Particle Canonical Variables and Guiding Center Hamiltonian up To

Home , Conjugate variables

Particle Canonical Variables and Guiding center Hamiltonian up

to second order in the Larmor radius

H. Vernon Wong

Institute for Fusion Studies

The University of Texas at Austin

Austin, TX 78712

Abstract

A generating function, expressed as a power series in the particle Larmor

radius, is used to relate an arbitrary set of magnetic eld line coordinates to

particle canonical variables. A systematic procedure is described for succes-

sively choosing the generating function at each order in the Larmor radius

so that the transformed particle Hamiltonian is independent of the Larmor

phase angle. The particle guiding center Hamiltonian up to second order in

the Larmor radius is thereby derived. The analysis includes nite equilib-

rium electrostatic elds and time dependent electromagnetic eld perturba-

tions. The transformations which relate magnetic ux coordinates to particle

canonical variables are also discussed.

I. INTRODUCTION

The orbits of charged particles in a large magnetic eld are most conveniently described using curvilinear eld line coordinates (, , ),where and label a eld line, is a third coordinate measuring ‘distance along the eld line’, and the magnetic eld is represented by B = ∇ ∇. To lowest order in the Larmor radius, the particle moves along a given eld

1 d d d 6 line ( dt =0, dt =0, dt = 0) and gyrates at the cyclotron frequency about the eld line. The particle dynamics is then separable into the three time scales characteristic of charged particle motion Thus, in formulating a Hamiltonian description of charged particle motion, it is advantageous to use eld line coordinates and to introduce a transformation to particle canonical variables such that [1]: (1) the coordinates (, ) constitute a pair of canonically conjugate variables describing slow perpendicular ‘drifts’; (2) the parallel motion (involving

d , dt ) is described by a second pair; (3) and the perpendicular gyration is described by a third pair. In this paper, we present a Hamiltonian formulation of charged particle guiding center motion in which we use a generating function to carry out the desired transformation. We begin with the single particle Hamiltonian in cartesian coordinates, and we introduce a generating function which transforms the ‘old’ set of cartesian canonical variables (x, px,y,py,z,pz) to a ‘new’ set of canonical variables denoted by (q1,p1,q2,p2,q3,p3). The generating function

G = G(p1,p2,p3,,,) is taken to be a function of the ‘new’ momenta (p1,p2,p3) and the ‘old’ coordinates (x, y, z) through (, , ). The form of the function G, similar to that introduced by Gardner [1], is carefully chosen so that: (1) the parallel and perpendicular dynamics (to lowest order) are decoupled when the motion is described in terms of the ‘new’ canonical variables; (2) the transformation equations relating , , to the new variables have a form which facilitates the expression of all quantities as a Taylor series expansion in the Larmor radius. In Secs. II and III, we describe a general procedure for determining the generating function. A particular case of our generating function where the coordinate was taken to be the distance s(x, y, z) along the eld line was previously discussed in Ref. [2]; alternative generating functions have been discussed by authors listed in Ref. [3]. Here, we consider the general case of an arbitrary choice of , and we extend our analysis to include: (a) nite electrostatic elds where the ‘electric drift’ velocity is of the order of the particle velocity; (b) small perturbations of the equilibrium magnetic elds; (c) second order Larmor radius

2 corrections. Initially, we derive in Sec. II the particle canonical variables and Hamiltonian for guiding center motion up to rst order in the Larmor radius. In Sec. III, we express the generating function in the form of a Taylor series expansion in the Larmor radius, with the lowest order terms determined in Sec. II. The additional higher order terms can then be systematically determined (order by order) by requiring the particle Hamiltonian at each order to be independent of the Larmor phase angle. We follow this procedure to evaluate the second order Larmor radius corrections to the guiding center Hamiltonian. In discussing toroidal equilibria, it is convenient to represent the magnetic eld in terms of magnetic ux coordinates , , where is the ux variable, and are periodic generalized poloidal and toroidal angular variables. However, magnetic ux coordinates can readily be related to magnetic eld line coordinates. We can therefore use the results of Sec. II and Sec. III to relate particle canonical variables to magnetic ux coordinates. We discuss these transformations in Sec. IV. Our procedure obviates the need to carry out a successive sequence of canonical transformations, and it is a relatively straightforward method for deriving particle canonical variables and Hamiltonian for guiding center motion in complex magnetic eld geometries up to second order in the Larmor radius. The guiding center equations of motion can also be obtained from the single particle La- grangian [4-8]. In an alternative formulation of guiding center motion in complex magnetic eld geometries, but involving non-canonical variables, Littlejohn [5] derived the guiding center Lagrangian and Hamiltonian by a systematic averaging procedure which can be car- ried out to any desired order in the Larmor radius. Other formulations have been discussed by Boozer [6], White et al. [7], and Meiss et al. [8] in the context of toroidal magnetic eld geometries.

3 II. MAGNETIC FIELD LINE COORDINATES

A. equilibrium magnetic elds

With eld line curvilinear coordinates (x, y, z),(x, y, z),(x, y, z), we take the electromagnetic magnetic vector potential A to be:

A = ∇ (1) and we have the following two representations of the magnetic eld B:

∂r B = ∇ ∇ = Bb (2) ∂

B = B (b∇ + b∇ + b∇) (3)

where B is the magnitude of the magnetic eld, the covariant components of the unit ∂r ∂r ∂r magnetic eld b = B/B are denoted by b = b ∂, b = b ∂ , b = b ∂, the position vector is r = r(, , ), and b = b ∇. Note that b b =1. We begin with the particle Hamiltonian written in cartesian coordinates

(p A)2 H = + (4) 2m

where p =(px,py,pz) and (x, y, z) are the canonical momenta and conjugate coordinate variables. We have set the electric charge e = 1 and the speed of light c = 1 for convenience.

We transform to new canonical variables (p1,q1,p2,q2,p3,q3) by introducing a generating

function G(p1,p2,p3,,,) of the old coordinates (x, y, z) and the new momenta (p1,p2,p3), where the dependence on (x, y, z) is through the eld line coordinates (, , ). The coordinate transformation equations are:

p = ∇G (5) ∂G qi = i =1, 2, 3. (6) ∂pi

The particle velocity v is given by:

4 mv = p A = ∇G ∇ ! ∂G ∂G ∂G = ∇ + ∇ + ∇. (7) ∂ ∂ ∂

In general, eld line coordinates are not orthogonal, and ∇ is not in the direction of the magnetic eld. It is desirable to resolve the velocity vector into components parallel and perpendicular to the magnetic eld, and to that end, we dene a set of unit orthogonal vectors 0, 1, 2:

∇ b ∇ = b= = . (8) 0 1 |∇| 2 |∇|

We express all vector quantities in terms of these unit vectors by making use of the identities

∇ ∇ B ∇ = ∇ + b ∇ |∇|2 |∇|2 1 b b ∇ = b ∇ ∇. b b b

Equation (7) for mv can then be rewritten in the equivalent form:

mv = { 0 ∇G} 0 + { 1 ∇G 1 ∇} 1 + { 2 ∇G 2 ∇} 2. (9)

The particular choice of the generating function G is motivated by the requirement that, to lowest order in a Larmor radius expansion, the particle dynamics is separated into the three time scales characteristic of charged particle motion in a strong magnetic eld, with one pair of canonically conjugate variables (p3,q3) describing fast cyclotron gyration, a second pair (p2,q2) describing parallel motion, and a third pair (p1,q1) describing slow drifts across the magnetic eld. To accomplish this goal, we seek coordinate relationships:

p1 q1 q2 and particle velocity v of the form:

