Hamilton-Jacobi Fractional Sequential Mechanics
Microsoft Word - H-J Fractional
 Hamilton-Jacobi Fractional Sequential Mechanics 
Eqab M. RABEI* and Bashar S. ABABNEH 
Department of Physics, Mutah University, Al-Karak, Jordan 
Abstract 
As a continuation of Rabei et al. work [11], the Hamilton- Jacobi partial 
differential equation is generalized to be applicable for systems containing fractional 
derivatives. The Hamilton- Jacobi function in configuration space is obtained in a 
similar manner to the usual mechanics. Two problems are considered to demonstrate 
the application of the formalism. The result found to be in exact agreement with 
Agrawal's formalism. 
Keywords: Fractional derivative, Fractional systems, Hamilotonian formalisms, 
Hamilton-Jacobi treatment. 
*eqabrabei@yahoo.com 
1-Introduction 
The Hamiltonian formulation of non-conservative systems has been developed 
by Riewe[1,2].He used the fractional derivative [3,4,5] to construct the Lagrangian and 
Hamiltonian for non-conservative systems. As a sequel to Riewe's work, Rabei et al. 
[6] used Laplace transforms of fractional integrals and fractional derivatives to develop 
a general formula for the potential of any arbitrary forces, conservative or non-
conservative. This led directly to the consideration of the dissipative effects in 
Lagrangian and Hamiltonian formulations. Besides, the canonical quantization of non-
conservative systems carried out by Rabei et al. [7]. 
Other investigations and further developments are given by Agrawal [8] .He 
presented the fractional variational problems and the resulting equations are found to 
be similar to those for variation problems containing integral order derivatives. This 
approach is extended to classical fields with fractional derivatives [9]. Besides, Kilmek 
[10] showed that the fractional Hamiltonian is usually not a constant of motion, even in 
the case when the Hamiltonian is not an explicit function of time. In addition, as a 
continuation of Agrawal’s work [8], Rabei et al. [11] achieved the passage from the 
Lagrangian containing fractional derivatives to the Hamiltonian. The Hamilton's 
equations of motion are obtained in a similar manner to the usual mechanics. 
In the present work, the Hamilton – Jacobi partial differential equation (HJPDE) 
is generalized to be applicable for systems containing fractional derivatives.   
The paper is organized as follows: In Sec. 2 Lagrangian and Hamiltonian 
formalisms with fractional derivatives are reviewed briefly. In Sec.3, Hamilton-Jacobi 
Partial differential equations with fractional derivatives is constructed, and two 
illustrative examples are given in Sec. 4. 
2- Hamiltonian Formalism with Fractional Derivative 
Several definitions of a fractional derivative have been proposed. These 
definitions include Riemann–Liouville, Grünwald–Letnikov, Weyl, Caputo, Marchaud, 
and Riesz fractional derivatives. Here; the problem is formulated in terms of the left 
and the right Riemann–Liouville fractional derivatives. 
The left Riemann–Liouville fractional derivative defined as 
∫ −−−⎟⎠
xa dfx
xfD τττ
αα )()(
)( 1                                                 (1) 
Which is denoted as the LRLFD and the right Riemann–Liouville fractional 
derivative reads as 
∫ −−−⎟⎠
bx dfx
xfD τττ
αα )()(
)( 1                                              (2) 
Which is denoted as the RRLFD. Here α is the order of the derivative such that 
nn ≤≤− α1  and Γ  represents the Euler gamma function. If α is an integer, these 
derivatives are defined in the usual sense, i.e. 
,....3,2,1,)()(,)()( =⎟
xfDxf
xfD bxxa             (3) 
The fractional operator 
xa D can be written as [13] 
D −= αα                                                                                                  (4) 
Where the number of additional differentiations n is equal to [α] +1, where [α] is 
the whole part of α. The operator αxa D is a generalization of differential and integral 
operators and can be introduced as follows: 
∫ − 0)Re()(
0)Re(1
0)Re(
                                                           (5) 
 Following to Agrawal [8], the Euler-Lgrange equations for fractional calculus 
of variations problem is obtained as 
                                                                  (6) 
Where L is the genaralized Lagrangian function of the form 
),,,( tqDqDqL btta
The generalized momenta are introduced as  
ββαα ∂
= ,                                                                          (7) 
And the Hamiltonian depending on the fractional time derivatives reads as  
LqDpqDpH btta −+=
α                                                                                   (8) 
In Ref [11], the Hamilton’s equations of motion are obtained in a similar manner 
to the usual mechanics. These equations read as,   
 ;    qD
,   ;     β
α pDpD
tabt +=∂
                           
