observables as fractional powers of derivative operators. As a result, we obtain a fractional generalization of the equation of motion. This fractional equation is exactly solved for the simple classical systems. The suggested fractional equations generalize a notion of classical systems to describe dissipative processes. Key words. Fractional derivative, Fractional equation, Classical dynamics

1. INTRODUCTION The analysis of non-integer order go es back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. There are many bo oks about fractional calculus and fractional differential equations [1, 2, 3]. Derivatives of fractional order, and fractional differential equations have found many applications in recent studies in physics (see for example [4, 5, 6] and references therein). The classical variables, which are also called observables, are defined as functions on the phase space. The dynamical description of system is given by an operator. The natural description of the motion is in terms of the infinitesimal change of the system. The infinitesimal operator of equation of motion is defined by some form of derivation of functions. Fractional derivative can be defined as a fractional power of derivative (see for example [7, 8]). It is known that the infinitesimal generator {H, ·} and L = G(q , p)p + F (q , p)p , which are used in the equation of motion, are derivations on an algebra of classical observables. A derivation of an algebra M is a linear map L, which satisfies L(AB ) = (LA)B + A(LB ) for all A, B M. In this paper, we consider a fractional derivative as a fractional power of derivative. As a result, we obtain a fractional generalization of the equation of motion. It allows us to generalized a notion of classical Hamiltonian systems. Note that some fractional generalization
Received October 1, 2008 1083-2564 $15.00 c Dynamic Publishers, Inc.


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of gradient systems has been suggested in [9], and a generalization of Hamiltonian systems is considered in [10]. The suggested fractional equation is exactly solved for a free particle, harmonic oscillator and damped oscillator. A classical systems that is presented by fractional equation can be considered as a dissipative system. Fractional derivatives can be used as a possible approach to describe an interaction between the system and an environment. Note that fractional dynamics can be considered with low-level fractionality by some generalization of metho d suggested in [11, 12, 13]. In section 2, the fractional power of derivative and the fractional equation are suggested. In section 3, the Cauchy problem for the fractional equation and the properties of time evolution described by it are considered. In section 4, the solution for simple examples of the fractional equation are derived. 2. FRACTIONAL DERIVATIVE AND FRACTIONAL EQUATIONS Let us consider the classical systems d qk = Gk (q , p), dt d pk = Fk (q , p) (k = 1, . . . , n). dt (1 )

A linear algebra M of classical observables is described by functions A = A(q , p) on the phase space R2n . Let L be a differential operator on M given by L = - G k (q , p ) + Fk (q , p) qk p .
k

(2 )

Here and later we mean the sum on the repeated index k from 1 to n. The equation of motion for the classical observable has the form d At = -LAt . dt Equations (1) are special cases of (3). If the functions Gk (q , p) and Fk (q , p) satisfy the Helmholtz conditions Gk Gl - = 0, pl pk Gl Fk + = 0, qk pl Fk Fl - = 0, ql qk (4 ) (3 )

(5 ) (6 )

then the classical system (3) is a Hamiltonian system. In this case, Gk and Fk can be represented in the form G k (q , p ) = H , pk Fk (q , p) = - H . qk


FRACTIONAL POWERS OF DERIVATIVES

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If H = H (q , p) is a continuous differentiable function, then the condition (4), (5) and (6) are satisfied. The equations of motion (3) for Hamiltonian system can be written in the form d At = -{H, At }, (7 ) dt where { , } is the Poisson brackets {A, B } = A B A B - . qk pk pk qk