1/2 1/2 m 0 v p2 mv⊥ = b (v b) p3B 1 q3B 2

5 Let the generating function be

2 (2) G = p1 + p2 + G (10)

where G(2) is yet to be determined. We introduce the symbol to identify terms proportional to the particle mass m and powers of the mass. The particle Larmor radius is proportional to m. Hence will be used as a convenient smallness parameter (which is set equal to unity at the end of the analysis) to order terms as a power series in the particle Larmor radius. Substituting G in Eq. (9), we have

m v = bp + 2 ∇G(2) 0 2 0 ! b 2 (2) m 1 v = p2 1 ∇ + 1 ∇ + 1 ∇G b 2 (2) m 2 v = 2 ∇ + 2 ∇G

where the new dependent variable = (/, p1/, p2,,,) is dened to be the following

function of /, p1/, p2,,,:

p1 b = + p2 (11) b

(2) We now consider G (,p2,p3,,,) to be a function of ,p2,p3,,,. The velocity components can therefore be expressed as follows:   m v = bp + D G(2)  0 2 0   !  (2) b ∂G (2) m v = p ∇ + ∇ + |∇| + D G  (12) 1 2 1 1 1  b ∂    (2)  m 2 v = 2 ∇ + D2G

where the operators Dj (j =0, 1, 2) are the components of the spatial gradient operator acting on G(2) but excluding the / dependence in , and they are dened by

∂G(2) D G(2) = ∇G(2)| + ∇| . (13) j j ,p2,p3 ∂ j /,p1,p2

For the coordinates qi, we obtain from Eq (6):

6  (2) ∂G ∂G   q1 = =  ∂p1 ∂   (2) (2)  1 ∂G b ∂G ∂G q2 = = + +  . (14) ∂p b ∂ ∂p  2 2   (2)  1 ∂G ∂G  q3 = 2 = ∂p3 ∂p3

(2) 1/2 1/2 We choose G such that in Eq. (12), m 1 v p3B , and m 2 v q3B .Thus we require: ! ∇ ∇ (2) |∇ | b |∇ |∂G 1/2 p2 + 2 + = p3B b |∇| ∂

(2) B 1/2 1/2 ∂G = q3B = B . |∇| ∂p3 Note that ∂2G(2) B1/2 = . ∂p3∂ |∇| Let G(2) be chosen as follows: ( ) 1/2 2 ∇ ∇ 2 (2) B b p2 b b G = p3 + p2 + 2 (15) |∇| b 2 |∇| 2 b b

2 where we have included an additional term independent of and p3 but proportional to p2 so that the transformation equation for q2 is of the form shown in Eq. (19). It should be noted that with this denition of G(2), the dependent variable is simply

1 ∂G B1/2 related to the canonical coordinate q , q = 2 = . 3 3 ∂p3 |∇| Substituting G(2) in Eq. (12) and Eq. (14), we obtain the following equations relating the particle velocity v and the coordinates , , to the canonical variables qi,pi:

v = v(0) + v(1) (16) where

(0) 1/2 1/2 mv = p2b 0 + p3B 1 q3B 2 (17)

(1) (2) (2) (2) mv = D0G 0 + D1G 1 + D2G 2 (18) and

7  b  = p p + ∇  1 2  b   b ∇ (19) = q1 + p2 +  b     = q2 + ∇ where is the Larmor radius vector m q p = b v(0) = 3 + 3 . (20) B B1/2 1 B1/2 2

(2) Evaluating DjG , we obtain B1/2 B1/2 B D G(2) = V p2p2 + p V p2p3 + V p2q3 + p V p3q3 + 2 V q3q3 j j 3 j |∇| j 3 |∇| j |∇|2 j

p2p2 p2p3 p2q3 p3q3 2 q3q3 = Vj + p3Vj + q3Vj + p3q3Vj + q3Vj (21) where ! p2 b b b b V p2p2 = 2 D D j 2 b j b b j b ( ) B1/2 b |∇| b ∇ ∇ b p2p3 D p2q3 D D Vj = p2 j Vj = p2 1/2 j + 2 j |∇| b B b |∇| b |∇| B1/2 1 |∇|2 ∇ ∇ V p3q3 = D V q3q3 = D j B1/2 j |∇| j 2 B j |∇|2 The transformed Hamiltonian is m m H = v(0) v(0) + + 2mv(0) v(1) + 3 v(1) v(1). (22) 2 2 We have here considered the equilibrium electrostatic potential to be of order (see Sec. 2b for the case of O(1)).

At this point, it is convenient to replace the coordinates q3,p3 with ‘action-angle’ variables

Q3,P3 for perpendicular particle gyration about the magnetic eld,where Q3 is the Larmor phase angle and P3 the magnetic moment. We transform to new canonical variables Qi,Pi dened by:   p = P q = Q  1 1 1 1    . (23) p2 = P2 q2 = Q2     1/2 1/2  p3 =(2P3) cos Q3 q3 =(2P3) sin Q3

8 Thus we have

(0) 1/2 1/2 mv = P2b 0 +(2P3B) cosQ3 1 (2P3B) sinQ3 2. (24)

The Hamiltonian equations of motion are  dQ 1 ∂H dP 1 ∂H  3 ˙ 3 ˙  Q3 = 2 P3 = 2  dt ∂P3 dt ∂Q3   dQ 1 ∂H dP 1 ∂H  2 Q˙ = 2 P˙ = (25) 2 2  dt ∂P2 dt ∂Q2   H H  dQ1 ∂ dP1 ∂  Q˙ 1 = P˙1 =  dt ∂P1 dt ∂Q1

dQ3 1 B dP3 dQ2 P2 To lowest order in , the equations of motion are: dt = m , dt =0, dt = m b b , P 2 dP2 2 ∂ P3 ∂B ∂ dQ1 dP1 dt = m b ∂b m ∂ + ∂, dt =0, dt = 0, and they describe perpendicular gyration and parallel motion along a eld line. The above coordinate transformations achieve the desired time scale separations in the particle dynamics, and they provide a starting point for an investigation of the particle motion in terms of a power series in the Larmor radius. We now assume that the particle Larmor radius || is small compared to the spatial scale lengths of the equilibrium electromagnetic elds and we proceed to carry out a Taylor series expansion in the Larmor radius.

Let 0,0,0 be dened by    0 = P1 P2 [bb ]  0,0,0    = Q + P [b b] (26) 0 1 2 0,0,0      0 = Q2

These equations implicitly determine 0 = 0(P1,Q1,P2,Q2), 0 = 0(P1,Q1,P2,Q2). Then, expanding in a power series in , we obtain from Eq. (19):

(1) = 0 + +

(1) = 0 +

9 where

(1) =[ ∇] (1) =[ ∇] (1) =[ ∇] . (27) 0,0,0 0,0,0 0,0,0

Note that

r(, , )=r0 + +

r0 = r(0,0,0)

and r0 is identiable as the particle guiding center position vector. The Taylor series expansion of the Hamiltonian is:

H = H(0) + 2 H(1) + ∇H(0) + (28) where " # P 2bb B H(0) = 2 + P + 2m m 3 h i r=r0 H(1) = m v(0) v(1) . r=r0

The functions on the right-hand side of these equations are evaluated at = 0, = 0,=

(0) The guiding center Hamiltonian is independent of the Larmor phase angle Q3. H is

(2) (1) independent of Q3, a fact guaranteed by our choice of G . However, H depends on Q3. In Sec. 3, we demonstrate that G can be expressed as a power series expansion in and can be chosen order by order so that H is independent of Q3 at each order in ; furthermore, the rst order guiding center Hamiltonian is the gyrophase average of H(1).

