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Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1860-1878 www.elsevier.com/locate/cnsns

Conservation laws and Hamilton's equations for systems with long-range interaction and memory
Vasily E. Tarasov
a

a,b,*

, George M. Zaslavsky

a,c

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA b Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia c Department of Physics, New York University, 2-4 Washington Place, New York, NY 10003, USA Received 25 April 2007; received in revised form 6 May 2007; accepted 6 May 2007 Available online 24 May 2007

Abstract Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether's theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time-space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form. ã 2007 Elsevier B.V. All rights reserved.
PACS: 05.45.Ða; 45.10.Hj; 11.10.Lm Keywords: Fractional differential equations; Noether's theorem; Long-range interaction; Long-term memory; Conservation laws

1. Introduction Different physical phenomena such as anomalous transport or random walk with infinite moments [1,2], dynamics of porous media [3,4], continuous time random walk [5-7], chaotic dynamics [8] (see also reviews [9,10]) can be described by equations with fractional integro-differentiation. Despite of fairly deep and comprehensive results in fractional calculus (see [11-14]) a possibility of their applications to physics needs to develop specific physical tools such as extension of fractional calculus to the areas as multi-dimension [11,17], multi-scaling [15,16], variational principles [18,19]. In this paper, we concentrate on two problems important for numerous physical applications: conservation laws and Hamiltonian type equations, both obtained from the corresponding fractional action principles.
* Corresponding author. Address: Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia. Tel.: +7 495 939 5989; fax: +7 495 939 0397. E-mail address: tarasov@theory.sinp.msu.ru (V.E. Tarasov).

1007-5704/$ - see front matter ã 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2007.05.017


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In Section 2, we derive the Noether's theorem for a Lagrangian that includes non-local space-time densities. The Noether's theorem was also discussed in [23,24]. Our new derivation shows in an explicit way how fractional derivative in time emerges from the specific type of the memory function, and how fractional derivative in space is related to a specific long-distance potential of interaction (Section 3.) In Section 4, these results are applied to a chain of nonlinear oscillators that is a subject of great interest in statistics and dynamics [25,26]. Finally, at Section 5, we derive a specific case of fractional Hamilton's equations. Different steps in this direction were performed in [29-31]. We consider the Lagrangian density as a functional without fractional derivatives but, instead, the differential 1-form has fractional differentials. Some examples are given for this type of systems. The main feature of this paper is the consideration of fractional type differentials or derivatives in both space-time coordinates. 2. Noether's theorem for long-range interaction and memory 2.1. Action and Lagrangian functionals Let us consider the action functional Z Z S Íu Ì d2 x d2 y L?u?xî; u?y î; ou?xî; ou?y îî;
R R

? 1î

where x Ì ?t; rî, t is time, r is coordinate, and y Ì ?t0 ; r0 î, ou?xî Ì ?ot u?t; rî; or u?t; rîî. The integration is carried out over a region R of the two-dimensional space R2 to which x belong. The field u(x) is defined in the region R of R2 . We assume that u(x) has partial derivatives o0 u? x î Ì ou? t ; r î ; ot o1 u? x î Ì ou? t ; r î ; or

which are smooth functions with respect to time and coordinate. Here L?u?xî; u?y î; ou?xî; ou?y îî is generalized density of Lagrangian. If L?u?xî; u?y î; ou?xî; ou?y îî Ì L?u?xî; ou?xîîd?x Ð y î; then we have the usual action functional Z d2 xL?u?xî; ou?xîî: S Íu Ì
R

? 2î

The variation of the action (1) is Z Z oL oL oL oL 2 2 h?xî? h?y î? dS Íu; h Ì ol h?xî? ol h? y î ; dx dy ou? x î o?ol u?xîî ou? y î o?ol u?y îî R R where l = 0, 1, ol = o/oxl and L Ì L?u?xî; u?y î; ou?xî; ou?y îî and h(x) = du(x) is the variation of the field u. The variation (3) can be presented as Z Z oL s oL s h?xî? ol h? x î ; d2 x d2 y dS Íu; h Ì ou? x î o?ol u?xîî R R where Ls Ì L?u?xî; u?y î; ou?xî; ou?y îî ? L?u?y î; u?xî; ou?y î; ou?xîî: We can define the functional Z 1 LÍx; u; ou Ì d 2 y Ls ; 2R

? 3î

? 4î

? 5î


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which will be called the Lagrangian functional. For (2), the functional (5) is equal to the usual Lagrangian density L?u; ouî. Using notations d LÍ x ; u ; o u Ì du?xî Z
R

d2 y

oL s ; ou? x î

dLÍx; u; ou Ì d?ol u?xîî

Z
R

d2 y

o Ls ; o?ol u?xîî

? 6î

the variation (4) is Z dLÍx; u; ou dLÍx; u; ou 2 h?xî? dS Í u; h Ì ol h? x î : dx du? x î d?ol u?xîî R

? 7î

The structure of (7) is non-local, i.e., it includes, through (5) and Ls a possibility of the long-term memory and long-range interaction. Let us derive the generalization of the Noether's theorem for this case in the way analogical to the local case (2). 2.2. Equations of motion First, we separate the variation that is linked in the variation of coordinates xl ! x0l Ì xl ? dxl ; and the variation caused by a change of the form of u, u?xî ! u0 ?xî Ì u?xî? du?xî: The variation h(x) = du(x) of u(x) that is not due to the variation in coordinates is called local. The total variation is Du?xî Ì u0 ?x0 îÐ u?xî Ì du?xî??ou=oxl îdx
l

