Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://theory.sinp.msu.ru/~tarasov/PDF/JPA2006-2.pdf Äàòà èçìåíåíèÿ: Wed Jun 14 20:04:11 2006 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 21:32:38 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 33

I

NSTITUTE OF

PHYSICS PUBLISHING

JOURNA

L OF

PHYSICS A: MAT

HEMATICAL AND

GENERAL

J. Phys. A: Math. Gen. 39 (2006) 8409­8425

doi:10.1088/0305-4470/39/26/009

Fractional variations for dynamical systems: Hamilton and Lagrange approaches
Vasily E Tarasov
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia E-mail: tarasov@theory.sinp.msu.ru

Received 7 February 2006, in final form 16 May 2006 Published 14 June 2006 Online at stacks.iop.org/JPhysA/39/8409 Abstract Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler­Lagrange equations are derived. Fractional equations are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives. PACS numbers: 45.20.-d, 45.20.Jj

1. Introduction The theory of derivatives of non-integer order [1, 2] goes back to Leibniz, Liouville, Riemann, Grunwald and Letnikov [2]. Derivatives and integrals of fractional order have found many applications in recent studies in mechanics and physics. In a fairly short period of time the list of such applications becomes long. For example, it includes chaotic dynamics [3, 4], mechanics of fractal media [5­8], quantum mechanics [9, 10], physical kinetics [3, 11­16], plasma physics [17, 18], astrophysics [19], long-range dissipation [20, 21], mechanics of nonHamiltonian systems [22, 23], theory of long-range interaction [24­26], anomalous diffusion and transport theory [3, 27, 28] and many other physical topics. In mathematics and theoretical physics, the variational (functional) derivative is a generalization of the usual derivative that arises in the calculus of variations. In a variation instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function. In this paper, we consider the fractional generalization of variational (functional) exterior derivatives. The main results are derived in sections 4.2, 5.2, 6.2 and 6.3. In sections 2, 3, 4.1, 5.1 and 6.1, brief reviews of fractional derivatives, differential forms, Hamiltonian systems are considered to fix notation and provide convenient references. In section 2, a brief review of differential forms is considered. In section 3, we consider Hamiltonian and fractional Hamiltonian systems [23]. In section 4, we define the fractional variations in Hamilton's
0305-4470/06/268409+17$30.00 © 2006 IOP Publishing Ltd Printed in the UK 8409


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approach to describe the motion. The fractional generalization of stationary action principle is suggested. In section 5, we discuss the fractional variations in Lagrenge's approach to describe the motion, and the fractional generalization of stationary action principle is suggested. In section 6, we consider the generalization of action principle to non-Hamiltonain systems. The fractional equations of motion with friction are discussed. Finally, a short conclusion is given in section 7. 2. Fractional derivatives and differential forms 2.1. Differential forms In this subsection, a brief review of differential forms [29, 30] is considered to fix notation and provide a convenient reference. Definition 1. A differential 1-form = F i (x ) dxi is called an exact 1-form in R n if the vector field F i (x ) can be presented as V , F i (x ) = - xi where V = V(x ) is a continuously differentiable function. (1)

(2)

In this case, the differential 1-form (1) is an exact form = -dV , where V = V(x ) is a continuously differentiable function (0-form). Here d is the exterior derivative [29]. The exterior derivative of the function V is the 1-form dV = dxi V / xi written in a coordinate chart (x1 ,...,xn ). For the k-form k and the l-form l , the exterior derivative obeys the relation (3) d(k l ) = (dk ) l + (-1)k k dl . Here k and l are integers. Note that d d = 0 for any form . If d = 0, then is called a closed form. In mathematics [29], the concepts of closed and exact forms are defined by the equation d = 0 for a given to be a closed form, and = dh for an exact form. It is known that to be exact is a sufficient condition to be closed. In abstract terms the question of whether this is also a necessary condition is a way of detecting topological information by differential conditions. Proposition 1. If a smooth vector field F = ei F i (x ) satisfies the relations F i F j - =0 xj xi on a contractible open subset X of R n , then (1) is the exact form such that V (x ) . = -dxi xi

(4)

(5)

Proof. Let us consider the form (1). The formula for the exterior derivative of (1)is 1 F i F j dxj dxi , d = - 2 xj xi where is the wedge product [29]. Therefore the condition for to be closed is (4). ` If F i = -V / xi , then the implication from exact' to `closed' is a consequence of the permutability of the second derivatives. For the smooth function V = V(x ), the second derivative commute, and equation (4) holds.