The time evolution of the Hamiltonian system is induced by the Hamiltonian H . It is interesting to obtain fractional generalizations of equations (3) and (7). We will consider here concept of fractional power for L. If L is a closed linear operator with an everywhere dense domain D (L), having a resolvent R(z , L) = (z LI - L)-1 on the negative half-axis, then there exists [16, 17, 18] the operator -(L) = sin


dz z
0

-1

R(-z , L) L

(8 )

defined on D (L) for 0 < < 1. The operator (L) is a fractional power of the operator L. It is known that the linear differential operator (2) is a closable operator [17]. Note that (L) (L) = (L) for , > 0, + < 1. As a result, we obtain the equation d At = -(L) At , (9 ) dt where t is dimensionless variable. This is the fractional equation of motion. We can define a fractional generalization of equation (7) by d At = -({H, ·}) At . (1 0 ) dt Note that ({H, ·}) cannot be presented in the form {H , ·} with a function H . Therefore, the fractional system described by (9) with L = {H, ·} are not Hamiltonian systems. The systems will be called the fractional Hamiltonian systems (FHS). Usual Hamiltonian systems can be considered as a special case of FHS. Note that another fractional generalization of Hamiltonian systems has been suggested in [10]. 3. SOLUTIONS OF FRACTIONAL EQUATIONS OF MOTION 3.1. Cauchy problem for fractional equations. If we consider the Cauchy problem for equation (3) in which the initial condition is given at the time t = 0 by A0 , then its solution can be written in the form At = t A0 .
+


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V. E. TARASOV

The operator t is called the evolution operator. It is not hard to prove that the following properties are satisfied: t s = t+s , (t, s > 0), 0 = I ,

where I A = A for all A M. As a result, the operators t form a semi-group. Then the operator L is called the generating operator, or infinitesimal generator, of the semi-group {t , t 0}. Let us consider the Cauchy problem for fractional equation (9) in which the initial condition is given by A0 . Then its solution can be presented [17, 18] as At () = t A0 , where the operators t , t > 0, form a semi-group which will be called the fractional () semi-group. The operator (L) is infinitesimal generator of the semi-group {t , t 0} that can be presented by 1 (L ) = (-)
() ()

ds s
0

--1

(t

()

- I ).

This is the Balakrishnan equation [16, 17]. 3.2. Properties of fractional evolution operator. Let us consider some proper() ties of temporal evolution described by a fractional semi-group {t , t 0}. (1) The operators t can be constructed in terms of the operators t by the Bo chner-Phillips formula [14, 15, 17]:
() t ()

=
0

dsf (t, s)s

(t > 0).

(1 1 )

Here f (t, s) is defined by f (t, s) = 1 2 i
a+i a-i

dz exp(sz - tz ),

(1 2 )

where a, t > 0, s 0, and 0 < < 1. The branch of z is so taken that Re(z ) > 0 for Re(z ) > 0. This branch is a one-valued function in the z -plane cut along the negative real axis. The convergence of this integral is obviously in virtue of the convergence factor exp(-tz ). By denoting the path of integration in (12) to the union of two paths r exp(-i), and r exp(+i), where r (0, ), and /2 , we can obtain f (t, s) = 1
0

dr exp(sr cos - tr cos())· (1 3 )

· sin(sr sin - tr sin() + ). If we have a solution At of equation (3), then formula (11) gives the solution


At () =
0

ds f (t, s)As ,

(t > 0 )

(1 4 )


FRACTIONAL POWERS OF DERIVATIVES

445

of fractional equation (9). As a result, we can obtain solution of fractional equation by using well-known solutions of usual equations. (2) In classical mechanics, the most important is the class of real operators. Let be a complex conjugation. If t is a real operator on M, then (t A) = t (A ) for all A D (t ) M. A classical observable is a real-valued function. If t is a real operator and A is a real-valued function A = A, then the function At = t A is real-valued, i.e., (t A) = t A. An operator, which is a map from a set of observables into itself, should be real. All possible dynamics, i.e., temporal evolutions of classical observables, should be described by real operators. Therefore the following statement () is very important. If t is a real operator, then t is real. The pro of will follows from the Bo chner-Phillips formula, which gives


(t A) =
0

()

ds f (t, s)(s A) ,

(t > 0).