? 8î

? 9î

in the first approximation with respect to dx. Let us consider the variation of (1) as Z Z 20 00 0 0 d x LÍx ; u ; o u Ð d2 xLÍx; u; ou: dS Í u; h Ì
R R

?10î

The elements of the two-dimensional volume in new and old coordinates are related through the formula d2 x0 Ì J ?x0 =xîd2 x; where J(x 0 /x) = detjox 0 l/oxmj is the Jacobian of the transformation. Using the well-known relation det A Ì exp Tr ln A; and the linear approximation oxl0 Ì dl ? om ? dx l î ; m ox m we get J ?x0 =xî Ì 1 ? ol ?dxl î: Here dl is the Kronecker symbol. m For the variation (10), we get Z Z dL dL du ? d?ol uî? ol ?Ldxl î ; dS Í u; h Ì d2 x?dL ? Lol ?dxl îî Ì d2 x du d? ol uî R R where the variations of L are defined in (6). Using d(olu) = ol(du), and dL dL dL ol du Ì ol du Ð o l du; d? ol uî d? ol uî d? ol uî ?11î

?12î


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we can rewrite (12) as Z Z dL dL dL 2 2 l Ð ol du : dx dS Íu; h Ì du ? d x ol L dx ? du d?ol uî d? ol uî R R The Gauss theorem gives Z Z dL dL du Ì du : d2 xol Ldxl ? dS l Ldxl ? d? ol uî d?ol uî R oR

?13î

?14î

We assume that at the boundary of the domain of integration the function u(x) is selected in a definite manner (the boundary condition). Then stationary (a minimum or a saddle point) values of S[u] from variational equations dS Íu; h Ì 0 with ÍduoR Ì 0; Ídx
oR

Ì0

at the boundary oR of the domain of integration, constitutes the necessary and sufficient condition for the real evolution of the field, that is, that u = u(x) represents the true dynamics under the given boundary conditions. The stationary action principle gives d LÍ x ; u ; o u dLÍx; u; ou Ð ol Ì 0; du? x î d?ol u?xîî ?15î

where the variations of L are defined in (6). This equation is the Euler-Lagrange equation for Lagrangian functional LÍx; u; ou. Let us consider three special cases of Eq. (15). (1) The absence of the memory and long-range interaction means that L?u?xî; u?y î; ou?xî; ou?y îî Ì L?u?xî; ou?xîîd?x Ð y î: Then (15) gives the usual Euler-Lagrange equation oL ? u; ouî oL ? u; ouî Ð ol Ì 0: ou? x î o?ol u?xîî (2) If the generalized Lagrangian density is L?u?xî; u?y î; ou?xî; ou?y îî Ì L?u?xî; ou?xîîc1 ?D; rîd?x Ð y î; where c1 ?D; rî Ì jrjDÐ1 C?Dî ? 0 < D < 1î ?16î

for a medium distributed on R1 with the fractional Hausdorff dimension D, then Eq. (15) has the form oL?u; ouî oL ? u; ouî Ð ol c1 ?D; rî c1 ?D; rî Ì 0: ?17î ou?xî o?ol u?xîî This is Euler-Lagrange equation for the field u?xî Ì u?t; rî in fractal medium. Examples of the field (wave) equations for fractal medium string and fractional hydrodynamics are considered in [33]. For example, 1 1 L?u?xî; ou?xîî Ì ?ot u?t; rîî2 Ð v2 ?or u?t; rîî2 ; 2 2 leads to the equation c1 ?D; rîo2 u?t; rîÐ v2 or ?c1 ?D; rîor u?t; rîî Ì 0; t that describes the propagation waves in fractal medium.


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(3) Consider the action functional Z Z 1 1 2 2 d x d y ot u?xîg0 ?x; y îot0 u?y îÐ or u?xîg1 ?x; y îor0 u?y îÐ V ?u?xî; u?y îî ; S Í u Ì 2 2 R R

?18î

where the kernels g0 ?x; y î and g1 ?x; y î are responsible for non-local time-coordinates dynamics and V is non-local interaction potential. Then the Lagrangian functional (5) is Z
R

LÍ x ; u ; o t u ; o r u Ì where

1 1 d y ot u?xîK 0 ?x; y îot0 u?y îÐ or u?xîK 1 ?x; y îor0 u?y îÐ U ?u?xî; u?y îî ; 2 2
2



?19î

1 K 0 ?x; y î Ì Íg0 ?x; y î? g0 ?y ; xî; 2 1 K 1 ?x; y î Ì Íg1 ?x; y î? g1 ?y ; xî; 2 U ?u?xî; u?y îî Ì V ?u?xî; u?y îî ? V ?u?y î; u?xîî: Assume that U ?u?xî; u?y îî Ì U ?u?xîîd?x Ð y î: In this case, the Euler-Lagrange functional Eq. (15) has the form (see also [22]) Z Z oU ?u?xîî Ì 0: d2 y ot K 0 ?x; y îot0 u?y îÐ d2 y or K 1 ?x; y îor0 u?y î? ou? x î R R

