The Conjugate Variable Method in Hamilton-Lie Perturbation Theory −Applications to Plasma Physics−

Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008)

Shinji TOKUDA Japan Atomic Energy Agency, Naka, Ibaraki, 311-0193 Japan (Received 21 February 2008 / Accepted 29 July 2008)

The conjugate variable method, an essential ingredient in the path-integral formalism of classical statistical dynamics, is used to apply the Hamilton-Lie perturbation theory to a system of ordinary differential equations that does not have the Hamiltonian dynamic structure. The method endows the system with this structure by doubling the unknown variables; hence, the canonical Hamilton-Lie perturbation theory becomes applicable to the system. The method is applied to two classical problems of plasma physics to demonstrate its effectiveness and study its properties: a non-linear oscillator that can explode and the guiding center motion of a charged particle in a magnetic field. c 2008 The Japan Society of Plasma Science and Nuclear Fusion Research Keywords: one-form, Hamilton-Lie perturbation method, conjugate variable, non-linear oscillator, guiding center motion, plasma physics DOI: 10.1585/pfr.3.057

1. Introduction Since the Hamilton-Lie perturbation method is a pow- We begin by considering the following two non-linear erful analytical tool that enables us to investigate the prob- oscillators lem deeply, it would be a significant contribution to the development of approximation methods in physics if it were dx dy = y, = −x − x3, (1) made applicable for equations such as Eq. (2), which do dt dt not have the Hamiltonian structure. This seems impossi- and ble since the Hamilton-Lie perturbation method assumes the Hamiltonian structure of the equations. However, that dx dy = y, = −x − y3, (2) is not the case. We can endow any ordinary differential dt dt equations with the Hamiltonian structure by doubling the where t is the time variable. When = 0, both oscillators unknown variables. This technique is well known as the reduce to an identical harmonic oscillator. When 0, the conjugate variable method, and plays an essential role in equations have quite different properties. Equation (1) has the path-integral formalism of classical statistical dynam- the variational principle in the phase space (x,y) [1], and ics [4]. Once we make the Hamiltonian for Eq. (2) by the performs the Hamiltonian motion under the Hamiltonian conjugate variable method, we can systematically apply the Hamilton-Lie perturbation method to the equation. In x2 y2 h(x,y) = + + x4. (3) the present paper, we report and also advocate the con- 2 2 4 jugate variable method for the Hamilton-Lie perturbation One of the properties peculiar to the Hamiltonian motion method. is, of course, the volume invariance of the flow of motion In Sec. 2, after briefly reviewing the conjugate variable in the phase space (x,y). Equation (2) does not have the method, we apply the Hamilton-Lie perturbation method 2 Hamiltonian structure and loses the invariant property of to Eq. (2) up to the second order O( ). We demonstrate the volume in the phase space (x,y). that the equations for the original variables are obtained Next, an important property of Eq. (1) from a practical in a closed form that does not include conjugate variables, point of view is that it can be analyzed by a powerful tool, although the conjugate variable method doubles the num- the Hamilton-Lie perturbation method [2]. This method, ber of variables. In Sec. 3, we apply the conjugate variable however, is not applicable for Eq. (2); therefore, different method to the guiding center motion of a charged particle approximation methods have been devised for Eq. (2), such in a magnetic field. This is a classical problem with a long as the multi-scale expansion method [3]. These methods history in plasma physics [5,6]. The most elegant and pro- abandon the viewpoint of Hamiltonian dynamics. found solution method will be the non-canonical perturbation method [7]. We show that when the conjugate variable author’s e-mail: [email protected]

057-1 c 2008 The Japan Society of Plasma Science and Nuclear Fusion Research Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) method is used, the guiding center problem can be solved 2.1 Harmonic oscillator with conjugate vari- by the canonical perturbation method. Our conclusions are ables given in Sec. 4, with discussion on further applications of We ﬁrst examine the case of = 0, the harmonic oscil- the present method to plasma physics. lator. Let us deﬁne the transformation from (x,y)to(a,θ) by

2. Perturbation Analysis of a Non- = = − Linear Oscillator by Introducing x a cos θ, y a sin θ. (14) Conjugate Variables Then we have

Following Ref. [4], we can construct a fundamental 1- qdx + pdy = [q cos θ − p sin θ]da form for Eq. (2) as follows − a[q sin θ + p cos θ]dθ, (15) γ = q(dx − ydt) + p[dy + (x + y3)dt] and = qdx + pdy − hdt. (4) (0) = − + Here, q(res. p) is called the conjugate variable with respect h a(q sin θ p cos θ). (16) to x(res.y); h is the Hamiltonian for the 1-form and is These results imply the transformation from (a,θ,q, p)to given by (a,θ,Q, P)deﬁnedby h = h(0) + h(1), (5) Q = Q(q, p, a,θ) = q cos θ − p sin θ, (17) h(0) = qy − px, (6) and and h(1) = −py3. (7) P = P(q, p, a,θ) = −a(q sin θ + p cos θ). (18)