                                                                                         
It is observed that the fractional Hamiltonian is not a constant of motion even 
though the Lagrangian does not depend on the time explicitly. 
3. Hamilton-Jacobi Partial Differential Equation with Fractional 
Derivatives 
In this section, the determination of the Hamilton-Jacobi partial differential 
equation for systems with fractional derivatives is discussed. According to Rabei et al. 
[11], the fractional Hamiltonian is written as 
( ) ),,,(,,, tqDqDqLqDpqDptppqH bttabtta βαββααβα −+=                            (9) 
Consider the canonical transformation with a generating function 
 ( )tPPqDqDF btta ,,,, 112 βαβα −−  
Then, the new Hamiltonian will take the form 
( ) ),,,(,,, tQDQDQLQDPQDPtPPQK bttabtta βαββααβα ′−+=                          (10) 
The old canonical coordinates βα ppq ,, , satisfy the fractional Hamilton’s 
principle that can be put in the form 
( ) 0
btta dtHqDpqDp
αδ                                                                  (11) 
At the same time the new canonical coordinates βα PPQ ,, , of course satisfy a similar 
principle.  
( ) 0
btta dtKQDPQDP
αδ                                                                  (12) 
The simultaneous validity of Eq. (11) and Eq. (12) does not mean of course that the 
integrands in both expressions are equal. Since the general form of the Hamilton’s 
principle has zero variation at the end points, both statements will be satisfied if the 
integrands connected by a relation of the form [12] 
KQDPQDPHqDpqDp bttabtta +−+=−+
α                              (13) 
Where the function F is given as:  
( ) QDPQDPtPPqDqDFF bttabtta 11112 ,,,, −−−− −−= ββααβαβα                              (14) 
The function F2 is called Hamilton’s principal function S for a contact transformation. 
( )tPPqDqDSF btta ,,,, 112 βαβα −−=                                                                        (15) 
Thus,  
btbttata
1111 −−−− −−−−= ββ
By using definitions of fractional calculus given in Eq. (4) then we have    
QDPQD
QDPQD
btbttata
αα −−−−= −− 11                           (16) 
Substituting the values of the 
 from Eq. (16) into the Eq. (13) we have  
KHqDpqDp bttabtta
11 −− −−+−=−+ ββααββ
α                  (17) 
Again using definitions of fractional calculus given in Eq. (4) we have the following 
form 
α 11                   (18) 
Substituting the values of the 
 from Eq. (18) into the Eq. (17) we get 
= ββαα                                                              (19) 
http://scienceworld.wolfram.com/physics/ContactTransformation.html
              
QD btta ∂
= −− 11 ,                                              (20) 
                           K
+                                                                                (21) 
We can automatically ensure that the new variables are constant in time by 
requiring that the transformed Hamiltonian K shall be identically zero, In other words, 
βα PPQ ,,  are constants. We see by putting K = 0 that this generating function must 
satisfy the partial differential equation. 
                         0=
H                                                                                  (22) 
This equation is called the Hamilton –Jacobi equation. Let us assume that 
21 , EPEP == βα  
Where 1E , 2E  are constants. Then the action function (15), can be expressed as 
                 ( )tEEqDqDSS btta ,,,, 2111 −−= βα                                                      (23) 
Further insight into the physical significance of Hamilton’s principal function S 
is furnished by an examination of the total time derivative, which can be computed 
from the formula 
           