?20î

?21î

It is an integro-differential equation, which allows us to derive field equations for different cases of the kernels K 0 ?x; y î and K 1 ?x; y î. In the absence of memory and for local interaction the kernels (20) are defined at the only instant t and point r, i.e., K 0 ? x ; y î Ì g 0 d? x Ð y î ; K 1 ? x ; y î Ì g 1 d? x Ð y î oU ?u?t; rîî Ì 0: ou?t; rî with some constants g0 and g1. Then Eq. (21) gives g0 o2 u?t; rîÐ g1 o2 u?t; rî? t r

For example g0 = g1 = 1, when U ?u?t; rîî Ì Ð cos u?t; rî; we get the sine-Gordon equation o2 u?t; rîÐ o2 u?t; rî? sin u?t; rî Ì 0: t r 2.3. Noether's current Let us derive the Noether's The second integral of (13) Z dL d d2 xol Ldxl ? d? ol uî R dL ?om uîÐ dl L; m d?ol uî current by using the action variation (13). can be presented as Z ! dL dL uÌ Ídu ??om uîdxm Ð ?om uîÐ dl L dxm : d2 xol m d? ol uî d? ol uî R ?22î

?23î

Using the total variation (9), Eq. (23), and the energy-momentum tensor hl Ì m ?24î


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1865

the variation (13) yields Z Z dL dL dL 2 2 l m Ð ol D u Ð hm dx : dx dS Íu; h Ì du ? d x ol du d?ol uî d? ol uî R R

?25î

Let us consider a continuous (topological) group of coordinate transformation x ! x0 Ì x0 ?x; aî, and let the field function u = u(x) admits the representation of this group: x ! x0 ; u?xî ! u0 ?x0 î: ?26î

The invariance of functional S[u] with respect to (26) means that Z Z d2 x0 LÍx0 ; u0 ; ou0 =ox0 Ì d2 xLÍx; u; ou=ox:
R R

The transformation (26) constitutes a group. Therefore, infinitesimal forms of transformations (26) are Dxm Ì X m das ; s D u Ì Y s da
s

?m Ì 0; 1; s Ì 1; ... ; mî;

?27î

where X m and Ys are the generators of the group of transformation correspondingly in the coordinate and the s field representations. The index s Ì 1; .. . ; m is defined by the representation of the group. For simple examples of transformation of the coordinate, time or scalar field, we have Dxm Ì X m da Du Ì Y da: ?m Ì 0; 1î; ?28î ?29î

Noether's theorem states that every continuous transformation of coordinate (28) and field function (29), which ensures that the variation of the action is zero admits a conservation law in the form of a continuity equation [27,28]. Substitution of (28), (29), and (15) into (25) gives Z dL Y Ð hl X m da: d2 xol dS Íu; h Ì ?30î m d? ol uî R In view of the fact that the variation of the parameter, da, is arbitrary, from (30) we get the conservation law ol J l Ì 0 ; where Jl Ì dL Y Ð hl X m d? ol uî
m

?31î

?32î

is the Noether's currents and hl is defined in (24). Eq. (31) means that there exists conservation law. Non-trivm iality of Eqs. (31) and (32) is that L and hl have non-local interaction and memory. m 3. Application of the Noether's theorem for long-range interaction and long-term memory 3.1. Lagrangian functional and energy-momentum tensor For l Ì 0; 1, x0 = t, and x1 = r, the Euler-Lagrange functional Eq. (15) is dLÍx; u; ot u; or u o dLÍx; u; ot u; or u o dLÍx; u; ou Ð Ð Ì 0: du? x î ot d?ot u?xîî or d?or u?xîî ?33î

The time and space variables in action can be separated to consider the field with power-law memory and long-range interaction. Let K 0 ?x; y î and K 1 ?x; y î have the form K 0 ?x; y î Ì d?r Ð r0 îK0 ?t; t0 î; K 1 ?x; y î Ì d?t Ð t îK1 ?r; r î;
0 0

?34î ?35î


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and U ?u?xî; u?y îî Ì U ?u?xîîd?x Ð y î; where x Ì ?t; rî, and y Ì ?t0 ; r0 î. Then the Lagrangian functional is Z Z 1 1 dt0 K0 ?t; t0 îot uot0 u Ð dr0 K1 ?r; r0 îor uor0 u Ð U ?uî: LÍx; u; ot u; or u Ì 2 2 ?36î

?37î

To present the long-term memory and long-range interaction, consider the kernels K0 ?t; t0 î and K1 ?r; r0 î in the power-law forms K1 ?r; r0 î Ì K1 ?jr Ð r0 jî Ì and K0 ?t; t î Ì with M?t Ð t0 î Ì 1 1 C?1 Ð bî ?t Ð t0 î
b 0

Ðg 1 cos?pa=2îC?2 Ð aî jr Ð r0 j

aÐ1

? 1 < a < 2î ;

?38î

&

M?t Ð t0 î; 0;

0 < t0 < t; t 0 > t ; t 0 < 0;

?39î

? 0 < b < 1î :