Fractional variations for dynamical systems

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2.2. Fractional differential forms If the partial derivatives in the definition of the exterior derivative d = dxi xi are allowed to assume fractional order, then a fractional exterior derivative is defined [31] by d = (dxi ) Di . x Here we use
x dy m f(y ) 1 , (7) -m+1 y m (m - ) 0 (x - y) where > 0, and m is the first whole number greater than or equal to . Equation (7) defines the Caputo fractional derivatives [5, 33­35] of order > 0.

(6)

D f(x ) = x

Definition 2. A differential 1-form = F i (x )(dxi )
i

(8) (9) Di x is a derivative of order .

is called an exact fractional form if the vector field F (x ) can be represented as F i (x ) = -Di V, x where V = V(x ) is a continuously differentiable function, and Using (6) the exact fractional form can be represented as = -d V = -(dxi ) Di V. x Therefore, we have (9). Note that equation (8) is a fractional generalization of the differential form (1). Obviously that fractional 1-form can be closed when the differential 1-form = 1 is not closed. The fractional analogue of proposition 1 has the form Proposition 2. If a smooth vector field F = ei F i (x ) on a contractible open subset X of R satisfies the relations Dj F i - Di F j = 0, x x then the form (8) is an exact fractional 1-form such that = -Di V(x ), x where V(x ) is a continuous differentiable function and Di x V = -F .
i n

(10) (11)

Proof. This proposition is a corollary of the fractional generalization of the Poincare lemma [32]. The Poincare lemma is shown [31, 32] to be true for the exterior fractional derivative. Note that we can generalize the definition of fractional exterior derivative by the equation
n

d =
i =1

(dxi )i Dxii ,

(12)

where = (1 ,2 ,...,n ), and consider the fractional differential 1-forms:
n

=
i =1

i (x )(dxi )i .

(13)

In this case, we can derive equations with derivatives of different orders i . For simplicity, we suppose that all i = .


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3. Hamiltonian systems In this section, a brief review of Hamiltonian systems [29] and fractional Hamiltonian systems [23] is considered to fix notation and provide a convenient reference. 3.1. Definition and properties of Hamiltonian systems Let us consider the canonical coordinates (q1 ,...,qn ,p1 ,...,pn ) in the phase space R 2n . We consider a dynamical system that is defined by the equations dpi dqi = Gi (q , p), = F i (q , p). (14) dt dt The definition of Hamiltonian systems can be realized in the following form [23, 38, 39]. Definition 3. The dynamical system (14) on the phase space R system if = Gi dpi - F i dqi
2n

is called a Hamiltonian (15)

is a closed form, d = 0. A dynamical system is called a non-Hamiltonian system if (15) is non-closed, d = 0. The exterior derivative for the phase space is defined as d = dqi +dpi . qi pi Here and later we mean the sum on the repeated index i from 1 to n. Proposition 3. If the right-hand sides of equations (14) satisfy the conditions Gi Gj - = 0, pj pi Gj F i + = 0, qi pj F i F j - = 0, qj qi (17)

(16)

then the dynamical system (14) is a Hamiltonian system. Proof. In the canonical coordinates (q , p), the vector fields that define the system have the components (Gi ,F i ), which are used in equation (14). Let us consider the 1-form (15). The exterior derivative of (15) is written by d = d(Gi dpi ) - d(F i dqi ). Then d = Gi F i F i Gi dqj dpi + dpj dpi - dqj dqi - dpj dqi . qj pj qj pj Gj F i + qi pj 1 Gj Gi - 2 pi pj 1 F i F j - 2 qj qi (18)

Here is the wedge product. Equation (18) can be presented in an equivalent form d = dqi dpj + dpi dpj + dqi dqj .

