Abstract The Hamilton- Jacobi partial differential equation is - PDF document

Fractional Hamilton-Jacobi Equation and WKB Approximation Eqab M. RABEI * Department of Physics, Mutah University, Al-Karak, Jordan Abstract 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. Wentzel, Kramers, Brillouin (WKB) approximation for fractional systems is investigated. In the fractional case, the wave function is constructed such that the phase factor is the same as the Hamilton's principle function "S". To demonstrate our proposed approach an example is investigated in details *eqabrabei@yahoo.com 1

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, Klimek [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. In addition the system is quantized using the WKB approximation. 2

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 n x ⎛ ⎞ 1 d ( ) ∫ α − α − = − τ τ τ n 1 ⎜ ⎟ D f ( x ) ( x ) f ( ) d , 1 a x Γ − α ( n ) ⎝ dx ⎠ , a Which is denoted as (LRLFD), and the right Riemann–Liouville fractional derivative reads as n b ⎛− ⎞ 1 d ( ) ∫ α − α − = τ − τ τ n 1 ⎜ ⎟ D f ( x ) ( x ) f ( ) d , 2 x b Γ − α ( n ) ⎝ dx ⎠ x Which, denoted as (RRLFD )? Here α is the order of the derivative such that represents the Euler gamma function. If α is an integer, these − ≤ α ≤ n 1 n Γ and derivatives are defined in the usual sense, i.e. α α ⎛ ⎞ ⎛− ⎞ d d α = α = α = D f ( x ) ⎜ ⎟ f ( x ) , D f ( x ) ⎜ ⎟ f ( x ) , 1 , 2 , 3 ,.... a x x b ⎝ dx ⎠ ⎝ dx ⎠ ( ) , 3 α a D The fractional operator can be written as [13] x 3

n d ( ) α α − = n D D , 4 a x a x n dx , Where the number of additional differentiations n is equal to [ α ] +1, where [ α ] is the α a D x whole part of α . The operator is a generalization of differential and integral operators and can be introduced as follows: ⎧ α d ⎪ α > Re( ) 0 α ⎪ dx ⎪ α = α = D 1 Re( ) 0 ⎨ a x ⎪ x ⎪ ∫ τ − α α < ( d ) Re( ) 0 ⎪ ⎩ a (5) The starting point of Agrawal is to consider the Lagrangian of the form α β L ( q , D q , D q , t ) a t t b Which, satisfies the following variation ∫ α β δ = L ( q , D q , D q , t ) dt 0 a t t b Then, the Euler-Lagrange equation for fractional calculus of variations problem is obtained as 4

∂ ∂ ∂ L L L ( ) α β + + = D D 0 , 6 t b a t ∂ ∂ α ∂ β q D q D q , a t t b The generalized momenta are introduced as [11] ∂ ∂ L L ( ) = = p , p , 7 α β ∂ α ∂ β D q D q a t t b And the Hamiltonian depending on the fractional time derivatives reads as ( ) α β = + − H p D q p D q L , 8 α β a t t b Calculating the total differential of this Hamiltonian we obtain ∂ ∂ L L = α + α + β + β − − α dH p d D q dp D q p d D q dp D q dq d D q α α β β a t a t t b t b ∂ ∂ α a t q D q a t ∂ ∂ L L ( ) β − − d D q dt , 9 β t b ∂ ∂ D q t t b Substituting the values of the momenta from Eq(7), we get ∂ ∂ L L ( ) α β = + − − dH dp D q dp D q dq dt , 10 α β a t t b ∂ ∂ q t Making use of the Euler-Lagrangian equation (6), We obtain ∂ L ( ) = α + β − β + ε − dH dp D q dp D q [ D p D p ] dq dt , 11 α β β β a t t b a t t b ∂ t This means that the Hamiltonian is a function of the form 5

= H H ( q , p , p , t ) α β Thus the total differential of this Hamiltonian reads as ∂ ∂ ∂ ∂ H H H H ( ) = + + + dH dq dp dp dt , 12 α β ∂ ∂ ∂ ∂ q p p t α β Comparing Eq.(11) and Eq.(12), we get the following Hamilton's Equations of motion ∂ H ∂ ∂ ∂ ∂ H H H L α β = + D p D p = α = β = − D q , D q α β t b a t ∂ ∂ a t ∂ t b q ∂ ∂ ; ; (13) p p t t α β It is observed that the fractional Hamiltonian is not a constant of motion even though the Lagrangian does not depend on the time explicitly. 6

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 ( ) ( ) α β α β = + − H q , p , p , t p D q p D q L ( q , D q , D q , t ), 14 α β α β a t t b a t t b . Consider the canonical transformation with a generating function ( ) α − β − 1 1 F D q , D q , P , P , t α β 2 a t t b Then, the new Hamiltonian will take the form ( ) ( ) ′ α β α β = + − K Q , P , P , t P D Q P D Q L ( Q , D Q , D Q , t ), 15 α β α β a t t b a t t b q , p α p , The old canonical coordinates , satisfy the fractional Hamilton’s β principle that can be put in the form t ( ) 2 ( ) ∫ δ α + β − = p D q p D q H dt 0 , 16 α β a t t b t 1 Q , P α P , At the same time the new canonical coordinates , of course satisfy a similar β principle. t ( ) 2 ( ) ∫ δ α + β − = P D Q P D Q K dt 0 , 17 α β a t t b t 1 The simultaneous validity of Eq. (16) and Eq. (17) does not mean of course that the integrands in both expressions are equal. Since the general form of the Hamilton’s 7

principle has zero variation at the end points, both statements will be satisfied if the integrands connected by a relation of the form [12] dF ( ) α β α β + − = + − + p D q p D q H P D Q P D Q K , 18 α β α β a t t b a t t b dt The function F can be given as: ( ) ( ) α − β − α − β − = 1 1 − 1 − 1 F F D q , D q , P , P , t P D Q P D Q , 19 . α β α β 2 a t t b a t t b The function F 2 called Hamilton’s principal function S for a contact transformation. ( ) ( ) α − β − = 1 1 F S D q , D q , P , P , t , 20 α β 2 a t t b Thus, dP dP dF dS d d β α − α − β − β − = − α − − − 1 1 1 1 D Q P D Q D Q P D Q α β a t a t t b t b dt dt dt dt dt dt By using definitions of fractional calculus given in Eq. (4) then we have dP dF dS dP ( ) β = − α − − α − β − − β α 1 1 D Q P D Q D Q P D Q , 21 α β a t a t t b t b dt dt dt dt dF from Eq. (21) into the Eq. (18) Substituting the values of the dt we have dP dP dS β α + β − = − + − α α − − β − 1 1 p D q p D q H K D Q D Q α β (22) a t t b a t t b dt dt dt But dP ∂ ∂ ∂ ∂ ∂ dP dS S d S d S S S β = α − + β − + + + 1 1 α D q D q α − a t β − t b ∂ ∂ ∂ ∂ ∂ 1 1 dt D q dt D q dt P dt P dt t α β a t t b 8