?40î

Then the variational derivatives of Lagrangian functional are dL oU ?u?t; rîî ÌÐ ; du ou? t ; r î dL Ì C Db u?t; rî ?0 < b < 1î; d?ot uî 0 t dL oaÐ1 Ì Ðg u? t ; r î d?or uî ojrjaÐ1 ? 1 < a < 2î ; ?41î ?42î ?43î

where C Db is Caputo derivative [14,12] defined by 0t Zt 1 ds on u? s ; r î Cb 0 Dt u?t ; r î Ì bÐn?1 C?n Ð bî 0 ?t Ð sî os n

?n Ð 1 < b < nî:

Substitution of (41)-(43) into (33) gives the fractional field Euler-Lagrange equation
C 0

Dtb?1 u?t; rîÐ g

oa oU ?u?t; rîî Ì 0 ? 1 < a < 2; 0 < b < 1î : a u?t ; rî? ou? t ; r î ojrj oU ?u?t; rîî Ì 0: ou? t ; r î

?44î

For a = 2 and b = 1, Eq. (44) is the usual field equation o2 u?t; rîÐ go2 u?t; rî? t r ?45î

For the potential U ?u?t; rîî Ì Ð cos u?t; rî, Eq. (45) gives the sine-Gordon equation, o2 u?t; rîÐ o2 u?t; rî? sin u?t; rî Ì 0; t r and Eq. (44) is a spatio-temporal fractional sine-Gordon equation ob u Ð oarj u ? sin u Ì 0; t j ?47î ?46î

where we used for abbreviation simplified notation for fractional derivatives. For the case b = 2, the equation was obtained in [20]. The energy-momentum tensor hl , can be presented in the form m hl Ì hdlm ? sl ; m m ?48î


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1867

where the first term is the diagonal part h Ì h0 ? h1 Ì 0 1 dL dL ?ot uî? ?or uîÐ 2L Ì U ?u?t; rîî: d? ot uî d? or uî ?49î

This represents a pressure-like quantity. The second term in the right hand side of (48) is a non-diagonal of hl : m sl Ì m where s 0 Ì s 1 Ì 0; 0 1 Ò Ó dL 0 s1 Ì ?or uî Ì C Db u?t; rî or u?t; rî; 0t d? ot uî " # dL o a Ð1 1 ? ot uî Ì Ð g u? t ; r î ot u? t ; r î : s0 Ì d? or uî ojrjaÐ1 ?51î ?52î ?53î dL dL ?om uîÐ ?oj uîdl ; m d? ol uî d? oj uî ?50î

For r 2 R3 the spatial components of sl , represent shear stress tensor. The value h represents a pressure-like m quantity, normal stress. 3.2. Homogeneity in time The homogeneity in time means invariance of action with respect to the transformation t ! t 0 Ì t ? a; Dxl Ì dl da; 0 with the generators X l Ì dl ; 0 Y Ì0 ?l Ì 0; 1î: ?56î r ! r; Du Ì 0 u ! u: ?l Ì 0; 1î ?54î ?55î

Then the infinitesimal transformations are

The Noether's current has two components dL l l l J Ì Ð h0 Ì Ð ot u Ð d0 L ? l Ì 0; 1î : d? ol uî Using x0 = t, and x1 = r, we get the continuity equation ot J 0 ? or J 1 Ì 0; where J0 Ì Ð dL ?ot uîÐ L d? ot uî Ð Ñ Ì Ð ot u?t; rîC Db u?t; rîÐ L 0t

?57î

?58î

! 1 1 oaÐ1 Cb u?t; rî? U ; Ì Ð ot u?t; rî0 Dt u?t; rî? gor u?t; rî aÐ1 2 2 oj r j and J1 Ì Ð dL o a Ð1 ?ot uî Ì got u?t; rî u?t; rî: aÐ1 d? or uî oj r j ! Ðo
r

?59î

?60î

As a result, the continuity equation is o
t

1 Cb 1 o a Ð1 ot u0 Dt u ? g or u u?U aÐ1 2 2 oj r j

g ot u

o

aÐ1 aÐ1

! u Ì 0; ?61î

oj r j


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where u Ì u?t; rî. Here 1 1 oaÐ1 H Ì ot uC Db u ? gor u u?U 0t a Ð1 2 2 ojrj is a fractional generalization of density of energy (density of Hamiltonian), and P Ì Ð g ot u o
aÐ1

?62î

u ?63î aÐ1 oj r j is a fractional generalization of density of momentum. For the case a = 2 and b = 1, we obtain the well-known relations [28]: 1 1 2 2 H Ì ?ot uî ? g?or uî ? U ; P Ì Ðgot uor u: 2 2 The continuity Eq. (61) can be presented in an usual form of the conservation of energy ot H ? o r P Ì 0 with the fractional generalizations of the energy and momentum given by (62) and (63). 3.3. Homogeneity of space The homogeneity of space means invariance of action with respect to transformations r ! r 0 Ì r ? a; Dxl Ì dl da; 1 with the generators X l Ì dl ; 1 Y Ì 0: ?68î t ! t; u ! u: ?66î ?67î The corresponding infinitesimal transformations are presented by Du Ì 0 ?64î

?65î

Then the Noether's current has the following two components dL ?or uîÐ dl L ? l Ì 0; 1î : J l Ì Ðhl Ì Ð 1 1 d? ol uî