Since Eq. (4) is a canonical 1-form, the equations of mo- Then, we obtain the 1-form of the harmonic oscillator in tion for the phase space coordinates (x, q,y,p) are obtained the phase space (a, Q,θ,P)givenby from the Hamilton equation (0) dx ∂h γ = Qda + Pdθ − h dt, (19) = = y, (8) dt ∂q with the Hamiltonian dq ∂h = − = p, (9) dt ∂x h(0) = P. (20) dy ∂h = = −x − y3, (10) dt ∂p Since h(0) is independent of a, Q and θ, the conjugate vari- and ables Q, a, P with respect to them are constants of motion. dp ∂h On the other hand, the equation of motion for θ is = − = −q + 3py2. (11) dt ∂y dθ ∂h(0) = = 1. (21) The equation of motion for any function of the phase space dt ∂P coordinates is also expressed by the Hamilton equation. Consequently, we have For example, the equation of motion for the energy of the oscillator, a(t) = a0,θ(t) = t + θ0, (22) 1 E = (x2 + y2), (12) 2 and is given by Q(t) = Q0, P(t) = P0, (23) dE = {E, h} = −y4 ≤ 0, (13) dt where a0, θ0, Q0 and P0 are arbitrary initial values, and where { , } is the Poisson bracket in the phase space from Eq. (14) we have (x, q,y,p). x(t) = a cos(t + θ ), (24) The 1-form, Eq. (4), appears to be only nominal and 0 0 therefore useless. It would be so if the only solution were and to solve the diﬀerential equations, Eqs. (8)-(11), directly.

However, the Hamiltonian structure provides other pow- y(t) = −a0 sin(t + θ0). (25) erful devices to solve the equation of motion: coordinate transformations and gauge functions in the phase space. The inverse transformations of Eqs. (17) and (18) are We will show them for the non-linear oscillator, Eq. (2), P q = Q cos θ − sin θ, (26) below. a 057-2 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) and conjugate variables. In contrast, the equations for Q and ﬃ P P, Eqs. (36) and (37), are linear with coe cients that are p = −Q sin θ − cos θ, (27) functions of time t determined implicitly by A and θ.We a also see that Eq. (34) predicts an explosion of amplitude A and q(t)andp(t) are easily obtained. Note that to perform for <0. the transformations of Eqs. (17) and (18), it is indispens- Equations (34)-(37) can be systematically solved by able to increase the variable (x,y)to(x, q,y,p). applying the canonical perturbation method, where the For applying the Hamilton-Lie perturbation method, it gauge functions are thoroughly exploited; a summary is is convenient to introduce given in Appendix A. It is shown that the canonical pertur- 1 bation method is applicable even if the amplitude A might A = a2, (28) 2 explode. and to replace Q/a with Q. The resultant 1-form remains 2.2 First-order canonical perturbation anal- canonical, ysis γ = QdA + Pdθ − h(0)dt, (29) Following Ref. [2], we determine the gauge function S (1) such that the Hamiltonian after the Lie transformation with (A,θ,Q, P) being canonical coordinates. is H(1) = h(1). From Eq. (A9) in Appendix A, the condi- Now, we return to Eq. (2) with 0. The perturbed tion for this yields the partial diﬀerential equation for S (1) Hamiltonian h(1) written in the phase space (A,θ,Q, P)is (1) ∂S (1) 3 = h˜ . (38) h(1) = −py3 = − QA2 + QA2 f (θ) + PAf (θ) ∂θ 2 c s = h(1) + h˜ (1), (30) Here, we have used the conditions that S (1) is independent of the time t, and have used and {S (1), h(0)} = {S (1), P} = ∂ S (1). (39) 3 θ h(1) = − QA2, (31) 2 Equation (38) is easily solved, where fc(θ)and fs(θ) are periodic functions (1) 2 S = QA Fs(θ) + PAFc(θ), (40) 1 fc(θ) = 2cos2θ − cos 4θ, (32) 2 where and 1 Fs(θ) = sin(2θ) − sin(4θ), (41) 1 1 8 fs(θ) = − sin 2θ + sin 4θ, (33) 2 4 and and represents the average with respect to θ. 1 1 The Hamilton equations of motion developed from the F (θ) = cos(2θ) − cos(4θ). (42) c 4 16 total Hamiltonian h = h(0) + h(1) are ¯ ¯ ¯ ¯ dA ∂h 3 The Hamiltonian in the phase space (A, θ, Q, P)afterthe = = − A2 + A2 f (θ), (34) dt ∂Q 2 c Lie transformation is