The guiding center Hamiltonian H(Q1,P1,Q2,P2,P3), to rst order in the Larmor radius, is therefore given by:

H = H(Q1,P1,Q2,P2,P3)=H0 + H1 + (29) where

10 (0) H0(P2,P3, r0)=H (30) D E (1) H1(P2,P3, r0)= H . (31)

The angular brackets hi denote averaging over the gyrophase Q3 I dQ h()i 3 (). 2

Evaluating the gyrophase-average, we obtain: " # P b P 2 { p2p2 q3q3 } 3 1/2 { p2p3 p2q3 } H1 = V0 + P3V0 + B V1 V2 (32) m m r=r * + 0 P b P 3bb 1 2 {V p2p2 + P V q3q3 } = 2 ∇ b (b ∇)b ∇ b ∇b m 0 3 0 2mB b P2P3b 2 ∇ 2 * + 2m P B1/2 P P b 3 {V p2p3 V p2q3 } = 2 3 b ∇b. m 1 2 m

The vector forms of the above equations were obtained by evaluating the vector components of the curl of ∇ and ∇. The parallel velocity component is

hD Ei (0) (1) (0) mVk(r0,P2,P3)=m b v + b v + b ( ∇)v r=r0 =[P b + {V p2p2 + P V q3q3 } + P b ∇b] . (33) 2 0 3 0 3 r=r0

The guiding center Hamiltonian can therefore be written in the form:

2 mVk P B(r ) H = H + H = + 3 0 + (r ) . (34) 0 0 1 2 m 0

The guiding center equations of motion are determined by:  ∂H ∂H  Q˙ = + P˙ = +  1 1  ∂P1 ∂Q1   1 ∂H 1 ∂H ˙ ˙ . (35) Q2 = + P2 = +  ∂P2 ∂Q2     P˙3 =0

11 In Appendix A, we demonstrate that the Hamiltonian equations of motion reproduce the correct guiding center drifts to rst order in the Larmor radius. Thus, for of order O(), we have

dr ∂r ∂r ∂r =˙ + ˙ +˙ dt ∂ ∂ ∂ 2 = Vkb + vD + O( )

where vD is the guiding center drift velocity

(bP )2 P 1 v = 2 b (b ∇)b + 3 b ∇B + b ∇ D mB mB B

Note that we have deleted the subscript “0” for convenience.

B. nite equilibrium electrostatic elds

In the previous section, was considered to be of order . Let us consider O(1) to be of order one. The perpendicular ‘electric’ drift is:

E b ∂r ∂r vE = = vE ∇ + vE ∇ B ∂ ∂ E∇ E∇ = v + v (36)

where vE is now of the order of the particle velocity. The electric eld is E = ∇(, , ), where the parallel component is ordered to be small (|b ∇||b ∇|) compared to the perpendicular component. We introduce the generating function G

2 (2) G = p1 + p2 + G

with G(2) modied so that:

1/2 E m 1 v = p3B + m 1 v +

1/2 E m 2 v = q3B + m 2 v +

12 where q = ∂G(2) . Note that the perpendicular velocity components of gyration about the 3 ∂p3 1/2 1/2 magnetic eld (p3b , q3B ) are dened in the frame of reference moving with the velocity vE. By following the procedure described earlier, we are led to the following choice for G(2): ! B1/2 b 2 ∇ ∇ G(2) = p + p + mvE + 3 ∇| 2 b 2 |∇|2 ! ! ! 1 b E b E mp2 E b E b p2 + mv p2 + mv + v v (37) 2 b b 2 b b where the dependent variable is dened by:

p1 b E = + p2 + mv . b

The additional terms independent of and p3 are included so that the transformation equation for q2 is of the form shown in Eq. (43). Substituting G(2) in Eq. (12), we obtain

v = v(0) + v(1) (38) where

(0) 1/2 1/2 mv = mu + p3B 1 q3B 2 (39)

(1) (2) (2) (2) mv = D0G 0 + D1G 1 + D2G 2 (40)

E mu = p2b b + mv

= p2∇ + mu∇ + mu∇ (41)

b E b E mu = p2 + mv mu = p2 + mv . b b

(2) Evaluating DjG , we obtain:

D (2) uu up3 uq3 p3q3 2 q3q3 jG = Vj + p3Vj + q3Vj + p3q3Vj + q3Vj (42) where

13 ! m2 mp b b V uu = (u D u u D u )+ 2 D vE vE j 2 j j 2 j b b ( ) B1/2 |∇| ∇ ∇ V up3 = m D u V uq3 = m D u + D u j |∇| j j B1/2 j |∇|2 j |∇| B1/2 1 |∇|2 ∇ ∇ V p3q3 = D V q3q3 = D . j B1/2 j |∇| j 2 B j |∇|2

Substituting G(2) in Eq. (14), we obtain   = p mu + ∇  1   

= q1 + mu + ∇  . (43)     E E ∇  = q2 + mb (bv bv )+

The transformed Hamiltonian H is:

m m H = + v(0) v(0) + 2mv(0) v(1) + 3 v(1) v(1). (44) 2 2

As before, we can now carry out an expansion in the Larmor radius. We omit the details. The guiding center position vector is:

r0 = r(0,0,0) where h i   E  0 = P1 P2bb + mv  r=r0   h i  E 0 = Q1 + P2bb + mv (45) r=r0  " #   m  E E  0 = Q2 + (bv bv )  2b r=r0 and the guiding center Hamiltonian, obtained by averaging over the gyrophase angle, is:

2 H = H0 + H1 + (46) where

14 m B H0 = + u u + P3 (47) D 2 E m (0) (1) H1 = mv v X P B1/2 uu q3q3 3 up3 uq3 = ( j u) Vj + P3Vj + (V1 V2 ) . (48) j=0,1,2 m

The functions on the right-hand side of these equations are evaluated at = 0, =

0,= 0. We ignore higher order Larmor radius terms involving . The parallel component of the particle velocity is: hD Ei (0) (1) (0) mVk = b v + b v + b ( ∇)v r=r0 uu q3q3 ∇ = P2b + V0 + P3V0 + P3b b.

The guiding center Hamiltonian can be written in the form:

H0 = H0 + H1 m B = + u u + P + P b ∇vE (49) 2 m 3 3

where

X 1 P uu q3q3 3 ∇ u = u + Vj + P3Vj j + (b b) b. j=0,1,2 m m In Appendix A, we demonstrate that the Hamiltonian equations generate the guiding center drifts to rst order in the Larmor radius:

E P3 m r˙ = Vkb + v + b ∇B + b (u ∇)u. mB B

C. small electromagnetic eld perturbations

Time dependent electromagnetic elds can be accommodated by introducing time dependent eld line coordinates. The generating function G now depends on time. The guiding center Hamiltonian is then obtained by following the same procedures described earlier, the only modication required being the addition to the transformed Hamiltonian of the term ∂G| ∂t p1,p2,p3,x,y,z.