?69î

Using x0 = t, and x1 = r, we get the continuity equation that corresponds to the homogeneity of one-dimensional space ot J 0 ? or J 1 Ì 0; where J0 Ì Ð and J1 Ì Ð dL oaÐ1 ?or uî? L Ì gor u?t; rî u?t; rî? L aÐ1 d?or uî oj r j ?72î dL ?or uî Ì Ðor u?t; rîC Db u?t; rî; 0t d?ot uî ?71î ?70î

1 1 o a Ð1 Ì ot u?t; rîC Db u?t; rî? gor u?t; rî u?t; rîÐ U : 0t aÐ1 2 2 oj r j As a result, the continuity Eq. (70) is Ð ot o Ð
C r u0

D u ?o

b t

Ñ

r

1 Cb 1 o a Ð1 ot u0 D t u ? g or u uÐU aÐ1 2 2 oj r j

! Ì0 ?73î

that can be interpreted as the momentum conservation law in the case of fractional dynamics.


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For b = 1 and a = 2, Eq. (73) 1 2 Ðot ?or uot uî ? or ?ot uî ? 2

gives 1 2 g?or uî Ð U 2 Ì 0: ?74î

Note that, in general, for integer and fractional values of a and b, J 0 6Ì J 1 , and the energy-momentum tensor 1 0 hl is not a symmetric with respect to l and m. m 3.4. Field translation invariance For the case U(u) = 0, the action (18) is invariant with respect to transformations x l ! x 0l Ì x l ; u ! u ? a: ?75î

The infinitesimal transformations are presented as Dxl Ì 0; Du Ì da ?76î

with the generators X l Ì 0; Y Ì 1: ?77î The Noether's current Jl Ì dL d? ol uî ? l Ì 1; 2î ?78î

has components J0 Ì J1 Ì dL Ì C D b u? t ; r î d ? ot u î 0 t ?0 < b < 1î; ?79î ?80î

dL o a Ð1 Ì Ðg u? t ; r î ? 1 < a < 2î : aÐ1 d? or uî oj r j

As a result, the continuity equation ol J l Ì o t J 0 ? o r J 1 Ì 0 has the form ot C Db u?t; rîÐ go 0t o
r aÐ1 aÐ1

oj r j

u? t ; r î Ì 0

? 1 < a < 2; 0 < b < 1î :

?81î

For a = 2 and b = 1, we get o2 u?t; rîÐ go2 u?t; rî Ì 0; t r which is the usual field Eq. (45) for U(u)= 0. Note that [14] ot C Db u?t; rî Ì C D 0 0t
1?b t

?82î

u?t; rî?

t Ðb ot u? 0; r î : C ?1 Ð b î

?83î

To have the relation ot C Db u?t; rî Ì C D1?b u?t; rî, the initial conditions ot u?0; rî Ì 0 should be applied. In gen0t 0t eral, Eq. (81) cannot be presented as
C 0

D

1?b t

u?t; rîÐ g

oa ? 1 < a < 2; 0 < b < 1î ; a u? t ; r î Ì 0 oj r j

?84î

which is the fractional field Eq. (44) for U(u)= 0.


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This shows that the conservation law (81), in general, does not coincide with the field equation as it is happened for integer derivatives, unless, we use special boundary and initial conditions. 4. Chain with long-range interaction It was shown in [20,21] how the long-range interaction between different oscillators can be described by the fractional differential equations in the continuous medium limit. In this section, the Noether's theorem will be applied to such kind of systems. 4.1. Equation of motion and Noether's currents Let us define the action as 0 1 ! Z ?1 ?1 ?1 X1 X B C _ S Í un Ì dt@ U ?un ?tî; um ?tîîA; u2 ?tîÐ V ?un ?tîî Ð 2n Ð1 n;mÌÐ1 nÌÐ1
m6Ìn

?85î

where un is displacement of the nth oscillator from the equilibrium, 1 U ?un ?tîî Ì g0 J a ?jn Ð mjî?un ?tîÐ um ?tîî2 ; 4 and J a ?jn Ð mjî Ì 1 jn Ð mj
a?1

?86î

?a > 0î:

?87î

The Lagrangian of the chain is ! ?1 X1 1 2 _ LÌ un ?tîÐ V ?un ?tîî Ð g 2 4 nÌÐ1 The equation of motion oL d oL Ð Ì0 _ o u n ? t î dt o u n ? t î for Lagrangian (88) has the form d2 oV ?un î ?g un ?tî? oun ? t î dt2
?1 X 0 n;mÌÐ1 m6Ìn ?1 X 0 n;mÌÐ1 m6Ìn

J a ?jn Ð mjî?un ?tîÐ um ?tîî2 :

?88î

?89î

J a ?jn Ð mjîÍum ?tîÐ un ?tî Ì 0:

?90î

A continuous limit of Eq. (90) can be defined by a transform operation from un(t)to u?x; tî [20,21]. First, define un(t) as Fourier coefficients of some function ^?k ; tî, k 2 ÍÐK =2; K =2, i.e. u ^? t ; k î Ì u
?1 X nÌÐ1

un ? t î e

Ðikxn

Ì FD fun ?tîg;