dθ ∂h 3 2 = = 1 + Afs(θ), (35) H = P¯ − Q¯A¯ , (43) dt ∂P 2 dQ ∂h = − = 3QA ¯ ¯ ¯ ¯ dt ∂A and the equations of motion in the phase space (A, θ, Q, P) − [Pfs(θ) + 2QAfc(θ)], (36) are dA¯ ∂H 3 and = = − A¯2, (44) dt ∂Q¯ 2 dP ∂h = − = −A[Pf(θ) + QAf (θ)]. (37) dθ¯ ∂H dt ∂θ s c = = 1, (45) dt ∂P¯ These equations of motion are exact, and prove the eﬀec- dQ¯ ∂H = − = 3Q¯A¯, (46) tiveness of the conjugate variable method and the transfor- dt ∂A¯ mations among phase space coordinates. The Hamiltonian h has a formal characteristic that h is linear in the conjugate and variables Q and P. Consequently, the equations for A and dP¯ ∂H = − = 0. (47) θ, Eqs. (34) and (35), are closed; they do not contain the dt ∂θ¯ 057-3 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008)

(2) The solutions of these equations are easily obtained; for P and Q, while ξ0 has powers of A higher than those in <0 h(1). Equation (55) can be re-arranged as

1 2 3 A¯(t) = A¯0 , (48) (2) PA (2) QA (2) (2) − / A¯ ||t ξ = − F (θ) − F (θ) + ξ , (56) 1 (3 2) 0 0 2 P 2 Q 0 where A¯0 is the initial value of A¯. On the other hand, the (2) (2) where FP (θ)andFQ (θ) are periodic functions whose av- solution (2) erages are zero, and the average of ξ is 2 0 ¯ = ¯ − 3 ¯ | | Q(t) Q0 1 A0 t , (49) 27 2 ξ(2) = PA2. (57) 0 64 becomes zero at the time when A¯ explodes (Q¯ 0 is the initial value of Q¯); this reflects the conservation property of the Therefore, when we choose the gauge function in phase volume in the (A,θ,Q, P) space. Eq. (A21) as Let us observe the motion on the original phase space PA2 QA3 (A,θ,Q, P). We need the generators g1 and g2 of the Lie S (2) = F(2)(θ)dθ + F(2)(θ)dθ, (58) (1) (1) 2 P 2 Q transformation (A,θ,Q, P) → (A¯, θ,¯ Q¯, P¯). They are the averaged Hamiltonian in the second order is ∂S (1) g1 = − = −A2F (θ), (50) (1) ∂Q s 3 27 H = P¯ − Q¯A¯2 − 2P¯A¯2. (59) 2 64 and Let us first notice that since the averaged Hamiltonian H ∂S (1) 2 = − = − is linear in P¯ and Q¯, they are not incorporated in the equa- g(1) AFc(θ). (51) ∂P tions for A¯ and θ¯. Next, the Hamiltonian H does not have The backward transformations are therefore given by θ¯ due to the averaging, the conjugate variable P¯ is the con- stant of motion. We also see that the second-order per- = ¯ − 1 = ¯ − ¯2 ¯ A A g(1) A A Fs(θ), (52) turbation does not influence the behavior of amplitude A¯, which remains Eq. (48), but influences the behavior of θ¯, and the equation of which is 2 θ = θ¯ − g = θ¯ − AF¯ s(θ¯). (53) (1) dθ¯ ∂H 27 √ = = 1 − 2A¯2, (60) By applying them to x(t) = 2A(t)cosθ(t), we have dt ∂P¯ 64 −1/2 and using Eq. (48), we have 3 2 2 x(t) = a 1 − a ||t cos(t) + O(,|| t), (54) 4 9 1 √ θ¯(t) = θ + t − (||A¯ ) − 1 . (61) 0 32 0 − | | ¯ whereweseta = 2A0. We found that the solution given 1 3/2 A0t by Eq. (54) accords with the solution given by the multi- The second-order perturbation delays the phase because scale expansion method [3]. dθ/¯ dt − 1 < 0, irrespective of the sign of ; the phase delay becomes remarkable as t approaches the explosion time. 2.3 Second-order canonical perturbation Finally, let us obtain the Lie transformation analysis (A,θ,Q, P) → (A¯, θ,¯ Q¯, P¯). The generators produced From Eq. (31) and Eq. (40), we have for Eq. (A22) in by S (2) are Appendix A ∂S (2) A3 3 3 g1 = − = − F(2)(θ)dθ, (62) {S (1), h(1)} = − F (θ){PA, QA2} = − F (θ)PA2. (2) ∂Q 2 Q 2 c 2 c Notice that the Poisson bracket results in a linear function and P of . Similar but lengthy algebra with Eq. (A22) gives ∂S (1) A2 g2 = − = − F(2)(θ)dθ. (63) 2 (2) ∂P 2 P (2) = − PA + ξ0 [3Fc(θ) fs(θ)Fs(θ) 2 Next, let us investigate the second-order effects of the gen- − + − 2 fc(θ)Fc(θ) Fc(θ) fs (θ) ( fs(θ)) ] j erators g(1). By substituting Eqs. (50) and (51) into the QA3 − [F (θ) f (θ) − f (θ) f (θ)]. (55) fourth term in the right-hand side of Eq. (A24), we get 2 c c s c ν 1 = ν − 2 Due to the properties of the Poisson bracket, the second- (g(1)∂ν)g(1) g(1)∂ν( A Fs) 3 2 order Hamiltonian is still linear in the conjugate variables = A [2(Fs) + Fc fc], (64) 057-4 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) and We first study the guiding center motion in a uniform magnetic field, and demonstrate that doubling the variables ν 2 = ν − (g(1)∂ν)g(1) g(1)∂ν( AFc) enables us to perform a canonical transformation in such a 2 = A [FsFc + Fc fs]. (65) manner that the guiding center motion looks straight on the transformed phase space [Eqs. (91) and (93)]. This is a 3 4 Here, g(1), g(1) that contain the conjugate variables do not nontrivial result and is the starting point for the perturba- contribute to the above results. Consequently, the back- tion analysis. In order to make the analysis intelligible, we ward Lie transformation does not mix the original variables adopt a simple configuration of magnetic fields, Eq. (98). A,θwith the conjugate variables Q, P. Equation (65) implies that the Lie transformation does 3.1 Guiding center motion in a uniform not yield any secular terms in θ, but Eq. (64) produces a magnetic field secular term Let a uniform magnetic field be given by