15 It is, however, often useful to have a time independent coordinate system based on the equilibrium elds and to represent the perturbing elds separately. We will adopt this representation of the electromagnetic elds, and we will assume that the perturbing elds are small (of order O()) compared to the equilibrium elds. Let the electromagnetic potentials be represented as follows:

= e (50)

A = (x, y, z)∇(x, y, z)+Ae

e e e = ∇ + A∇ + A∇ + A∇ (51)

where e(, , , t) and Af(, , , t) are the perturbed electromagnetic potentials. B ∇ ∇ We again dene unit vectors 0, 1, 2 by Eq. (8), where 0 = b = B and B = . We introduce the generating function G

2 (2) G = p1 + p2 + G

with G(2) chosen so that

1/2 E m 1 v = p3B + m 1 v +

1/2 E m 2 v = q3B + m 2 v + .

e E ∇ e ∂A The ‘electric’ drift v , given by Eq. (36) with E = ∂t , is ordered to be O() for convenience, and the perpendicular velocity of gyration dened relative to vE. We again follow the procedure described earlier, and we are led to the following choice for G(2): ! B1/2 2 ∇ ∇ m2 G(2) = p + mu + u u 3 |∇| 2 |∇|2 2 m(p Ae )b + 2 (b ue b ue ) (52) 2

where the dependent variable is here dened to be:

p = 1 + mu (53)

16 e e e and u, u are the covariant components of u, while u,u are the covariant components of u:

e f f E mu = A b Ab + mv = mu∇ + mu∇ (54)

e mu = p2b b + mu = p2∇ + mu∇ + mu∇. (55)

The particle velocity v is:

v = p A = ∇G ∇ A.e (56)

Substituting for G in Eq. (56), we obtain

v = v(0) + v(1) (57) where

(0) e 1/2 1/2 mv = b (p2 A) 0 + p3B 1 q3B 2 (58)

(1) (2) (2) (2) E mv = D0G 0 + D1G 1 + D2G 2 + mv . (59)

(2) Evaluating DjG :

D (2) uu up3 uq3 p3q3 2 q3q3 jG = Vj + p3Vj + q3Vj + p3q3Vj + q3Vj (60) where ( !) m2 m b b V uu = (u D u u D u )+ D P Ae ue ue j 2 j j 2 j 2 b b ( ) B1/2 |∇| ∇ ∇ V up3 = m D u V uq3 = m D u + D u j |∇| j j B1/2 j |∇|2 j |∇| B1/2 1 |∇|2 ∇ ∇ V p3q3 = D V q3q3 = D . j B1/2 j |∇| j 2 B j |∇|2

Substituting for G in Eq. (6), we obtain

= p1 mu + ∇

= q1 + mu + ∇

e e = q2 + mb (bu bu)+ ∇.

17 The transformed Hamiltonian is   " # m ∂G(2) H = v(0) v(0) + + 2 mv(0) v(1) +  2 ∂t p1,p2,p3,x,y,z m +3 v(1) v(1). (61) 2

As before, we can now carry out an expansion in the Larmor radius. We again omit the details. The guiding center position vector is:

r0 = r(0,0,0) where

0 = p1 mu

0 = q1 + mu

e e 0 = q2 + mb (bu bu) (62) and the guiding center Hamiltonian H, obtained by averaging over the gyrophase angle, is:

2 H = H = H0 + H1 + (63) where

(0) b b e 2 B e H0 = H = P2 A + P3 + (64) 2m *" # m + D E ∂G(2) H = mv(0) v(1) + 1 ∂t p1,p2,p3,x,y,z *" # + e 1/2 (2) P2 A b P B ∂G = (V uu + P V q3q3 )+ 3 (V up3 V uq3 )+ . (65) m 0 3 0 m 1 2 ∂t p1,p2,p3,x,y,z The parallel component of the particle velocity is:

e uu q3q3 ∇ mVk =(P2 A)b + V0 + P3V0 + P3b b (66)

The guiding center Hamiltonian (ignoring time derivative terms assumed to be small) can therefore be written in the form:

18 H0 = H0 + H1 2 mVk P B(r ) P = + 3 0 + e + 3 b ∇Af + (67) 2 m m

Linearizing with respect to the perturbed elds, we have f E r( , , )=[r( , , )]e e + b A + mv + 0 0 0 0 0 0 =0,A=0 B

and the linearized guiding center Hamiltonian is h i f f H0 = H0 =0, A =0 + H0 e

f e e f P3 f H = A b Vk A v + b ∇A + 0 D m

III. SECOND ORDER LARMOR RADIUS CORRECTIONS

A. Equilibrium elds

We discuss second order corrections to the guiding center Hamiltonian by modifying our generating function G to include higher order terms in our expansion parameter . Let G be represented by the following power series expansion in :

X n (n) G = p1 + p2 + G (,p2,p3,,,) (68) n=2 where G(2) is given by Eq. (15), the dependent variable is dened by Eq. (11),

p1 b = + p2 b

and G(3),G(4), etc., are yet to be determined. Substituting for G in Eq. (6) , we obtain the

coordinates qi:  X (n)  ∂G ∂G  q = = n 1  1  ∂p1 n=2 ∂  ( )  X (n) (n)  1 ∂G n1 b ∂G ∂G q2 = = + +  . (69) ∂p2 b ∂ ∂p2  n=2   1 ∂G B1/2 X ∂G(n+1)  q = = + n1  3 2 |∇ |  ∂p3 n=2 ∂p3

19 Substituting for G in and Eq. (9), we obtain the velocity v:  P  m v = bp + n 1D G(n)  0 2 n=2 0   ( )  P (n+1)  ∂G  m v = p B1/2 + n 1 D G(n) + |∇|  1 3 n=2 1 ∂  . (70) X  B n1D (n)  m 2 v = + 2G  |∇|  n=2  ( )  P ∂G(n+1)  1/2 n1 D (n) 1/2  = q3B + n=2 2G + B  ∂p3

The dependent variable is no longer simply related to the canonical variable q3.How-

1 ∂G ever, can be expressed in terms of q ,p ,p ,,, by solving equation q = 2 as a 3 2 3 3 ∂p3 power series in . We obtain:

(0) (1) = (q3,,,)+ (q3,p2,p3,,,)+ (71) where

|∇| (0) = q B1/2 3 |∇ | (3) (1) ∂G | = 1/2 =(0) B ∂p3

Substituting for in Eq. (69) and expanding about = (0), we obtain for , , to order 2:  (3) b 1 ∇ ∂G  = p p + ∇ 2 +  1 2 1/2  b B ∂p3   ∇ (3) ∇ (3)  b ∇ 2 ( 1 ) ∂G 2 ( 2 ) ∂G = q1 + p2 + ( ) + +  (72) b B1/2 ∂p B1/2 ∂q  3 3   ∂G(3) ( ∇) ∂G(3) ( ∇) ∂G(3)  ∇ 2 2 1 2 2  = q2 + ( ) 1/2 + 1/2 + ∂p2 B ∂p3 B ∂q3 Similarly, we obtain from Eq. (70) the particle velocity:

v = v(0) + U (1) + 2U (2) + (73) where

20 (0) 1/2 1/2 mv = p2b 0 + p3B 1 q3B 2 (74) (3) (3) (1) (1) 1/2 ∂G 1/2 ∂G mU = mv + B 1 + B 2 (75) ∂q3 ∂p3 (4) (4) (2) (2) 1/2 ∂G 1/2 ∂G mU = mv + B 1 + B 2 (76) ∂q3 ∂p3 and

(1) (2) (2) (2) mv = D0G 0 + D1G 1 + D2G 2 (77) (3) (3) (2) (3) ∂G ∂ (1) ∂G ∂ (1) m j v = DjG + (m j v ) (m j U ). (78) ∂q3 ∂p3 ∂p3 ∂q3

Note that all quantities are now evaluated at = (0), and that

(3) (3) (3) ∂G ∂ (2) DjG = DjG + DjG ∂q3 ∂p3 where the operator Dj is dened by:

(3) ∇ (3)| DjG = j G q3,p1,p2,p3 .