?91î

where xn = nDx, and Dx = 2p/K is a distance between nearest particles in the chain, and Z 1 ?K =2 un ? t î Ì dk ^?t; k îeikxn Ì FÐ1 f^?t; k îg: u Du K ÐK =2

?92î

Secondly, in the limit Dx ! 0(K ! 1) replace un ?tî Ì ?2p=K îu?xn ; tî ! u?x; tîdx,and xn = nDx = 2pn/K ! x. In this limit, Eqs. (91), (92) are transformed into the integrals


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~? t ; k î Ì u u? t ; x î Ì

Z 1 2p

?1

dxe
Ð1

Ðikx

u?t; xî Ì Ffu?t; xîg Ì lim FD fun ?tîg;
Dx!0

?93î ?94î

Z

?1

Ð1

dk eikx ~?t; k î Ì FÐ1 f~?t; k îg Ì lim FÐ1 f^?t; k îg: u u Du
Dx!0

Applying (91) to (90) and performing the limit (93), we obtain o2 u? t ; x î oa u? t ; x î oV ? uî Ì 0 ? 0 < b < 2; 1 < a < 2î ; ? ga ? a ot 2 oj x j ou?t; xî where ga Ì 2g0 ?Dxîa C?Ðaî cos pa 2 ?96î ?95î

is the renormalized constant. Eq. (95) were considered in [20,21]. Consider a continuous transformation of time and field t ! t0 ; un ?tî ! u0n ?t0 î; ?97î

which form a continuous group with generators X and Yn such that the infinitesimal transformations are Dt Ì X da; Dun Ì Y n da: ?98î

The corresponding Noether's current is JÌ where hÌ
?1 X nÌÐ1 ?1 X nÌÐ1

oL Y n Ð hX ; _ oun ? t î

?99î

oL _ un ?tîÐ L _ oun ? t î

is the energy. Then JÌ
?1 X nÌÐ1

oL _ ÍY n Ð X un ?tî ? LX : _ oun ? t î

?100î

The equation of conservation law is d J Ì 0: dt In the next subsections, we consider examples of the conservation laws. 4.2. Homogeneity of time The homogeneity of time means the invariance of action with respect to the transformation (compare to (54)) t ! t ? a; un ! un : D un Ì 0 Y n Ì 0: ?102î ?101î

Its infinitesimal form is Dt Ì da; with generators X Ì 1; ?104î ?103î


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Then the Noether's current is J t Ì Ðh Ì Ð
?1 X nÌÐ1 ?1 X oL _n _ u2 ?tî? L un ?tî? L Ì Ð _ oun ? t î nÌÐ1 ?1 X 0 n;mÌÐ1 m6Ìn

?1 ?1 X1 X 1 _n ÌÐ V ?un ?tîî Ð g u2 ?tîÐ 2 4 nÌÐ1 nÌÐ1

J a ?jn Ð mjî?un ?tîÐ um ?tîî ÐH ;

2

where H is the Hamiltonian. The conservation law dH/dt = 0 is in the continuous limit gives the equation ! Z1 Z ?1 d 1 1 oa d 2 dr Íot u?t; rî ? V ?u?t; rîî ? ga ; u?t; rî u? t ; r î Ì d r H Ì 0; ?105î a dt Ð1 2 2 dt Ð1 oj r j where 1 < a < 2, and the density H of Hamiltonian (62) is introduced. One can compare this equation with Eq. (65) for b = 1. Both results coincide if the boundary conditions
r!Ö1

lim P Ì 0

?106î

is applied. 4.3. Translation invariance For V(un) = 0, the action (85) is invariant with respect to the transformations t ! t; D t Ì 0; X Ì 0;
?1 X nÌÐ1

un ! un ? a; Dun Ì da; Y n Ì 1:
?1 X oL _ Ì un ? t î P ; _ oun ?tî nÌÐ1

?107î ?108î ?109î

or in the infinitesimal form

with the generators

Then the Noether's current is Jr Ì ?110î

where P is the total momentum. The conservation law is dP/dt =0. The continuous limit of this conservation law gives the equation Z ?1 d drot u?t; rî Ì 0: dt Ð1

?111î

Let us compare this equation with Eq. (81) that is derived for scalar field u?t; rî. Integration of (81) with respect to coordinate r gives Z ?1 Z ?1 oaÐ1 ot drC Db u?t; rîÐ ga dror u?t; rî Ì 0 ?1 < a < 2; 0 < b < 1î: ?112î 0t aÐ1 ojrj Ð1 Ð1 Then Z o
t Ð1 ?1

drC Db u?t; rîÐ g 0t

o
a

a Ð1 aÐ1

!?1 u?t; rî
Ð1

oj r j

Ì 0:

?113î

As a result, Eq. (81) coincides with (111), for b = 1, if we use the boundary conditions
r!Ö1

lim

o

aÐ1 aÐ1

oj r j

u? t ; r î Ì 0:

?114î


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5. Fractional Hamilton's equations 5.1. De Donder-Weyl Hamiltonian The idea of variation of fields based on a manifestly covariant version of the Hamiltonian formalism in field theory known in the calculus of variation of multiple integrals [36,37] has been proposed by Born and Weyl [38]. The mathematical study of geometrical structures underlying the related aspects of the calculus of variations and classical field theory has been undertaken recently by many authors (see for example [37,39,40]). The De Donder-Weyl Hamiltonian form of the field equations [36] are ol u ? x î Ì where pl Ì oL o? ol uî ?116î oH ; opl ol pl ?xî Ì Ð oH ; ou ?115î

is called the multi-momenta, and H?u; pl î Ì ?ol uîpl Ð L ?117î

is called the De Donder-Weyl Hamiltonian density function, L Ì L?u; ol uî is Lagrangian density. These equations are known to be equivalent to the Euler-Lagrange field equations if L is regular in the sense that ! o2 L det 6Ì 0: ?118î o?ol uîo?ol uî 5.2. Hamilton's equations of integer order Let us consider Hamiltonian systems in the extended phase space of coordinates ?xl ; u; pl î. The evolution of fields is defined by stationary states of the action functional Z S Íu; p Ì Ípl ol u Ð H?u; pîd2 x; ?119î where the Hamiltonian density H is defined by (117), both u and p are assumed to be independent functions of x Ì ?t; rî. In classical field theory, the evolution of the field u(x) is derived by finding the condition for which the action integral (119) is stationary (a minimum or a saddle point). The action functional (119) can be rewritten as Z S Íu; p Ì xl ; ?120î where xl Ì pl du Ð H?u; pîdxl : is the Poincare-Cartan 1-form or the action 1-form. The stationary action condition dS = 0 leads to dxl Ì 0: Here the exterior derivative is d Ì ?dxm îDxm ??duîDu ??dpm îDpm ; where we put new notation for the derivatives Dxm Ì o ; ox m Du Ì o ; ou Dpm Ì o : opm ?124î ?123î ?122î ?121î


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The De Donder-Weyl Hamiltonian form of field equations in (122) can be obtained from the condition that the derivative is exterior one. This condition is equivalent to the stationary action principle dS Íu; p Ì 0. Then Eq. (121) gives dxl Ì d?pl duîÐ d?Hdxl î Ì ?Dxm pl îdxm ^ du ??Du pl îdu ^ du ??Dpm pl îdpm ^ du Ð?Dxm Hîdxm ^ dxl Ð?Du Hîdu ^ dx Ð?Dpm Hîdpm ^ dxl : Using dp ^ dx = Ðdx ^ dp, and ?Dxm Hî, Dup = 0, we get dx Ì ?Dx p ? Du Hîdx ^ du Ð?Dp pdu Ð Dp Hdxî^ dp; where the matrix form of the equation is used. The relation Dp p = 1 gives dx Ì ?Dx p ? Du Hîdx ^ du Ð?Du Ð Dp Hdxî^ dp: From (122), we have du Ð Dp Hdx Ì 0; As the result, we obtain ou Ì Dpl H; ox l op l Ì ÐDu H; oxl ?129î Dx p Ì ÐDu H: ?128î ?127î ?126î
l

?125î

which are the well-known De Donder-Weyl Hamilton's equations (115). 5.3. Fractional Hamilton's equations The fractional generalization of the form (121) can be defined [34,32] by x?aî Ì pDa u Ð H?u; pîda x: s s ?130î

It will be called the fractional Poincare-Cartan 1-form or simply the fractional action 1-form. We can consider the fractional exterior derivative of the form (130), and use da x?aî Ì 0 to obtain the fractional field equations. Here the fractional exterior derivative is da Ì da xm Dam ? da uDa ? da pm Da m ; s s s x u p where Dam , Da , D x u x Ì ?t; rî, we use
a pm

?131î da x Ì CÐ1 ?a ? 1îda xa ; s

?132î

can be fractional derivatives of different types [14]. For example, for xl 2 R2 , such that

Da Ì ?C Da0 ; Da1 î; r x t0 t where t0 C Da0 is Caputo fractional derivative, and Da1 is the Riesz derivative. Fractional differential forms have t r been suggested in [34] and it is used to describe dynamical systems [32,35]. Then, by some transformations (see Appendix), one can obtain ! Ò Ó p1Ða da u Ð Da Hda x ^ da p: ?133î da x?aî Ì Da p ? Da H da x ^ da u Ð s s s s x u p C ? 2 Ð aî s Using (133) and (131), we get p1Ða da u Ð Da Hda x Ì 0; s p C?2 Ð aî s For the case u = u(x), da u Ì da xDa u: s s x ?135î Da p Ì ÐDa H: x u ?134î


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As the result, we obtain Dal u Ì ?pl î x
a Ð1

C?2 Ð aîDa l H; p

Dal pl Ì ÐDa H: x u

?136î

These equations are the fractional generalization of De Donder-Weyl Hamilton's equations (129). As an example, we can consider H?u; pî Ì Using Da m ?pl î Ì p
b

2Ða 0 2 2Ða 1 2 ?p î Ð ? p î ? U ? uî : 2 2 C?b ? 1î ?pl î C?b ? 1 Ð aî

?137î

bÐa l dm

;