Here we use the skew-symmetry of dqi dqj and dpi dpj with respect index i and j . It is obvious that conditions (17) lead to the equation d = 0. Equations (17) are called the Helmholtz conditions [23, 36, 38, 39] for the phase space. Proposition 4. The dynamical system (14) on the phase space R 2n is a Hamiltonian system that is defined by the Hamiltonian H = H(q , p) if the form (15) is an exact form = dH , where H = H(q , p) is a continuous differentiable unique function on the phase space.


Fractional variations for dynamical systems

8413

Proof. Suppose that the form (15)is H H = dH = dpi + dqi . pi qi Then the vector fields (Gi ,F i ) are H H , F i (q , p) = - . (19) Gi (q , p) = pi qi If H = H(q , p) is a continuous differentiable function, then conditions (17) are satisfied, and (14) is a Hamiltonian system. Substitution of (19)into(14)gives dpi dqi H H , (20) = =- dt pi dt qi As the result, the equations of motion are uniquely defined by the Hamiltonian H. 3.2. Fractional Hamiltonian systems Fractional generalization of Hamiltonian systems has been suggested in [23]. The fractional analogue of the form (15) can be defined by = Gi (dpi ) - F i (dqi ) . Let us consider the equations of motion At qi = Gi (q , p), Bt pi = F i (q , p), (22) where At and Bt are the linear (or nonlinear) operators that have derivatives (integer or fractional order) with respect to time. As the simple examples, we can consider the total time derivatives At = Bt = d/dt and equation (14). For the fractional derivatives At = Bt = Dt , Dt qi = Gi (q , p), Dt pi = F i (q , p).
2n

(21)

(23) is called a fractional (24)

Definition 4. The dynamical system (22) on the phase space R Hamiltonian system if (21) is a closed fractional form d = 0,

where d is the fractional exterior derivative. The system is called a fractional nonHamiltonian system if (21) is a non-closed fractional form, i.e., d = 0. The fractional exterior derivative for the phase space R d = (dqi ) Di + (dpi ) D i , q p > 0.
2n

is defined as (25)

Let us consider a fractional generalization of the Helmholtz conditions. Proposition 5. If the right-hand sides of equations (22) satisfy the conditions D j Gi - D i Gj = 0, p p Di Gj + D j F i = 0, q p Dj F i - Di F j = 0, q q (26) then the dynamical system (22) is a fractional Hamiltonian system. Proof. This proposition has been proved in [23]. Proposition 6. The dynamical system (22) on the phase space R 2n is a fractional Hamiltonian system with the Hamiltonian H = H(q , p) if (21) is an exact fractional form = d H, where H = H(q , p) is a continuous differentiable function on the phase space. (27)


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Proof. Suppose that the fractional form (21) is = d H = (dpi ) D i H + (dqi ) Di H. p q Then Gi (q , p) = D i H, p and equation (22) gives At qi = D i H, p Bt pi = -Di H. q (28) These equations describe the motion of fractional Hamiltonian systems. 4. Hamilton's approach 4.1. Hamilton's equations of integer order Let us consider Hamiltonian systems in the extended phase space M 2n+1 = R 1 â R n â R n of coordinates (t ,q ,p). The motion of systems is defined by the stationary states of the action functional S [q, p] = [p q - H(t , q , p)]dt, (29) F i (q , p) = -Di H, q

where H is a Hamiltonian of the system, q = dq/dt and both q and p are assumed to be independent functions of time. In classical mechanics, the trajectory of an object is derived by finding the path for which the action integral (29) is stationary (a minimum or a saddle point). In Hamilton's approach the action functional (29) can be written as S [q, p] = where h = p dq - H dt. (31) The form (31) is called the Poincare­Cartan 1-form or the action 1-form. The Poincare­Cartan form looks like the integrand of the action or the Lagrangian. However, it is a differential form on the extended phase space M 2n+1 of (t ,q ,p), not a function. Once we integrate it over a curve C in M 2n+1 , we get the action S [q, p] =
C

h ,

(30)

h =

B A

[p dq - H(t , q , p) dt ].