(3) (3) Hereafter G = G (q3,p2,p3,,,). The transformed Hamiltonian is m H = v(0) v(0) + + 2mv(0) U (1) 2 m +3 U (1) U (1) + mv(0) U (2) + . (79) 2

We now expand all quantities in a Taylor series in the Larmor radius. Thus, dening

0,0,0 by Eq. (26), we have for , , :   = + (1) + 2(2) +  0    (1) 2 (2) (80) = 0 + + +     (1) 2 (2)  = 0 + + + where

(1) = ∇(1) = ∇(1) = ∇ (81)

21  ∇ (3)  (2) b(2) 1 ∂G  =  B1/2 ∂p  3   ∇ ∂G(3) ∇ ∂G(3) (2) b(2) 1 2 = +  (82) B1/2 ∂p B1/2 ∂q  3 3   ∇ ∂G(3) ∇ ∂G(3)  (2) b(2) 1 2  = 1/2 + 1/2 B ∂p3 B ∂q3

!  b  b(2) ∇ ∇  = p2  b  !   b b(2) ∇ ∇ = + p2  . (83) b    ∂G(3)  b(2) = ∇( ∇)  ∂p2 The Taylor series expansion of the Hamiltonian is: H = H(0) + 2 H(1) + ∇H(0) + 3 H(2) + ∇H(1) ! 1 ∂H(0) ∂H(0) ∂H(0) +3 ( ∇)2H(0) + (2) + (2) + (2) (84) 2 ∂ ∂ ∂ where

H(0) b b 2 B 2 2 = p2 + (p3 + q3)+ (85) 2m 2m ! B ∂G(3) ∂G(3) H(1) = mv(0) v(1) + p q (86) m 3 ∂q 3 ∂p 3 3 ! (4) (4) (2) m (1) (1) (0) (2) B ∂G ∂G H = U U + mv v + p3 q3 . (87) 2 m ∂q3 ∂p3

The functions on the right-hand side of these equations are evaluated at = 0, =

0,= 0. Hereafter we delete the subscript 0 for convenience.

At this point, we introduce the canonical variables Qi,Pi dened by Eq. (23). We now ad- dress the problem of determining G(3),G(4), etc. so that the particle Hamiltonian H is independent of the Larmor phase angle. Let the guiding center Hamiltonian H(Q1,P1,Q2,P2,P3) be expressed as the following power series in :

2 3 H = H = H0 + H1 + H2 + (88)

(0) Since H is independent of the Larmor phase angle Q3, we have to lowest order in :

22 (0) H0 = H . (89)

2 The terms of order O( ) in Eq. (84) are not independent of Q3. However, it should be noted that there is a term in H(1) (see Eq. (86)), involving the as yet unknown function G(3),

(3) which is proportional to the derivative of G with respect to the Larmor phase angle Q3 ! B ∂G(3) ∂G(3) B ∂G(3) p3 q3 = . m ∂q3 ∂p3 m ∂Q3 We therefore have the freedom to choose G(3) so that B ∂G(3) is equal to the other terms m ∂Q3 2 of order O( ) in Eq. (84) which depend on Q3: (3) B ∂G c = H1 (90) m ∂Q3 c where H1 is given by: D E c (0) (1) (0) (0) (1) H1 = mv v + ∇H mv v . (91)

(3) 2 With this choice of G , we can then set H1 equal to the remaining terms of order O( ) which are independent of Q3: D E (1) (0) H1 = H + ∇H (92) D E = mv(0) v(1) (93)

H1 is the rst order correction to the guiding center Hamiltonian, and it has already been evaluated in earlier sections. Explicit expressions for G(3) are evaluated in Appendix B. This procedure in which G(n+1) is properly chosen, so that the term of order O(n)in

Eq. (84) is independent of Q3, can be repeated at each order in . The guiding center

Hamiltonian at each order is then obtained by equating Hn1 to the corresponding terms of

n order O( ) in Eq. (84) which are independent of Q3. Thus, the second order correction to the guiding center Hamiltonian, obtained from Eq. (84), is: m (1) (1) (0) (2) H2 = U U + mv v 2 D E 1 + ∇H(1) + ( ∇)2H(0) * + *2 + * + ∂H(0) ∂H(0) ∂H(0) + (2) + (2) + (2) (94) ∂ ∂ ∂

23 In AppendixC,wemakeuseofthedierential equation determining G(3) to express the

equation for H2 in a more convenient form: m D E m ∂ D E H = v(1) v(1) + ∇mv(0) v(1) Hc Hc 2 2 2B ∂P 1 1 * 3 ! + 1 ∂ ∂ ∂ + ( ∇)2H(0) + b(2) + b(2) + b(2) H(0) . (95) 2 ∂ ∂ ∂

(2) In this form, it will be noted that the evaluation of H2 involves only G and the explicit solution for G(3) is not required.

B. Perturbed elds

The above discussion of second order corrections can be extended to include the presence of perturbed elds. However, it should be recalled that in Sec. II, the spatial scale lengths of the perturbed elds were implicitly considered to be similar to that of the equilibrium elds since no subsidiary ordering was introduced to distinguish between the two scale lengths. For short wave length perturbations, let

Af = Af(/1/2,/1/2,,t)

e (1) up3 up3 uq3 uq3 and similarly for . In Eq. (59) for v , we then have terms p3V1 ,p3V2 ,q3V1 ,q3V2 which are of order O(1/1/2). For example: ! p B1/2 b up3 3 ∇ e e p3V1 = 1 A A + O(1). |∇| b

We therefore nd it convenient to modify the generating function as follows:

X 2 (2) 3/2 (5/2 n (n) G = p1 + p2 + G + G + G (96) n=3 where the additional term G(5/2) is included so that not only are terms of order O()inthe Hamiltonian H independent of the phase angle (guaranteed by the choice of G(2)) but so also are terms of order of O(3/2). G(2) is given by Eq. (52) and is dened by Eq. (53):

p = 1 + bb p + Ae + mvE. 2

24 It will be veried below that a proper choice of G(5/2) is given by:

p 2B1/2 G(5/2) = 1/2 3 (V up3 + V uq3 )+1/2 (V up3 V uq3 ) . (97) 2|∇| 1 2 2|∇|2 2 1

1 ∂G As before, we express in terms of q ,p ,p ,,, by solving equation q = 2 as a 3 2 3 3 ∂p3 power series in :

(0) (1) = (q3,p2,,,)+ (q3,p2,p3,,,)+ (98) where " # ∂ (2) 1/2 (5/2) q3 = G + G ∂p3 =(0)

" # " # ∂2 ∂G(3) (1) G(2) + 1/2G(5/2) = . ∂∂p3 =(0) ∂p3 =(0) We substitute Eq. (98) for in the transformation equations for , , , Eq. (6), and in the equations for the particle velocity components, Eq. (56), and we expand about = (0) to obtain equations equivalent to Eq. (72) and Eq. (73). Omitting the details, we obtain the transformed Hamiltonian:   " # m  ∂ 1  H = v(0) v(0) + e + 2 mv(0) U (1) + G(3/2) + G(2) 2  ∂t 1/2  p1,p2,p3,x,y,z  " #  m ∂ +3  U (1) U (1) + mv(0) U (2) + G(3)  + 2 ∂t p1,p2,p3,x,y,z where (0) e 1/2 1/2 mv = p2 A b 0 + p3B 1 q3B 2 (99) (3) (3) (1) (1) 1/2 ∂G 1/2 ∂G mU = mv + B 1 + B 2 (100) ∂q3 ∂p3 (4) (4) (2) (2) 1/2 ∂G 1/2 ∂G mU = mv + B 1 + B 2 (101) ∂q3 ∂p3 and (1) (2) 1/2 (5/2) E m j v = Dj G + G + m j v (102) |∇| ∂G(5/2) B1/2 ∂G(5/2) + 1/2 j,1 + 1/2 j,2 (103) ∂ ∂p3 (3) (3) (2) (3) ∂G ∂ (1) ∂G ∂ (1) m j v = DjG + m j v m j U (104) ∂q3 ∂p3 ∂p3 ∂q3

