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

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The Conjugate Variable Method in Hamilton-Lie Perturbation Theory −Applications to Plasma Physics− Plasma and Fusion Research: Regular Articles Volume 3, 057 (2008) The Conjugate Variable Method in Hamilton-Lie Perturbation Theory −Applications to Plasma Physics− 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 cen- ter 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 de- velopment 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 perturba- tion 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.
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