2 83 2(Fs) + Fc fc = , (66) = = 64 B Bb Be3, (71) and then the averaged solution of A is where e3 is the unit vector along the z-axis. Also, let e1 and e be unit vectors along the x-andy- axes, respectively. 83 2 A(t) = A¯(t) 1 + (A¯)2 , (67) Then, the Hamiltonian in Eq. (70) reads 128 H = u q + Ωy · (u × b), (72) in the original phase space. j j Although the conjugate variable method doubles the where Ω = eB/m. Following the theory of guiding center variables, we obtain the equations of A and θ and their motion [7], we introduce the auxiliary vectors Lie transformations unmixed with the conjugate variables

P and Q until the second order. These are the consequences a = e1 cos θ − e2 sin θ, (73) of the Poisson bracket’s properties and should be valid for any orders, although a strong proof is not provided here. and If mixing occurred at each order, the conjugate variable = − − method would be complicated, and therefore, would not c e1 sin θ e2 cos θ, (74) be an eﬀective tool. where the angle θ is deﬁned by u · e 3. Analysis of Charged Particle Mo- = −1 1 θ tan · . (75) tion u e2

In the Cartesian coordinate system (x1, x2, x3), the The inverse transformations for Eqs. (73) and (74) are equation of motion for a charged particle with mass m and e = a θ − c θ, electrical charge e in a magnetic ﬁeld B (the Lorentz equa- 1 cos sin (76) tion) is and

dx j du j e = u j, = (u × B) j, j = 1, 2, 3. (68) e = −a sin θ − c cos θ. (77) dt dt m 2