25 It may now be veried by substitution of Eq. (97) for G(5/2) that the terms of O(1/1/2) in mv(0) v(1) are independent of the Larmor phase angle:

(p2 + q2)B1/2 mv(0) v(1) = 3 3 (V up3 V uq3 )+O(1). 2m 1 2

Proceeding as before, we express all quantities in a Taylor series expansion in the Larmor

b(2) b(2) b(2) radius. For , , , we have Eq. (80) with 0,0,0 dened by Eq. (62) and , , dened by:  1 ∇ ∂G(5/2)  b(2) ∇ ∇ 1  = ( mu) 1/2 1/2  B ∂p3   (5/2) (5/2)  b 1 1 ∇ ∂G 1 2 ∇ ∂G  (2) ∇ ∇  = ( + mu) 1/2 1/2 + 1/2 1/2  B ∂p3 B ∂q3  . (105) b b ∂ 1  b(2) ∇ ∇ e e (5/2) (3)  = ( + m u m u) 1/2 G + G  b b ∂p2    1 ∇ ∂G(5/2) 1 ∇ ∂G(5/2)  1 2  1/2 1/2 1/2 1/2 B ∂p3 B ∂q3 For the Taylor series expansion of the Hamiltonian, we have 1 H = H(0) + 2 H(1) + ∇H(0) + 3 H(2) + ∇H(1) + ( ∇)2H(0) ! 2 ∂H(0) ∂H(0) ∂H(0) 1 +3 (2) + (2) + (2) + 4 ( ∇)2H(1) (106) ∂ ∂ ∂ 2 where

(0) b b e 2 B e H = (P2 A) + P3 + 2m " m # ! (3) (3) H(1) (0) (1) ∂ 1 (3/2) (2) B ∂G ∂G = mv v + 1/2 G + G + p3 q3 ∂t m ∂q3 ∂p3 p1,p2,p3,x,y,z ! (4) (4) (2) m (1) (1) (0) (2) B ∂G ∂G H = U U + mv v + p3 q3 2 m ∂q3 ∂p3

For the guiding center Hamiltonian H up to second order in the Larmor radius, we have:

2 3 H = H0 + H1 + H2 + (107) where

26 b b e 2 B e H0 = (P2 A) + P3 + (108) 2m *"m # + D E ∂ 1 H = mv(0) v(1) + G(3/2) + G(2) (109) 1 ∂t 1/2 p1,p2,p3,x,y,z m H = U (1) U (1) + mv(0) v(2) 2 2 D E (1) 1 2 + ( ∇)H + ( ∇) (H0 + H1) * 2 ! + ∂ ∂ ∂ + (2) + (2) + (2) (H + H ) (110) ∂ ∂ ∂ 0 1 and G(3) is determined by integrating the equation: (3) ∂G c B = H1 (111) ∂Q3 where Hc is given by: 1 " # ∂ 1 Hc = v(0) v(1) + G(3/2) + G(2) 1 ∂t 1/2 p1,p2,p3,x,y,z + ∇(H0 + H1) H1 (112)

In Eq. (110) and Eq. (112), we have added the term H1 which is formally of order but which includes perturbed elds Af = Af(/1/2,/1/2,,t), etc., with scale length variations which can be much smaller than the equilibrium scale lengths.

(3) It is again convenient to use Eq. (111) for G to rewrite H2 in the manner described in Appendix C. Omitting the details, we have: m D E m ∂ D E H = v(1) v(1) + ∇mv(0) v(1) Hc Hc 2 2 2B ∂P 1 1 * 3 ! + 1 ∂ ∂ ∂ + ( ∇)2 (H + H ) + b(2) + b(2) + b(2) (H + H ) 2 0 1 ∂ ∂ ∂ 0 1 * !+ c H1 ∂ + ∇ m (H0 + H1) B . (113) B ∂P3 (2) (5/2) In this form, it will again be noted that the evaluation of H2 involves only G and G ; the explicit solution for G(3) is not required.

IV. MAGNETIC FLUX COORDINATES

Toroidal equilibria are most conveniently described using magnetic ux coordinates , , where is the toroidal ux and , are generalized poloidal, toroidal angles. Since

27 magnetic ux coordinates can be related to eld line coordinates, we can utilize our previous analysis to relate magnetic ux coordinates to particle canonical variables by coordinate transformation. We discuss in this section the required coordinate transformations for straight eld line

ux coordinates in which = (the toroidal ux), or = p (the poloidal ux), and also for non-straight eld line ux coordinates. To avoid unnecessary complications, we consider specically equilibrium magnetic elds with nested ux surfaces.

A. Toroidal ux : =

Let the magnetic eld B of toroidal equilibria be represented in terms of magnetic ux coordinates by ( ) B = ∇ ∇ . (114) q()

These ux coordinates are related to eld line coordinates , , by

= = /q(). (115)

The third coordinate can be taken to be the toroidal angle = , the poloidal angle = , the helical angle = L + N where L and N are integers and L + N =0,orany6 other suitable function of and . To be specic, let us discuss the choice of = L + N. We now consider r = r(, , ) to be a function of , , , and we have: ! ∂r ∂r 1 dq ∂r ∂r → N L ∂ ∂ (Nq + L) q d ∂ ∂ ! ∂r q ∂r ∂r → N L ∂ (Nq + L) ∂ ∂ ! ∂r 1 ∂r ∂r → + q ∂ (Nq + L) ∂ ∂

A Hamiltonian description of guiding center motion can then be obtained by using Eq. (26) to relate magnetic ux coordinates , , to the particle canonical variables

P1,Q1,P2,Q2:

28   = P P b (Nb Lb )  1 2   ( )   (Nq + L) dq  = + Q + P b b (Nb Lb )  q() 1 2 q q2 d (116)  (Nq + L)   = + Q1 + P2b b +  q(P1) q     = L + N = Q2

where (b,b,b ) and (b ,b ,b ) are the corresponding covariant and contravariant components of the unit vector b respectively. Note that

q = b /b b b + b b =1 (Nq + L)b q b = b = . q (Nq + L)b

These equations implicitly determine = (Q1,P1,Q2,P2), = (Q1,P1,Q2,P2), =

(Q1,P1,Q2,P2), and the particle guiding center is located at r(, , ).

The parallel velocity component Vk, determined by Eq. (33), is: 2 b P2 1 mVk = P b + ∇ b (b ∇b) ∇ b ∇b 2 2B b P +P b ∇b 3 ∇ . 3 2 2 2

The guiding center Hamiltonian H = H0 + and the equations of motion are determined by Eq. (34) and Eq. (35) respectively.

The coordinates and are periodic, and so also is = Q2 = L+N. The coordinate Q1 is not periodic. However, the equations of motion are invariant under the transformations:

→ +2` → +2n ! 2n P dq Q → Q +2` 1 2 (Nb Lb )b 1 1 q q d

Q2 → Q2 +2`L +2nN.

Solving Eq. (116) for and in terms of (Q1,P1,Q2,P2), we obtain approximately:

= P1 P2b (Nb Lb )

= (P1,Q1,Q2)+P2Nb b

= (P1,Q1,Q2) P2Lb b

29 where (Q2 + q(P1)NQ1) (Q2 LQ1) = = q(P1) (L + q(P1)N) (L + q(P1)N) r = r( = P1, = , = ).