(32)

The integration is taken from A to B in the extended phase space M 2n+1 . Now suppose we integrate from A to B along two slightly different paths and take the difference to get a close loop integral. To evaluate this integral we can use Stokes's theorem [29]. In the language of differential forms, Stokes's theorem is written as =
M M

d.

(33)

Here, M is an n-dimensional compact orientable manifold with boundary M and is a (n - 1)-form; `d' is its exterior derivative. Note that M can be a submanifold of a larger space, so that Stokes's theorem actually implies a whole set of relations including the familiar Gauss and Stokes laws of ordinary vector calculus. Applying equation (33) to the difference of actions computed along two neighbouring paths with (q , t ) fixed at the endpoints, we get S [q, p] =


dh =


(dp dq - dH dt),

(34)


Fractional variations for dynamical systems

8415

where denotes the surface area in the extended phase space bounded by the two paths from A to B. The principal of stationary action states that S = 0 for small variations about the true path, with (q , t ) fixed at the end points. This will be true for arbitrary small variations, if and only if dh = 0 for the tangent vector along the extremal path. We can consider the exterior derivative of Poincare­Cartan 1-form, and derive the equations of motion from the condition dh = 0. Using this condition, we get Hamilton's equations of motion. This condition is equivalent to the stationary action principle S [q, p] = 0. Proposition 7. The exterior derivative of the Poincare­Cartan 1-form (31) is defined by the equation dh = [Dt p + Dq H ]dt dq - [dq - Dp H dt ] dp, where , Dq = , Dp = . t q p Proof. The exterior derivative of the form (31) can be calculated from the equation Dt = dh = d(p dq) - d(H dt) = Dt p dt dq + Dq p dq dq + Dp p dp dq - Dt H dt dt - Dq H dq dt - Dp H dp dt. Using dt dt = 0, dp dt = -dt dp, and Dq p = 0, we get dh = [Dt p + Dq H ]dt dq - [Dp p dq - Dp H dt ] dp. The relation Dp p = 1gives dh = [Dt p + Dq H ]dt dq - [dq - Dp H dt ] dp. (39) (38) (36) (35)

(37)

Stationary action principle in Hamilton's approach. The trajectory of a Hamiltonian system can be derived by finding the path for which the Poincare­Cartan 1-form h is closed, i.e., dh = 0. Using the stationary action principle (40), we get the equations of motion dq - Dp H dt = 0, Dt p = -Dq H. (41) As the result, we obtain dq dp = Dp H, = -Dq H, dt dt which are the well-known Hamilton's equations. 4.2. Fractional Hamilton's equations The fractional generalization of the form (31) can be defined by h, = p(dq) - H(dt) . (43) Note that h, is a fractional 1-form that can be called a fractional Poincare­Cartan 1-form or fractional action 1-form. We can consider the fractional exterior derivative of the form (43), and use d h, = 0 to obtain the fractional equations of motion. (40)

(42)


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VETarasov

Proposition 8. The fractional exterior derivative of the fractional form (43)is
d h, = Dt p + Dq H (dt) (dq) -

p 1- (dq) - Dp H(dt) (2 - )



(dp) .

(44)

Proof. The fractional exterior derivative of the form (43) is calculated by using the rule


D (f g ) = x
k =0

k

D-k f x

kg , x k

and the relation k [(dx) ] = 0 (k x k For example, we have


1).

d [Ai (dxi ) ] =
k =0

(dxj )

k


Dj-k Ai x 0

k (dxi ) k xj



= (dxj ) (dxi ) where (-1)k-1 (k - ) . = k (1 - ) (k +1) As the result,

Dj Ai = Dj Ai (dxj ) (dxi ) , x x

d h, = d (p(dq) ) - d (H (dt) ) = Dt p (dt) (dq) + Dq p (dq) (dq) + Dp p (dp) (dq) - Dt H (dt) (dt) - Dq H (dq) (dt) - Dp H (dp) (dt) .