Using (13), it is easy to see that f (t, s) = f (t, s) is a real-valued function. Then (t A) = t A leads to

(t A) = t (A ) for all A D (t ) M.
()

¯ (3) Let t be a operator on M. An adjoint operator of t is a operator t on M , such that ¯ (t (A)|B ) = (A|t (B )) for all B D (t ) M and some A M . The scalar pro duct on M can be defined by (A|B ) = dq dp [A(q , p)]B (q , p)
R
2n

¯ Then an operator t is called adjoint if ¯ dq dp [(tA)(q , p)] B (q , p) =
R
2n

dq dp [A(q , p)](t B )(q , p).
R
2n

¯ Let us give the basic statement regarding the adjoint operator. If t is an adjoint operator of t , then the operator


¯ () t =
0 ()

¯ dsf (t, s)s ,

(t > 0),

is an adjoint operator of t . We prove this statement by using the Bo chner-Phillips formula: ¯ () ¯ dq dp (t A) B = ds f (t, s) dq dp (s A) B =
R
2n

0

R

2n

=
0

ds f(t, s)
R
2n

dq dp A(s B ) =
R
2n

dq dp A(t B ).

()


446

V. E. TARASOV

¯ The semi-group {t , t > 0} describes a temporal evolution of the distribution function ¯ t (q , p) = t 0 (q , p) by the Liouville equation d ¯ t (q , p) = -Lt (q , p), dt where ¯ L = L + (q , p), (q , p) =
k =1 n

Gk Fk . + qk pk

¯ () If < 0, then the system is called dissipative. The semi-group {t , t > 0} describes the evolution of the density function ¯ () t (, q , p) = t 0 (q , p) by the fractional equation d ¯ t = -(L) t . dt This is the fractional Liouvil le equation. () ¯ (4) It is known that t is a real operator if t is real. Analogously, if t is a ¯ () real operator, then t is real. (5) Let t , t > 0, be a positive one-parameter operator, i.e., t A 0 for A 0. Using the Bo chner-Phillips formula and the property f (t, s) 0 (s > 0), it is easy to prove that
()

t A 0 (A 0), are also positive.

()

i.e. the operators t

4. EXAMPLES OF FRACTIONAL EQUATIONS OF MOTION 4.1. Fractional free motion of particle. Let us consider equation (3) for free particle. Then 12 p H= p , L = {H, ·} = q , 2m m -1 where p is dimensionless variable and m has the action dimension. For A = q , and A = p, equation (3) gives 1 d d qt = pt , pt = 0 . dt m dt The well-known solutions of these equations are qt = q0 + t p0 , m pt = p0 . (1 5 )


FRACTIONAL POWERS OF DERIVATIVES

447

Using these solutions and the Bo chner-Phillips formula, we cab obtain solutions of the fractional equations d 1 qt = - (pq ) qt , dt m in the form qt () = t q0 =
0 ()

d pt = 0 . dt p t ( ) = p 0 ,

(1 6 )

dsf (t, s)qs ,

where qs is given by (15). Then qt () = q0 + where b (t) =
0

1 b (t)p0 , m


pt = p0 ,

dsf (t, s) s.

If = 1/2, then we have t b1/2 (t) = 2 and
0

1 ds e- s

t2 /4s

=

t2 , 2 (1 7 )

t2 p0 , pt = p0 . 2m2 These equations describe a fractional free motion for = 1/2. qt (1/2) = q0 -

1 2 m 2 2 p+ q, (1 8 ) 2m 2 where t and p are dimensionless variables. For A = q , and A = p, equation (3) gives H= 1 d d qt = pt , pt = -m 2 qt . dt m dt The well-known solutions of these equations are p0 sin( t), qt = q0 cos( t) + m pt = p0 cos( t) - m q0 sin( t). (1 9 )

4.2. Fractional equation for harmonic oscillator. Let us consider equation (3) for harmonic oscillator. Then L = {H, ·}, where

(2 0 )

Using (20) and the Bo chner-Phillips formula, we can obtain solution of the fractional equations d d qt = -({H, ·}) qt , pt = -({H, ·}) pt , (2 1 ) dt dt where H is defined by (18). It can be written in the form 1 d qt = - m2 2 q p - pq dt m 1 d pt = - m2 2 q p - pq dt m


qt , pt .