?138î

and C(z + 1) = zC(z), Eq. (136) gives Da0 u Ì p0 ; Da1 u Ì Ðp1 ; t r Da0 p0 ? Da1 p1 Ì ÐDau U ?uî: t t u After substitution of (139) into (140), we obtain ?Da0 î u?t; rîÐ?Da1 î u?t; rî? Dau U ?uî Ì 0: t r u For a0 = a1 = au = 1, Eq. (141) is the usual wave equation o2 u?t; rîÐ o2 u?t; rî? t r oU ? uî Ì 0: ou? t ; r î
2 2

?139î ?140î

?141î

For U(u) = 0, Eq. (141) has the form ?Da0 î u?t; rîÐ?Da1 î u?t; rî Ì 0: t r
2 2

?142î

The solution u?t; rî of this equation is a linear combination of the solutions u1 ?t; rî and u2 ?t; rî of the equations Da0 u1 ?t; rîÐ Da1 u1 ?t; rî Ì 0; t r Da0 u2 ?t; rî? Da1 u2 ?t; rî Ì 0: t r ?143î

For a1 = 1, there exists a relation between the Dirac solutions and the fractional extension of D'Alembert expression that is considered in [41]. Using the property Da Da u Ì D2a u; xx x and we get the fractional equation D
2a0 t

?144î

u Ð D2a1 u ? Dau U ?uî Ì 0: r u C?2î u C?2 ? au î
2a1 r

?145î

For a special case, U ? uî Ì m Eq. (145) gives D
2a0 t 2 1?au

:

?146î

u?t; rîÐ D

u?t; rî? m2 u?t; rî Ì 0:

?147î

This is a fractional generalization of Klein-Gordon equation (see also [20,30]). Note that Eq. (147) is not Lorentz invariant equations. To obtain fractional relativistic equations, the fractional power of D'Alembertian should be used [42] (see also Section 28 of [11]). The causality principle [28,43] also should be taken into account. For U ?uî Ì sin?u?t; rîî; ?148î


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Eq. (145) gives D
2a t
0

uÐD

2a1 r

u ? sin?u ? au p=2î Ì 0:

?149î

This equation is fractional sine-Gordon equation that was considered in [20] for a0 = 1 and au = 2. 6. Conclusion Exploiting a variational principle to obtain fractional dynamics seems to be a fairly powerful tool that permits a universal consideration of situations with fractal time and space. In this paper, we have demonstrate it by deriving different equations of using generalized Noether's theorem. Such equations are similar to the regular conservation laws, although the presence of fractional time derivatives reflects a dissipation. Similar variation of action can be used to derive Hamiltonian type equations although the situation here is not uniquely defined and some freedom of choosing the differential action 1-form leaves different possibilities. In fact the introduction of Hamiltonian type Eqs. (136) needs a more transparent discussion. It is known that Hamiltonian description of systems can be constructed even for the case of presence of dissipation if one use fractional derivatives (see for example [29,30] and references therein) or interaction of the system with random forces (environment [44] or stochastic processes [31,45]). Nevertheless, such unusual construction of generalized Hamiltonian type equations doesn't mean the existence of usual preserving variables. Instead, one arrive to some generalized ``conservation laws'' in the form ot q ? divJ Ì 0; ?150î

where q and J are defined through the equations with long-range interaction and long-term memory. Eq. (150) has a meaning of conservation of some flow, but it does not imply a unique interpretation depending on the type of the problem, boundary-initial conditions, performed limits in its derivations, etc. The same can be mentioned about the generalized Hamiltonian type Eq. (136). We consider them more as a possible formal tool since no specific application is introduced. Acknowledgement This work was supported by the Office of Naval Research, Grant No. N00014-02-1-0056, and the NSF Grant No. DMS-0417800. Appendix To prove the proposition (133), we use the rule 1 Xa os g a Dx ?fgî Ì ?DaÐs f î s ; x ox s sÌ 0 and the relation [14] os a Íd x Ì 0 ox s s ? s P 1î

for integer s, where a ?Ð1îkÐ1 aC?k Ð aî : Ì C?1 Ð aîC?k ? 1î k For example, we have 1 Ò Ó X am a al d Al d s x Ì ds x ^ ?D s sÌ 0
a aÐs xm

os al am a Al î s d x Ì ds x ^ ds x o?xm î s

l

a 0

Ð Ñ ?Dam Al î Ì Dam Al da xm ^ da xl : x x s s


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As a result, da x?aî Ì da ?pda uîÐ da ?Hda xî s s Ì ?Da pîda x ^ da u ??Da pîda u ^ da u ??Da pîda p ^ da u Ð?Da Hîda x ^ da x Ð?Da Hîda u ^ da x x u p x u s s s s s s s s s s Ð?Da Hîda p ^ da x: p s s Here Da0 x a ds p
a0 t a x1 a a

?151î

Ì C D is Caputo fractional derivative, and D Ì o =ojrj is the Riesz fractional derivative [14]. Using t0 ^ da x Ì Ðda x ^ d a p s s s

and Dx H?u; pî Ì 0, Da p Ì 0 for Riesz and Caputo derivatives, we can rewrite Eq. (151) in the form u Ò Ó Ò Ó da x?aî Ì Da p ? Da H da x ^ da u Ð ?Da pîda u Ð?Da Hîda x ^ da p: ?152î x u p p s s s s s Substitution of Da p Ì p p1Ða ; C?2 Ð aî ?153î

into Eq. (152) gives (133). References
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