By introducing the conjugate variables q j(res.yj)for Then we have x (res. u ), the fundamental 1-form for the Lorentz equa- j j = + tion is constructed as u u⊥ c u b, (78) u × b = −u⊥ a, (79) γ = q jdx j + y jdu j −Hdt, (69) and where the Hamiltonian H is = = − e da cdθ, dc adθ. (80) H = u q + y · (u × B). (70) j j m Here, let us introduce the guiding center coordinates Here, the summation convention is implied for the repeated (R, u⊥,θ,u ) by the transformation index. Since the Hamiltonian H is linear in B, H is eas- = + u⊥ = ily decomposed into unperturbed and perturbed parts when x R Ω a, R (X, Y, Z). (81) expressing B = B0 + B˜ . Therefore, as long as the unperturbed Hamiltonian made by B0 is solvable, the canoni- Then we have cal perturbation method can be systematically applied to = + du⊥ + u⊥ Eq. (69). dx dR Ω a Ω dθc, (82) 057-5 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) and 3.2 Canonical perturbation analysis of the guiding center motion du = du⊥ c + du b − u⊥ adθ. (83) We adopt a simple model where the magnetic fields These are well-known procedures; however, by applying consists of them in Eqs. (69) and (72), we have B = Bb + B˜ (x), (98) q · dx + y · du = q · dR + y du qa and B˜ is assumed to be independent of z and perpendicular + + yc du⊥ Ω to b, qc + − y u⊥dθ, (84) Ω a B˜ (x) · b = 0. (99) and Following Ref. [8], we define the transformation from the H = + Ω qc − u q Ω ya u⊥, (85) Cartesian coordinate system to the guiding center coordinate system (R, u ,μ,θ,q,y , P, M) using the uniform field; where the transformation is given by Eq. (81), and the vectors a c q = q · b, qa = q · a, qc = q · c, (86) and are defined by Eqs. (73) and (74). Consequently the 1-form remains Eq. (91), whereas the Hamiltonian consists and y , ya and yc are similarly defined. Equations (84) and of (85) imply the transformations H = H (0) + h, (100) qa + = u⊥ Ω yc Ω P, (87) where the unperturbed Hamiltonian is and (0) qc − = H = u q + ΩM, (101) Ω ya u⊥ M. (88) and the perturbed Hamiltonian is given by These are the transformations (ya,yc) → (P, M) u⊥ q = − a h = Ωy · (u × B˜ (x)). (102) yc Ω P Ω , (89) and Here, the perturbed magnetic field is normalized to the uni-

M qc form magnetic field B. ya = − + , (90) u⊥ Ω Let LB be the characteristic length over which the perturbed magnetic field varies, and assume that the Larmor and we get a canonical 1-form radius ρ is much shorter than LB such that = · + + + −H γ q dR y du Pdμ Mdθ dt, (91) ρ = O(), (103) where LB 2 u⊥ ˜ μ = (92) and we expand B(x)as 2Ω ˜ = ˜ + u⊥ ·∇ ˜ is the magnetic moment, and the Hamiltonian is given by B(x) B(R) Ω (a )B(R). (104) H = + Ω u q M. (93) Consequently, we have Consequently, the transformation from (x, u, q, y)to h = h(1) + 2h(2), (105) (R, u ,μ,θ,q,y , P, M) is a canonical transformation that has a trivially soluble Hamiltonian; the equations are where the first-order Hamiltonian reads dR ∂H = = u b, (94) h(1) = Ωy · (u × B˜ (R)), (106) dt ∂q du ∂H = = 0, (95) and the second-order Hamiltonian reads dt ∂y (2) dμ ∂H h = u⊥y · u × (a ·∇)B˜ (R) . (107) = = 0, (96) dt ∂P Based on the assumption, the perturbed field is written as and dθ ∂H B˜ (R) = B˜ (X, Y)e + B˜ (X, Y)e . (108) = = Ω. (97) x 1 y 2 dt ∂M These results again demonstrate the effectiveness of the 3.3 First-order perturbation analysis conjugate variable method. The perturbed Hamiltonian h(1) can be 057-6 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) easily expressed by the canonical coordinates Here, we used the assumption that S (1) is independent of (R, u ,μ,θ,q,y , P, M), both time and Z. Equation (123) gives the solution (1) P u u⊥ u h = Ωu⊥ y − u B˜ (R,θ) − MΩ B˜ (R,θ) (1) = − + ˜ Ω a c S q M Ba(R,θ) u⊥ Ω u⊥ + u (q1B˜x(R) + q2B˜y(R)) (109) u − u⊥ y − P B˜ c(R,θ). (124) = h˜ (1) + h(1), (110) Ω and The averaged Hamiltonian is therefore (0) (1) (1) H = H + h , (125) h = u (q1B˜ x(R) + q2B˜ y(R)). (111)