448

V. E. TARASOV

It is easy to see that these equations with = 1 give Eqs. (19). The solutions of fractional equations (21) have the forms


qt () =

() t q0

=
0

dsf (t, s)qs ,

pt () = t p0 =
0

()

dsf (t, s)ps .

(2 2 )

Substitution of (20) into (22) gives qt () = q0 C (t) + p0 S (t), m (2 3 ) (2 4 )

pt () = p0 C (t) - m q0 S (t), where


C (t) =
0

ds f (t, s) cos( s),

S (t) =
0

ds f (t, s) sin( s).

Equations (23) and (24) describe solutions of fractional equations (21) for classical harmonic oscillator. If = 1/2, then C
1/2

t (t) = 2 t (t) = 2



ds
0

cos( s) - e s3/2 sin( s) - e s3/2

t2 /4s

,

S

1/2

ds
0

t2 /4s

.

These functions can be presented through the Macdonald function (see [19], Sec. 2.5.37.1.) such that C
1/2

(t) =

t2 4 t2 4

1/4

e+

i/8

K

-1/2

2e+

i/4

t2 4 t2 4

+ e-

i/8

K

-1/2

2e-

i/4

t2 4 t2 4

,

1/4

S

1/2

(t) = i

e+

i/8

K

-1/2

2e+

i/4

- e-

i/8

K

-1/2

2e-

i/4

,

where > 0, and K (z ) is the Macdonald function [1], which is also called the mo dified Bessel function of the third kind. Note that fractional oscillators are an ob ject of numerous investigations (see for example [20, 21, 22, 23, 24, 25]) because of different applications.


FRACTIONAL POWERS OF DERIVATIVES

449

4.3. Fractional equation for damp linear friction d 1 qt = pt , dt m where < . The solution of (25) has qt = e-
t

ed oscillator. Let us consider oscillator with d pt = -m 2 qt - 2 pt , dt the form 1 p0 sin m 2 - 2t 2 - 2t , . (2 6 ) (2 5 )

q0 cos p0 cos

2 - 2t +

pt = e-

t

2 - 2 t - m q0 sin
2

The fractional equations has the form d qt = - (m dt d pt = - (m dt It is easy to see that these equations Bo chner-Phillips formula, we obtain t qt () = q0 C
p q qt , m p 2 q + 2 p)p - q pt . m with = 1 give Eqs. (25). Using (26) and the he solutions

q + 2 p)p -

,

(t) +

1 p0 S m

,

(t), (t), (2 7 )

p t ( ) = p 0 C where C
,

,

(t) - m q0 S

,

(t) =
0

ds f (t, s) e- t cos( 2 - 2 s), ds f (t, s) e- t sin( 2 - 2 s ). (2 8 )

S

,

(t) =
0

These equations describe solutions of the fractional damped motion of classical harmonic oscillator. REFERENCES
[1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach, New York, 1993). [2] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999). [3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations (Elsevier, Amsterdam, 2006). [4] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, Oxford, 2005). [5] A. Carpinteri, F. Mainardi, (Eds), Fractals and Fractional Calculus in Continuum Mechanics (Springer, Wien, 1997). [6] G. M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport," Phys. Rep. 371 (2002) 461­580. [7] C. Martinez, M. Sanz, The Theory of Fractional Powers of Operators (Elsevier, New York, 2000). [8] V. E. Tarasov, "Fractional derivative as fractional power of derivative," Int. J. Math. 18(3) (2007) 281­299.


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