B. Poloidal ux p: = p

Let the magnetic eld B be represented by

B = ∇p() ∇{q() } (117)

∂p where p is the poloidal ux and ∂ =1/q(). These coordinates are related to eld line coordinates , , by

= p = q() (118)

If the third coordinate is taken to be = L + N,wehave: ! ∂r ∂r q dq ∂r ∂r → q N L ∂ ∂ (Nq + L) d ∂ ∂ ! ∂r 1 ∂r ∂r → N L ∂ (Nq + L) ∂ ∂ ! ∂r 1 ∂r ∂r → + q . ∂ (Nq + L) ∂ ∂

These magnetic ux coordinates are then related to particle canonical variables

P1,Q1,P2,Q2 by   b  p = P1 P2 (Nb Lb )  q   n o   dq  = q() Q1 P2 b b(Nq + L)+b d (Lb Nb)  (119)   = q(P ) Q P b b (Nq + L)+  1 1 2      L + N = Q2

30 C. Non-straight eld line ux coordinates

For more general magnetic ux coordinates: ( ) B = ∇ ∇ + (, , ) (120) q()

where is periodic in and . Thus:

= = /q()+(, , ). (121)

If the third coordinate is taken to be = L + N,wehave: ! ∂ dq ∂r ∂r q( + 2 ) ∂r ∂r → + ∂ q d N + L ∂ ∂ (Nqg + Lg ) ∂ ∂ 1 2 ! ∂r q ∂r ∂r → N L ∂ (Nqg + Lg ) ∂ ∂ 1 2 ! ∂r 1 ∂r ∂r → g2 + qg1 ∂ (Nqg1 + Lg2) ∂ ∂

∂ ∂ where g1 =1+∂ and g2 =1 q ∂ . These magnetic ux coordinates are then related to particle canonical variables by:  P q(Nb Lb )  2  = P1  qg1b + g2b      ∂ dq   (Nqg1 + Lg2) q( ∂ + q2 d ) + (, , )=Q + P b (Nb Lb )  . (122) 1 2    q qg1b + g2b qg1b + g2b     = L + N = Q2

D. Boozer coordinates [6]

In order to establish a connection with the canonical variables for guiding center motion introduced by Boozer, we consider the case of = and = . We use the following

generating function G(Qi, Pi) to transform from ‘old’ canonical variables Qi,Pi to ‘new’

canonical variables Qi, Pi:

2 G = P1Q1 + P2 + p(P1)/ Q2 + P3Q3

31 where ∂ (P ) 1 p 1 = . ∂P1 q(P1) The transformation equations are

∂G ∂G Q2 P1 = = P1 Q1 = = Q1 + ∂Q1 ∂P1 q(P1) 1 ∂G 1 ∂G P2 = = P2 + p(P1)/ Q2 = = Q2 ∂Q2 ∂P2 1 ∂G 1 ∂G P3 = 2 = P3 Q3 = 2 = Q3. ∂Q3 ∂P3

From Eq. (116), we have    = P1 P2 + p(P1)/ b b     (123) = Q1 + P2 + p(P1)/ b b      = Q2

where we note that P2 + p(P1)/ = P2 = mVk/b + is proportional to the parallel momentum and hence is of order O().

We solve Eq. (123) to obtain P1 in terms of P2,,,. The coordinates , , can then be related to canonical variables Qi, Pi by:  b   P1 = + P2 + p()/ +  b    b  Q = P + ()/ +  1 2 p b . (124)   P + ()/ = mVkb +  2 p      Q2 =

In Boozer ux coordinates, b is proportional to the Prsch-Schluter current, and hence to the plasma beta. If we ignore in Eq. (124) the term proportional to b (assuming low plasma beta), we recognize Qi, Pi to be the canonical variables introduced by Boozer.

32 V. DISCUSSION

We have discussed a Hamiltonian formulation of the guiding center motion of charged particles in a strong magnetic eld. We introduced a set of particle canonical variables in terms of which the charged particle dynamics was separated into a fast cyclotron gyration about the magnetic eld line, parallel motion along and slow perpendicular drifts across the eld line. We described a systematic procedure for nding the appropriate generating function which determined the transformation equations relating eld line coordinates to these canonical variables. The generating function was expressed in the form of a power series expansion in the particle Larmor radius, and was chosen so that the transformed Hamiltonian was independent of the Larmor phase angle at each order in the Larmor radius expansion. We included in our analysis the presence of a large equilibrium electrostatic eld and also small perturbations of the equilibrium electromagnetic elds. We derived the particle guiding center Hamiltonian up to second order in the Larmor radius. Toroidal magnetic eld congurations are most conveniently described with magnetic ux coordinates. By making use of the relationships between magnetic ux coordinates and magnetic eld line coordinates, we extended our analysis to relate particle canonical variables to straight and non-straight eld line magnetic ux coordinates.

Acknowledgments

This work was supported by the U.S. Dept. of Energy Contract No. DE-FG03-96ER- 54346. The author is grateful to H.L. Berk for many useful discussions.

33 Appendix A

In the limit of large equilibrium electrostatic elds, where the eld line coordinates , , (delete the subscripts) are related to canonical variables by Eq. (45) and the guiding center

Hamiltonian H = H0 + is given by Eq. (49), the Hamiltonian equations of motion (see Eq. (35)) are " # ∂H ∂H0 ∂u ∂ ∂u ∂ Q˙ 1 = = m + m + (A1) ∂P1 ∂ ∂ ∂ ∂ ∂ " r=r#0 ∂H ∂H0 ∂u ∂ ∂u ∂ P˙1 = = m + m + (A2) ∂Q1 ∂ ∂ ∂ ∂ ∂ " # r=r0 1 ∂H ∂H0 Q˙ 2 = = + ∇ b ∇H0 ∂P2 ∂P2 B " r=r0 # m ∂ ∂ (∇ b ∇u ) (∇ b ∇u ) + (A3) B ∂ ∂ " # r=r0 1 ∂H ∂H0 ∂u ∂ ∂u ∂ P˙2 = = m + m + (A4) ∂Q2 ∂ ∂ ∂ ∂ ∂ r=r0 where u and u are given by Eq. (41)

E mu = p2b b + mv = p2∇ + mu∇ + mu∇ (A5)

The time derivative of is:

b ˙ = P˙1 P˙2 mu ∇u b m ∂ m ∂ = ∇ b ∇H ∇ b ∇u + ∇ b ∇u mu ∇u + B 0 B ∂ B ∂

From Eq. (A5), we have

u (∇u)=u (∇u ∇ + ∇u ∇)

= u ∇ ∇u u ∇u ∇ + u ∇ ∇u u ∇u ∇ and

34 ∇ b {u (∇u)} ( ) ( ) ∂r ∂r = Bu ∇u + ∇ b b ∇ ∇ b ∇u ∇ b b ∇ ∇ b ∇u ∂ ∂

Thus, ˙ can be expressed as follows: ! m B m ˙ = ∇ b ∇ + u u + P ∇ b {u (∇u)} + B 2 m 3 B E = v ∇ + vD ∇ where P m v = 3 b ∇B + b (u ∇)u. D mB B Similarly, the time derivative of is:

˙ b = Q˙1 + P˙2 + mu ∇u b m ∂ m ∂ = ∇ b ∇H ∇ b ∇u + ∇ b ∇u + mu ∇u + B 0 B ∂ B ∂ m = ∇ b ∇H ∇ b {u (∇u)} + B 0 B E = v ∇ + vD ∇