Here, and the equations of motion are

˜ = ˜ − ˜ dX Ba(R,θ) Bx(R)cosθ By(R)sinθ, (112) = u B˜ , (126) dt x and dY = u B˜ , (127) dt y B˜ c(R,θ) = −B˜ x(R)sinθ − B˜ y(R)cosθ. (113) dZ = u , (128) dt These mean du ∂H = = 0, (129) dt ∂y B˜ (R) = Bã(R,θ)a + B˜ c(R,θ)c. (114) dμ ∂H = = 0, (130) Let us make the equations of motion from H = H (0)+h(1). dt ∂P They are and dX ∂H = = u B˜ , (115) dθ ∂H dt ∂q x = = Ω. (131) 1 dt ∂M H dY = ∂ = ˜ u By, (116) Since the Hamiltonian H does not include the conjugate dt ∂q2 dZ ∂H variables y and P, the corresponding variables u and μ = = u , (117) (and consequently the kinetic energy E = Ωμ + u2/2) are dt ∂q conserved. du ∂H = = Ωu⊥ B˜ a = Ω(u × B˜ ) · b, (118) dt ∂y 3.4 Second-order perturbation analysis dμ ∂H = = −u⊥u B˜ , (119) When we consider the averaged Hamiltonian, dt ∂P a Eq. (A22) is reduced to and (2) = − (2) + 1{ (1) ˜ (1)} dθ ∂H u ξ0 h S , h , (132) = = Ω 1 − B˜ c . (120) 2 dt ∂M u⊥ since {S (1), h(1)} is periodic with respect to θ, and con- The first three equations express the guiding center’s mo- sequently its average vanishes. Our work is then re- tion along the magnetic field lines, and the other three duced to calculating {S (1), h˜ (1)} and h(2). For calculating equations are identical to the equation {S (1), h˜ (1)}, it is convenient to express h˜ (1) and S (1) as du = Ωu × (b + B˜ ) (121) h˜ (1) = h B˜ + h B˜ , (133) dt a a c c expressed in terms of u , μ and θ. and The Poisson bracket in Eq. (A9) is, for the present (1) = + problem, S S aB˜ a S cB˜ c. (134)

(1) (1) (1) (1) (0) ∂S ∂S ∂S Here {S , H } = u + Ω − q . (122) ∂Z ∂θ ∂y P ha = Ωu⊥ y − u , (135) Therefore, we obtain an equation for S (1) that yields the Ω u Hamiltonian averaged over θ hc = −MΩ , (136) u⊥ (1) (1) ∂S ∂S (1) u⊥ u u⊥ hc Ω − q = h˜ . (123) S a = − q + M = −q + , (137) ∂θ ∂y Ω u⊥ Ω Ω 057-7 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) and Applying the transformation formula, Eqs. (89) and (90), (2) u for Eq. (154) yields h expressed in the phase space coor- = − − = ha S c u⊥ y Ω P Ω . (138) dinates (R, u ,μ,θ,q,y , P, M)

2 Using Eqs. (133) and (134), we have ⊥ (2) = (u ) Ω − h Ω Wa( y Pu ) {S (1), h˜ (1)} = {S B˜ , h B˜ } + {S B˜ , h B˜ } a a a a a a c c − + u⊥ + Mu Wc Ω u (q1W1 q2W2), (155) + {S cB˜ c, haB˜ a} + {S cB˜ c, hcB˜ c}. (139) and averaging it gives Averaging over θ operates on B˜ a and B˜ c, resulting in Mu 2 2 1 2 (2) = − = (B˜ ) = (B˜ ) = |B˜ | , (140) h Mu Wc Jz. (156) a c 2 2 and From Eqs. (142) and (156), we obtain the averaged second- order Hamiltonian B˜ aB˜c = 0. (141) 1 H(2) = −ξ(2) = h(2)− {S (1), h˜ (1)} After lengthy algebraic computation for the right-hand side 0 2 of Eq. (139), the details of which are shown in Appendix B, Mu 1 2 M 2 = J − |B˜ | q u − (u ) , (157) we have 2 z 2 2μ (1) (1) 1 2 M 2 {S , h˜ } = |B˜ | 2q u − (u ) . (142) and then the total Hamiltonian 2 μ H = H(0) + H(1) + 2H(2), (158) Now let us rewrite h(2) as

(2) where h = u⊥y · [u × W] . (143) (0) = + Ω Here, H q u M, (159)

W = (a ·∇)B˜ (R) and

= W1e1 + W2e2 (144) (1) H = u (q1B˜ x(R) + q2B˜ y(R)). (160) = Wa a + Wc c, (145) Again, we can observe that the Hamiltonian H retains W1 = (cos θ∂x − sin θ∂y)B˜ x(R), (146) linearity in the conjugate variables (q,y , P, M). Conse- W2 = (cos θ∂x − sin θ∂y)B˜ y(R), (147) quently, the equations of motion for (R, u ,μ,θ) do not con- = − Wa W1 cos θ W2 sin θ, (148) tain any conjugate variables; they are and dX ∂H = = u B˜ (R), (161) dt ∂q x W = −W sin θ − W cos θ. (149) 1 c 1 2 dY ∂H = = u B˜ y(R), (162) Averaging them over θ gives dt ∂q2 2 dZ = ∂H = − | ˜ |2 W1 = W2 = 0, (150) u 1 B , (163) dt ∂q 2 1 W = (∂ B˜ + ∂ B˜ ) = 0, (151) du ∂H a 2 x x y y = = 0, (164) dt ∂y and dμ ∂H = = 0, (165) 1 1 dt ∂P W = (∂ B˜ − ∂ B˜ ) = − J , (152) c 2 y x x y 2 z and where Jz is the current density 2 dθ ∂H 2 Jz (u ) 2 = = Ω + u + |B˜ | . (166) Jz = ∂xB˜y − ∂yB˜ x. (153) dt ∂M 2 2μ