α 11                                       (24) 
By using Eq. (19) we have 
                        HqDpqDp
btta −+=
And using Eq. (9) we have 
                                   L
Thus 
                                  ∫=
LdtS                                                                                (25) 
If we restrict our considerations to the time -independent Hamiltonians, then the 
Hamilton-Jacobi function can be written in the form 
     ( ) ( ) ( )tEEfEqDWEqDWS btta ,,,, 21212111 ++= −− βα                                          (26) 
Where W  is called Hamilton’s characteristic function and the function, f  takes the 
following form: 
                                      ( ) EttEEf −=,, 21  
 Making use of equations (19) and (20) we obtain: 
                
= ββαα                                               (27) 
            2
11 , λλ βα =
QD btta                                 (28) 
Here 1λ , 2λ  are constants.    
                                                                                         
The physical significance of W can be understood by writing its total time derivative 
                            qD
=                                                              (29) 
Comparing this expression to the results of substituting Eq. (27) into Eq. (29) we see 
that 
∫∫ −=⇒=⇒= qDdpWqdtDpWqDpdt
tatata
α            (30) 
Again one may show that 
                                  ∫ −= qDdpW bt 12 ββ                                                            (31) 
4. Illustrative Examples 
To demonstrate the application of our formalism, let us discuss the following models: 
As a first model consider the lagrangian given by Agrawal [8]  
                                                      ( )20
qDL t
The (HJPDE) for this Lagrangian is calculated as  
                                                   ( ) 0
1 2 =
Using Eq. (27) we obtain 
                                         0
1 =−⎟⎟
− EqD
Solving this equation we have 
                                                 qDEW t
−= α  
Thus 
                                       Ep 2=α  
Making use of Eq. (26) we obtain the function S as: 
                                   EtqDES t −=
Eq. (28) leads to 
                       1
1 λαα =−=
= −− tqD
QD tt  
Thus 
                                  ( )110 2 λα +=− tEqDt  
                                α
α pEqDt == 20  
This is the same result obtained by Rabei et al. [11], which is equivalent to Agrawal 
formalism [8].  
As a second model consider the Lagrangian given by Rabei et al. [11] 
                                     ( ) ( ) qDqDqDqDL tttt βαβα 102120
++=  
The Hamiltonian is calculated as 
                                            ( ) ( )22
βα ppH ==  
Thus, the Hamilton-Jacobi partial differential equation reads as: 
                                          ( ) 0
1 2 =
Making use of Eq. (26) we have 
                                          0
1 =−⎟⎟
− EqD
Thus, 
                                            qDEW t
−= α  
Again the (HJPDE) can be written as 
                                        ( ) 0
1 2 =
Then 
                                       0
2 =−⎟⎟
− EqD
Which leads to 
                                      qDEW t
−= β  
Thus the Hamilton-Jacobi action function can be written as 
                               EtqDEqDES tt −+=
Where 
                             E
= −αα  
                               E
= −ββ  
Using Eq. (28) we get 
                              1
1 λβαα =−+= −−− tqD
QD ttt  
Thus 
                              ( )11110 2 λβα +=+ −− tEqDqD tt  
                                        EqDqD tt 210 =+
Then 
                                        qDqDp tt
α 10 +=  
                                          qDqDp tt
β 10 +=  
These Leads to 
                              0))(( 1010 =++ qDqDDD tttt
This result is in exact agreement with Rabei et al. [11]. 
5- Conclusion 
In This work we have studied the Hamilton-Jacobi partial differential equation 
for systems containing fractional derivatives. A general theory to solve the Hamilton-
Jacobi partial differential equation is proposed for systems containing fractional 
derivatives under the condition that this equation is separable. The Hamilton-Jacobi 
function is determined in the same manner as for usual systems. Finding this function 
enables us to get the solutions of the equations of motion. 
In order to test our formalism, and to get a somewhat deeper understanding, we 
have examined two examples of systems with fractional derivatives. The result found 
to be in exact agreement with Lagrangian formulation given by Agrawal [8] and with 
Hamiltonian formulation given by Rabei et al. [11]. 
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