The time derivative of is: ! ˙ ∇ b E b E ˙ = Q2 mu v v + . b b

From Eq. (48) for H0,wehave: ( ) ∂H0 m b b = Vkb ∇ + u ∇u u ∇u ∂P 2 b b 2 ( ) ! m b b m b b + u u ∇ u u ∇ + u ∇ vE vE 2 b b 2 b b ( ) b b = Vkb ∇ + m u ∇u u ∇u b b ! ∇ b E b E +mu v v . b b

Thus, substituting Eq. (A3) for Q˙ 2, we obtain for ˙:

m ˙ = b Vk + ∇ b ∇H ∇ b {u (∇u)} + B 0 B E = Vkb ∇ + v ∇ + vD ∇

35 The guiding center velocity is therefore:

∂r ∂r ∂r r˙ =˙ + ˙ +˙ ∂ ∂ ∂ E 2 = Vkb + v + vD + O( )

36 Appendix B

For the equilibrium elds discussed in Sec. IIA, the velocity v(1) determined by Eq. (18) is:

(1) D (2) p2p2 p2p3 p2q3 p3q3 2 q3q3 m j v = jG = Vj + p3Vj + q3Vj + p3q3Vj + q3Vj .

c We then obtain from Eq. (91) for H1:

D E c (0) (1) (0) (0) (1) H1 = mv v + ∇H mv v p q = 3 p bV p2p3 + B1/2V p2p2 + 3 p bV p2q3 B1/2V p2p2 m 2 0 1 m 2 0 2 p q (q2 p2) + 3 3 p bV p3q3 + B1/2V p2q3 B1/2V p2p3 + 3 3 p bV q3q3 B1/2V p2p3 B1/2V p2q3 m 2 0 1 2 2m 2 0 1 2 q p2 p q2 q3B1/2 + 3 3 B1/2V p3q3 + 3 3 B1/2V q3q3 B1/2V p3q3 3 V q3q3 m 1 m 1 2 m 2 q p + 3 ∇H(0) + 3 ∇H(0) (B1) mB1/2 1 mB1/2 2

We integrate Eq. (90) for G(3):

B h i G(3),P = Hc m 3 1 using the following identities:

[p3,P3]=q3 [q3,P3]=p3 (B2)

2 2 2 2 [p3q3,P3]=p3 q3 [q3 p3,P3]=4p3q3 (B3) h i 3 2 3 2 p3,P3 = 3q3p3 [q3,P3]=3p3q3 (B4) 2 2 p q2 + p3,P = q3 q p2 + q3,P = p3 (B5) 3 3 3 3 3 3 3 3 3 3 3 3

2 2 where P3 =(q3 + p3)/2 and the Poisson bracket is here dened with respect to the canonical variables q3,p3 or Q3,P3:

∂f ∂g ∂f ∂g ∂f ∂g ∂f ∂g [f,g]= = . ∂q3 ∂p3 ∂p3 ∂q3 ∂Q3 ∂P3 ∂P3 ∂Q3

We obtain:

37 n o n o (3) p2p3 1/2 p2p2 p2q3 1/2 p2p2 BG = q3 p2b V0 + B V1 + p3 p2b V0 B V2 (q2 p2) n o p q n o 3 3 p3q3 1/2 p2q3 1/2 p2p3 3 3 q3q3 1/2 p2p3 1/2 p2q3 p2b V0 + B V1 B V2 + p2b V0 B V1 B V2 4 2 p3 q3 n o 2 + 3 B1/2V p3q3 3 B1/2V q3q3 B1/2V p3q3 p q2 + p3 B1/2V q3q3 3 1 3 1 2 3 3 3 3 2 p q + 3 ∇H(0) 3 ∇H(0) (B6) B1/2 1 B1/2 2

38 Appendix C

The second order Larmor radius correction to the guiding center Hamiltonian H2 is determined by Eq. (94). Substituting Eq. (74), Eq. (75), Eq. (78) for v(0), U (1), v(2), we obtain: D E m D E H(2) = U (1) U (1) + mv(0) v(2) 2 * + * + m p B1/2 q B1/2 = v(1) v(1) + 3 D G(3) 3 D G(3) 2 m 1 m 2 * + * + ∂G(3) ∂ ∂G(3) ∂ + m (v(0) v(1)) m (v(0) v(1)) ∂q3 ∂p3 ∂p3 ∂q3 *  ! ! + * !+  (3) 2 (3) 2 (3) (3) (3) B ∂G ∂G B ∂G ∂ ∂G ∂G +   p3 q3 . (C1) 2m ∂q3 ∂p3 m ∂p3 ∂q3 ∂q3 ∂p3

The generating function G(3) is determined by (Eq. (90)):

B h i G(3),P = Hc m 3 1

2 2 where P3 =(q3 + p3)/2 and the Poisson bracket is here dened with respect to the canonical variables q3,p3 or Q3,P3:

∂f ∂g ∂f ∂g ∂f ∂g ∂f ∂g [f,g]= = . ∂q3 ∂p3 ∂p3 ∂q3 ∂Q3 ∂P3 ∂P3 ∂Q3

The terms in Eq. (C1) involving G(3) can be expressed as follows: D E D E (3) (3) p3D1G q3D2G Dh iE D h iE (3) (3) (3) = (q3D1G + p3D2G ),P3 (q3D1 + p3D2) G ,P3 * + Hc = mB1/2 ∇ 1 B

Dh iE G(3),mv(0) v(1) Dh iE (3) c (0) = G , H1 ∇H * + Dh iE Hc = G(3), Hc + 1 ∇B 1 B * + * + (3) (3) ∂G 1 ∇H(0) ∂G 1 ∇H(0) 1/2 2 + 1/2 1 ∂q3 B ∂p3 B

39 * + ∂G(3) ∂ h i G(3),P ∂p ∂q 3 3 *3 + * + 1 ∂G(3) ∂ h i 1 ∂G(3) ∂ h i = G(3),P G(3),P 2 ∂p ∂q 3 2 ∂q ∂p 3 * 3 ("3 # )+ 3 *3 (" # )+ 1 ∂G(3) ∂G(3) ∂G(3) 1 ∂G(3) ∂G(3) ∂G(3) + ,P3 + ,P3 + 2 ∂p3 ∂q3 ∂p3 2 ∂q3 ∂p3 ∂q3 * ! + * ! + Dh iE (3) 2 (3) 2 1 (3) b 1 ∂G 1 ∂G = G , H1 + 2 2 ∂p3 2 ∂q3

Substituting in Eq. (C1), we have:

D E D E Dh iE (2) m (1) (1) c 1 (3) c H = v v + ∇H1 + G , H1 2* + *2 + (3) (3) ∂G 1 ∇H(0) ∂G 1 ∇H(0) 1/2 2 + 1/2 1 ∂q3 B ∂p3 B

For the additional terms in Eq. (94), we have:

D E D E ∇H(1) = ∇ ∇H(0) * ! + ∂ ∂ ∂ (2) + (2) + (2) H(0) ∂ ∂ ∂ * + * + 1 ∂G(3) 1 ∂G(3) = ∇H(0) + ∇H(0) B1/2 ∂p 1 B1/2 ∂q 2 * 3 ! 3+ ∂ ∂ ∂ + b(2) + b(2) +ˆ(2) H(0) . ∂ ∂ ∂

The second order guiding center Hamiltonian H2 can therefore be expressed in the form: m D E m ∂ D E H = v(1) v(1) + ∇(mv(0) v(1)) Hc Hc 2 2 2B ∂P 1 1 * 3 ! + 1 ∂ ∂ ∂ + ( ∇)2 H(0) + b(2) + b(2) + b(2) H(0) (C2) 2 ∂ ∂ ∂ where Dh iE D E 1 (3) c m ∂ c c G , H1 = H1H1 . 2 2B ∂P3

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