Applying Eqs. (148) and (149) to Eq. (143), we have It is important to study whether the conjugate variable method gives the same results as those of the usual canoni- × = − + u W u⊥Wa b u Wa c u Wc a, cal (or non-canonical) Hamilton-Lie perturbation method and then or the multi-scale expansion method. However, this would require rigorous investigation with deep mathemat- (2) 2 h = y (u⊥) Wa + u⊥u (yaWc − ycWa). (154) ical analysis, and therefore remains open to future study. 057-8 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008)

4. Summary where ξ(2) is given by It has been shown that the conjugate variable method (2) (2) (1) 1 2 (0) enables us to apply the Hamilton-Lie perturbation method ξ = γ −L(1)γ + (L(1)) γ . (A7) 2 systematically to any system of ordinary differential equa- (2) tions that does not have the Hamiltonian dynamic struc- In this appendix, we give the expression for ξ assuming ( j) ture. Although the method doubles the unknown variables, that the 1-form is canonical. In the following, let h be ( j) the equations for the original variables are derived without the Hamiltonian in the corresponding form γ . mixing with the conjugate variables at each order in the In the first-order canonical perturbation analysis, the j canonical perturbation analysis. generators g(1) for space variables are given by Several problems are open to future study. One impor- g j = ∂ S (1) Jkj, (A8) tant problem is to determine the equivalence of the method (1) k proposed in the present work with other methods, that is, where S (1) is the gauge function. The transformed Hamil- whether the results by the conjugate variable method are tonian H(1) reads the same as those by other methods, such as the usual (1) (1) (1) (0) (1) canonical (or non-canonical) perturbation method, and the −H = ∂tS + {S , h }−h , (A9) classical multi-scale expansion method. Another inter- where the Poisson bracket {S (1), h(0)} is defined by esting problem in plasma physics is to apply the present (1) (0) (1) kj (0) method to eigenvalue equations related to MHD stabil- {S , h } = ∂kS J ∂ jh . (A10) ity analysis, such as the MHD ballooning mode equation. Studies on these problems will be reported in the near fu- In Hamilton-Lie perturbation analysis, the gauge function (1) (1) ture. S is determined such that the Hamiltonian H becomes simple. It was found that ξ(2) in Eq. (A7) can be expressed in a simple form, since γ(1) and γ(2) are canonical. Acknowledgment We start with the calculation of L γ(1) in Eq. (A7). The author thanks Dr. M. Kikuchi, the Unit Manager (1) Let us tentatively write the gauge function defined accord- of the Division of Advanced Plasma Research, Fusion Re- ing to the first-order perturbation analysis, Eq. (A9), as φ. search and Development Directorate at Japan Atomic En- Then the generators read ergy Agency, for his encouragement. j kj g = ∂kφ J ( j 0), (A11) Appendix A: Summary of Canonical and the generator for the time is g0 = 0. The Lie derivative Hamilton-Lie Perturbation Analysis of γ(1) with respect to g j results in

Following Ref. [2], let a 1-form be given by (1) [Lφγ ] j = 0, (A12) γ = γ(0) + γ(1) + 2γ(2) + ··· , (A1) and 1 2N 0 (0) in a phase space (z , ··· , z ) with z := t,andletγ be (1) kj (1) [Lφγ ]0 = −∂kφJ ∂ jh . (A13) integrable. Recall that, with respect to the generators gν for the Lie transformation, the Lie derivative Lg for a 1-form Consequently, we have

γ is deﬁned by (1) (1) Lφγ = −{φ, h }dt. (A14) L = ν = ··· ( gγ)μ g ωνμ,μ,ν 0, , 2N. (A2) 2 (0) By calculation of (L(1)) γ in Eq. (A7) using a procedure Here similar to that used for Eq. (A14), we have (0) j (0) Ω = Lφγ = ∂ jφ dz −{φ, h }dt. (A15) ωνμ = ∂νγμ − ∂μγν (A3) j kj Next, letg ¯ = ∂kψJ be generators deﬁned by another is the Lagrangian tensor; the inverse of it, which is ex- function ψ, and operate Lψ on Ω.Thenwehave pressed by Jkj, is the Poisson tensor. The 1-form Γ after (0) (0) the Lie transformation LψLφγ = −{ψ, {φ, h }}dt. (A16)

Γ = γ(0) + Γ(1) + 2Γ(2) + ··· , (A4) By applying Eq. (A16) in Eq. (A7), we have L 2 = −{ { (0)}} reads ( φ) γ0 φ, φ, h dt. (A17)