(45)

Here D are the Riesz or Caputo fractional derivatives [5, 33­35]. Note that the Riemann­ Liouville fractional derivative leads us to dependence of independent coordinates [22, 23]:
Dq p = pDq 1 = 0.

(46)

Therefore the fractional equations are more complicated for Riemann­Liouville derivatives. Using (dt) (dt) = 0,(dp) (dt) = -(dt) (dp) and Dq p = 0 for Riesz and Caputo derivativs, we can rewrite equation (45) in the form
d h, = Dt p + Dq H (dt) (dq) - Dp p (dq) - Dp H (dt)

(dp) .

(47)

Substitution of p1- (2 - ) into equation (47) gives (44).
Dp p =

(48)

Fractional action principle in Hamilton's approach. The trajectory of a dynamical system can be derived by finding the path for which the fractional Poincare­Cartan 1-form h, is a fractional closed form, i.e., d h, = 0. (49)

Here, we consider only fractional Hamiltonian systems. The non-Hamiltonian systems are considered in section 6.


Fractional variations for dynamical systems

8417

Using (44) and (49), we get p 1- (dq) - Dp H(dt) = 0, (2 - ) As the result, we obtain dq dt
Dt p = -Dq H.

(50)

= (2 - )p

-1

Dp H,

Dt p = -Dq H.

(51)

These equations are the fractional generalization of Hamilton's equations. For the fractional Poincare­Cartan 1-form h, = p (dq) - H(dt) , equations (51)are dq dt


(52)

=

( +1 - ) p ( +1)

-

Dp H,

Dt p = -Dq H.

(53)

Note that we cannot use the rule of differentiating composite functions for fractional derivative Dt p . Therefore equations (53) with = 1 are more complicated than equation (51). 5. Lagrange's approach 5.1. Lagrange's equations of integer order Suppose L(t , q , v ) is a Lagrangian of dynamical system, where q is coordinate and v is the velocity. Let us consider the variational problem on the extremum of the action functional S0 [q, v ] = L(t , q , v ) dt, (54)

under the additional condition q = v, (55)

where both q and v are assumed to be independent functions of time. In this case, p play the role of independent Lagrange multipliers. Obviously, the indicated variational problem is equivalent to the problem on the extremum of the action, S [q, v , p] = [L(t , q , v ) + p(q - v)]dt, (56)

where already all the variables q, v , p have to be varied. The corresponding Lagrange's equations are q = v, p= L , q p= L . v (57)

We can introduce the extended Hamiltonian in the space of variables (t ,q ,p,v ) as H (t ,q ,p,v ) = pv - L(t , q , v ). The corresponding extended Poincare­Cartan 1-form is defined by h = p dq - H dt = p dq + L dt - pv dt. Proposition 9. The exterior derivative of the form (59) is defined by the equation dh = [Dt p - Dq L]dt dq - [dq - v dt ] dp - [p - Dv L]dv dt. (60) (59) (58)


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Proof. The exterior derivative of (59)is d
h

= d(pd q ) - d(H dt) = Dt p dt dq + Dq p dq dq + Dp p dp dq + Dv p dv dq (61) - Dt H dt dt - Dq H dq dt - Dp H dp dt - Dv H dv dt. (62)

Using dt dt = 0,d q dt = -dt dq and Dq p = 0,Dv p = 0, equation (61)gives dh = [Dt p + Dq H ]dt dq - [Dp p dq - Dp H dt ] dp - Dv H dv dt. From (58), we get Dq H = Dq [pv - L] = -Dq L, Dp H = Dp [pv - L] = vDp p - Dp L(t , q , v ) = vDp p = v, Dv H = Dv [pv - L] = pDv v - Dv L = p - Dv L. As the result, we obtain (60). Stationary action principle in Lagrange's approach. The trajectory of a dynamical system can be derived by finding the path for which the form (59) is closed , i.e., dh = 0. From equations (60) and (63), we get Dt p - Dq L = 0, dq - v dt = 0, p - Dv L = 0. (64) It is easy to see that equation (64) coincides with Lagrange's equations (57) that can be presented as d Dq L - Dv L = 0. (65) dt v =q As the result, we obtain L d L = 0, i = 1,...,n, - qi dt q i which are the Euler­Lagrange equations for the Lagrangian system. 5.2. Fractional Lagrange's equations Suppose L(t , q , v ) is a Lagrangian of the system, and the extended Hamiltonian is defined by H (t ,q ,p,v ) = pv - L(t , q , v ), where is a positive power. Let us define h