(1) (0) (1) (1) From Eqs. (A14) and (A17), and from the assumption that Γ = −L(1)γ + dS + γ , (A5) γ(2) = −h(2)dt, we observe that ξ(2) is canonical, and obtain and the second-order Hamiltonian

(2) (2) (1) (1) 1 (1) (1) (0) Γ(2) = −L γ(0) + S (2) + ξ(2), ξ = −h +{S , h }− {S , {S , h }}. (A18) (2) d (A6) 0 2 057-9 Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008)

Here, let us return to Eq. (30). The gauge function S (1) is {S cB˜ c, haB˜ a} = B˜ cB˜ a{S c, ha} = 0, (B3) determined such that and {S (1), h(0)} = h˜ (1). (A19) 1 2 {S cB˜ c, hcB˜ c} = |B˜ | {S c, hc}. (B4) We have therefore in Eq. (A18) 2 1 By adding Eqs. (B1), (B2), and (B4), we get {S (1), h(1)}− {S (1), {S (1), h(0)}} 2 (1) (1) 1 2 1 {S , h˜ } = |B˜ | [{S a, ha} + {S c, hc} = {S (1), h(1)} + {S (1), h˜ (1)}. (A20) 2 2 + ∂M(S ahc)] . (B5) The second-order Hamiltonian is Next, from Eqs. (136) and (138), we get − (2) = { (2) (0)} + (2) H S , h ξ0 , (A21) { } = − 1 { } + 1 { } S a, ha Ω q u⊥, ha Ω hc, ha , and 1 and ξ(2) = −h(2) + {S (1), h(1)} + {S (1), h˜ (1)}. (A22) 0 2 { } = 1 { } S c, hc Ω ha, hc . We can apply the same procedure as the ﬁrst-order analysis for Eq. (A21) to determine the gauge function S (2) that Then Eq. (B5) is further reduced to makes the transformed Hamiltonian H(2) as simple as pos- (1) (2) j | ˜ |2 { } sible. From S and S we get the generators g and (1) (1) B q u⊥, ha (1) {S , h˜ }= − +∂M(S ahc) . (B6) j 2 Ω g(2), and we can construct the successive Lie transforma- μ μ tion from (¯z )to(z ) Finally, by substituting

μ 2 μ z = exp(− L(2))exp(−L(1))I (¯z), (A23) {q u⊥, ha} = −Ωq u , (B7) μ = μ where I (¯z) z¯ is the coordinate function. By expanding and the right-hand side to O(2), we have 2 (u ) = − j = j − j μ − 2 j μ ∂M(S ahc) q u M, (B8) z z¯ g(1)(¯z ) g(2)(¯z ) μ 2 + [(gν ∂ )g j ](¯zμ). (A24) in Eq. (B6), we obtain Eq. (142). 2 (1) ν (1)

j When we substitutez ¯ (t), the solutions of the second-order [1] E.C.G. Sudarshan and N. Mukunda, Classical Dynamics: A analysis, into the right-hand side of Eq. (A24), we obtain Modern Perspective (Rober E. Krieger Publishing Company, the solution z j(t) in the original phase space. Note that Florida, U.S.A, 1983). when S (1) is used for the fourth term of Eq. (A24), we have [2] J.R. Cary and R.G. Littlejohn, Ann. Phys. 151, 1 (1983). [3] C.M. Bender and S.A. Orszag, Advanced Mathematical ν j (1) (1) μν σ j Methods for Scientists and Engineers (McGraw-Hill, New (g ∂ν)g = ∂μS ∂ν∂σS J J . (A25) (1) (1) York, U.S.A, 1978) Chap.11. [4] B. Jouvet and R. Phytian, Phys. Rev. A 19, 1350 (1979). (1) (1) Appendix B: Calculation of {S , h˜ } [5] T.G. Northrop, The Adiabatic Motion of Charged Particles By calculating the Poisson bracket in the right-hand (Interscience, New York, U.S.A, 1963). side of Eq. (139) and using Eqs. (140) and (141), we obtain [6] A.I. Morozov and L.S. Solov’ev, Motion of Charged Parti- cles in Electromagnetic Fields, in Reviews of Plasma Physics 1 Vol.2, 201-297, Edited by M.A. Leontovich, Consultants Bu- {S B˜ , h B˜ } = |B˜ |2{S , h }, (B1) a a a a 2 a a reau, New York (1966). [7] R.G. Littlejohn, J. Plasma Phys. 29, 111 (1983). 1 2 {S aB˜ a, hcB˜ c} = |B˜ | ∂M(S ahc), (B2) [8] T.S. Hahm, W.W. Lee and A. Brizard, Phys. Fluids 31, 1940 2 (1988).

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