(63)

(66)

(67) (68)

= p(dq) - H (dt) = p(dq) + L(dt) - pv (dt) ,

which is a fractional generalization of the extended Poincare­Cartan 1-form (59). Proposition 10. The fractional exterior derivative of the fractional 1-form (68) is defined by d h
= Dt p - Dq L (dt) (dq) - Dp p [(dq) - v (dt) ] (dp) - pDv v - Dv L (d v ) (dt) .

(69)


Proof. The fractional exterior derivative of the form (68) gives d h
= d [pd q ] - d [H (dt) ] = Dt p (dt) (dq) + Dq p (dq) (dq) + Dp p (dp) (dq) + Dv p (dv) (dq) - Dt H (dt) (dt) - Dq H (dq) (dt) - Dp H (dp) (dt) - Dv H (dv) (dt) .

(70)


Fractional variations for dynamical systems

8419

Using (dt) (dt) = 0,(dq) (dt) = -(dt) (dq) and Dq p = 0,Dv p = 0 for Riesz or Caputo fractional derivatives, we can rewrite (68)as d h
= Dt p + Dq H (dt) (dq) (dp) - Dv H (dv) (dt) . - Dp p (dq) - Dp H (dt)

(71)

Using the extended Hamiltonian (67) and properties of Riesz and Caputo derivatives, we have
Dq H = Dq [pv - L] = -Dq L, Dp H = Dp [pv - L] = v Dp p - Dp L(t , q , v ) = v Dp p, Dv H = Dv [pv - L] = pDv v - Dv L.

As the result, we obtain (69). Fractional action principle in Lagrange's approach. The trajectory of fractional dynamical systems can be derived by finding the path for which the fractional extended Poincare­Cartan 1-form (68) is a closed form, i.e., d h
t

= 0. (dq) - v (dt) = 0, ( +1) v ( +1 - )
- pDv v - Dv L = 0.

(72) (73)

Using (69) and (72), we have
D p - Dq L = 0,

From the relation
Dv v =

,

(74)

where > -1, we get (dq) ( +1 - ) v , p= (dt) ( +1) Substitution of the third equation from (75) into the first one gives ( +1 - ) - Dq L - Dv L v =(q) = 0. Dt v ( +1) Equation (76) has the dependence
Dt p = Dq L,

v =

-

Dv L.

(75)

(76)

v = (q) .

(77)

It is easy to see that equation (76) looks unusually even for = 1. Therefore we use = for the Hamiltonian (67) and the form (68). Using = in equations (67), (68) and 1 , (78) Dv v = (2 - ) we have the fractional extended Lagrange's equations (dq) (Dt p) = Dq L, v = , p = (2 - )Dv L. (79) (dt) Substituting the third equation from (80) into the first one, we obtain
Dq L - (2 - )Dt Dv L v =q

= 0,

(80)

that is fractional Euler­Lagrange equations. As the result, the fractional equations of motion for Lagrangian systems are presented by ^i E L(t , q , q) = 0, i = 1,...,n, (81)


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VETarasov

where
^i E = Dqi - (2 - )Dt Dq i .

(82)

For = 1, equations (81) are the Euler­Lagrange equations (66). 6. Fractional non-Hamiltonian systems 6.1. Helmholtz conditions In this subsection, we consider the brief review of Helmholtz conditions [36, 37] to fix notation and provide a convenient reference. The Helmholtz equations give the necessary and sufficient conditions for equations to be the Euler­Lagrange equations that can be derived from the stationary action principle. Proposition 11. The necessary and sufficient conditions for Fi (t , q , q , ...,q Fj Fi - - qj qi
i (m) qj (N )

) = 0, dk dt k

i = 1,...,n F

(83)

to be equations that can be derived from the stationary action principle are
N

(-1)
k =1 k

k

j )

qi(k

= 0,

(84)

F

N

-

(-1)
k =m

k dk-m m dt k-m

F

j )

qi(k

= 0,

m = 1,...,N

(85)

where qi(k) = dk qi /dt k ,i,j = 1,...,n and k k! = , m m!(k - m)! d = + dt t
N

qi(k
k =1

)



k

qi(k-1)

.

(86)

Proof. This proposition is proved in [37]. For simple example, let us consider the equations Fi (t , q , q) = 0. Conditions (84) and (85) have the form Fj Fi d Fj = 0, - + qj qi dt q i Fi Fj + = 0, qj qi where d = + qi . dt t qi Equations (89)gives 2 Fi = 0, i,j , = 1,...,n. qj q These conditions are satisfied for the linear dependence Fi with respect to q , i.e., Fi = Cij (t , q )q j + Di (t , q ) = 0, i = 1,...,n. (90) i, j = 1,...,n, (87)

(88) (89)

(91)

(92)


Fractional variations for dynamical systems

8421

Corollary 1. The necessary and sufficient conditions to derive equations (92) from the stationary action principle have the form Cij = -Cji , Cij Cj Ci + + = 0, q qi qj Cij Di Dj + = 0. - t qj qi (93) (94)

(95)

Proof. Substitution of equation (92) into equations (88) and (89) yields (93), (94) and (95). Corollary 2. The necessary and sufficient conditions for ¨ Fi (t , q , q, q) = 0, i = 1,...,n (96) to be equations that can be derived from stationary action principle are Fj Fi - = 0, ¨ ¨ qj qi Fi Fj d + -2 qj qi dt Fj Fi d - + qj qi dt where d ¨ + qi . = + qi dt t qi qi (100) Fj ¨ qi F q
j i

(97)

= 0, - d2 dt 2 Fj ¨ qi = 0,

(98)

(99)

Note that using equation (98) condition (99) can be rewritten in the more symmetric form Fj Fi 1d - - qj qi 2 dt 6.2. Non-Lagrangian systems Definition 5. A dynamical system is called non-Lagrangian system if the equations of motion (83) cannot be represented in the form
N

F Fi - qj q

j i

= 0.

(101)

(-1)
k =0

k

dk L(t , q , q, . . . , q dt k q (k)
(N )

(N )

) = 0,
(k )

(102)

with some function L = L(t , q , q , ...,q

), where q

= dk q/dt k .

It is well known that the equations of second order cannot be presented as ^ E i L(t , q , q) = 0, i = 1,...,n, ^ i is the Euler­Lagrange operator where E ^ E i = Dqi - Dt Dq i .

(103) (104)


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In the general case, Lagrange's equations have the additional term Qi (t , q , q) which is a ^ generalized non-potential force. This force cannot be presented as Qi = E i U for some function U = U(t , q , q). In general, the Euler­Lagrange equations [44]are ^ E i L(t , q , q) + Qi = 0. (105)

If we consider non-potential forces and non-Lagrangian systems, then the non-holonomic variational equation suggested by L I Sedov [40­43] should be used instead of the stationary action principle. 6.3. Non-Hamiltonian systems and friction force In general, the phase-space equations of motion cannot be presented in the form q i = Dpi H, pi = -Dqi H, pi = -Dqi H + F i (t ,q ,p), (106) where H = H(t , q , p) is a smooth function. Hamilton's equations are written as q i = Dpi H + Gi (t ,q ,p), (107) where H = H(t , q , p) is a Hamiltonian of the system. For example, H(t , q , p) = T(p) + U(q ), where T(p) is a kinetic energy and U(q ) is a potential energy of the system. The functions Gi (t ,q ,p) and F i (t ,q ,p) describe the non-potential forces which act on the system. For mechanical systems, we can consider Gi (t ,q ,p) = 0. If the functions Gi (t ,q ,p) and F i (t ,q ,p) do not satisfy the Helmholtz conditions (17), then (107) is a non-Hamiltonian system. In general, the exterior derivative of the Poincare­Cartan 1-form is not equal to zero (dh = 0). This derivative is equal to differential 2-form that is defined by non-potential forces = F i (t ,q ,p) dt dqi - Gi (t ,q ,p) dt dp = -pi dt dqi .
i i

(108)

for the non-Hamiltonian system (107). For example, the linear friction force F = -pi gives (109)

Proposition 12. The differential 2-form of non-potential forces is a non-closed form. Proof. If differential 2-form is a closed form (d = 0) on a contractible open subset X of R 2n , then the form is the exact form such that a function h = h(t , q , p) exists, and = dh. In this case, we have a new Poincare­Cartan 1-form h = h + h, such that d = 0, and the system is Hamiltonian. There is a generalization of the stationary action principle for the systems with nonpotential forces. Action principle for non-Hamiltonian systems. The trajectory of a non-Hamiltonian system can be derived by finding the path for which the exterior derivative of the action 1-form (31) is equal to the non-closed 2-form (108), i.e., dh = . (110)

Equations (35), (108) and (110) give the equations of motion (107) for the nonHamiltonian system.


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6.4. Fractional generalization of non-Hamiltonian systems Let us define a fractional generalization of the form (108) by = F i (t ,q ,p)(dt) (dqi ) - Gi (t ,q ,p)(dt) (dpi ) . (111)

This form allows us to derive fractional equations of motion for non-Hamiltonian systems. Fractional action principle for non-Hamiltonian systems. The trajectory of a fractional system subjected by non-potential forces can be derived by finding the path for which the fractional exterior derivative of the fractional action 1-form (43) is equal to the non-closed fractional 2-form (111), i.e. d h = . Using (44), (111) and (112), we get p 1- (dq) - Dp H(dt) = G(t , q , p)(dt) , (2 - ) As the result, we obtain dq dt
Dt p = -Dq H + F(t , q , p). (113)

(112)

= (2 - )p

-1

Dp H + G(t , q , p),

Dt p = -Dq H + F(t , q , p).

(114)

These equations are the fractional generalization of equations of motion for non-Hamiltonian systems. 7. Conclusion In this paper, we define a fractional exterior derivative for calculus of variations. Hamiltonian and Lagrangian approaches are considered. Hamilton's and Lagrange's equations with fractional derivatives are derived from the stationary action principles. We prove that fractional equations can be derived from the action which has only integer derivatives. Derivatives of non-integer order appear by the fractional variation of Lagrangian and Hamiltonian. Application of fractional variational calculus can be connected with a generalization of variational problems. The gradient systems (GS) form a restricted class of ordinary differential equations. Equations for GS can be defined by one function--potential. Therefore the study of GS can be reduced to research of potential. As a physical example, the ways of some chemical reactions are defined from the analysis of potential energy surfaces [45, 46]. The fractional gradient systems (FGS) have been suggested in [23]. It was proved that GS are a special case of such systems. FGS include a wide class of non-gradient systems. For example, the Lorenz and Rossler equations are fractional gradient systems [23]. Therefore the study of non-gradient systems which are FGS can be reduced to research of potential. We can assume that the ways of some chemical reactions with dissipation and systems with deterministic chaos can be considered by the analysis of fractional potential surfaces. Let us note the interesting property of potential surfaces for systems with strange attractors. The surfaces of the stationary states of the Lorenz and Rossler equations separate the threedimensional Euclidean space into some number of areas [23]. We have eight areas for the Lorenz equations and four areas for the Rossler equations. This separation has the interesting property: all regions are connected with each other [23]. Beginning movement from one of the areas, it is possible to appear in any other area, not crossing a surface. Any two points from different areas can be connected by a curve which does not cross a surface.


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The fractional variations can be used to define the fractional generalization of gradient type equations that have wide application for the description dissipative structures [47, 48]. The fractional gradient type equations are generalization of FGS [23] from ordinary differential equations into partial differential equations. We plan to realize this generalization in the next paper by using the de Donder­Weyl Hamiltonian and the Poincare­Cartan n-form. References
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