Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-center motion by a spatial symmetry. Here, we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analog of the Grad–Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field.

The concept of quasi-symmetry was introduced by Boozer (1983) and then distilled into a design principle for stellarators by Nührenberg and Zille (1988). In its strongest sense, it means integrability of first-order guiding-center motion by a spatial symmetry. An excellent survey of the subject was provided by Helander (2014), assuming magnetohydrostatic (MHS) fields, that is, magnetohydrodynamic equilibrium with isotropic pressure and no mean flow.

A fundamental step was made by Burby and Qin (2013), who stated necessary and sufficient local conditions for integrability of guiding-center motion in terms of a continuous symmetry of three differential forms derived from the magnetic field and made clear that quasi-symmetry can be separated from the issue of whether the magnetic field is MHS or not.

Perturbative calculations of Garren and Boozer (1991), however, make it look very likely that the only possibility for exact quasi-symmetry for MHS fields with bounded magnetic surfaces is axisymmetry. Our paper gives first steps to deciding whether or not this is true.

In this paper, we prove many consequences of quasi-symmetry and thereby restrictions on possible quasi-symmetric fields. In the case of a quasi-symmetric MHS field, we derive a generalization of the axisymmetric Grad–Shafranov (GS) equation.

Burby and Qin (2013) built in an assumption that a quasi-symmetry must be a circle action. Here, we relax this requirement, though prove that under some mild conditions, it is actually a circle action.

We write many equations using differential forms. For those unfamiliar with differential forms, the book of Arnol’d (1978) (Chap. 7), is a classic, and there is a tutorial (MacKay, 2019) specifically for plasma physicists.

Throughout the paper, we will assume enough smoothness that the equations we write make sense, at least in a weak sense.

We consider non-interacting charged particles in a steady, smooth (at least C1) magnetic field B in 3D satisfying div B = 0, with B ≠ 0 in the region of interest.

The (non-relativistic) motion of a particle of mass m, charge e, and position q in a magnetic field B on oriented Euclidean R3 has a formulation as a Hamiltonian system of three degrees of freedom (DoF),

iVω=dH
(1)

for the vector field V=(q̇,ṗ) on the cotangent bundle T*R3, with Hamiltonian function and symplectic form (non-degenerate closed 2-form) given by

H(q,p)=|p|22m,
(2)
ω=dϑeπ*β.
(3)

Here, p is a cotangent vector at qR3 (applied to a tangent vector ξ to R3, it produces p(ξ) = pξ), |p| is its Euclidean norm, ϑ is the tautological 1-form on T*R3 defined by ϑ(q,p)(δq, δp) = p(δq), π : (q, p)↦q is the natural map from T*R3 to R3, π* is the pullback by π, and β = iBΩ for volume-form Ω on R3 corresponding to the Euclidean metric and chosen orientation. Note that div B = 0 is equivalent to = 0.

One could allow time-dependent B, electric fields, arbitrary oriented Riemannian 3-manifold, and relativistic effects, but to focus ideas, we avoid all of these (the cases with electrostatic fields and relativity are treated in  Appendix A).

If the perpendicular speed v is less than rBB|, where rB is the radius of curvature of the field lines and ΩB = −e|B|/m is the “gyrofrequency,” then there is a locally unique “guiding center” X within rB of q and “gyro-radius vector” ρ perpendicular to B(X) and smaller than rB such that

v=emB(X)×ρ+vb(X),
(4)
q=X+ρ,
(5)

where v=q̇, b = B/|B|, and v = vb. Indeed, the above formulas provide a local diffeomorphism from (X, ρ, v) to (q, v) for |ρ| < rB.

If B varies slowly on the length-scales of ρ and vB, then rotation of ρ about B(X) is an approximate symmetry of the particle motion. There is a corresponding adiabatic invariant,

μ=mv22|B(X)|=12|eΩB(X)||ρ|2,
(6)

called the “magnetic moment.”

If one neglects the variation of μ with time, one can reduce charged particle motion by gyro-rotation (Littlejohn, 1983) to obtain a Hamiltonian system of 2DoF with state (X, v) and

H=12mv2+μ|B(X)|,
(7)
ω=eπ*βmd(vπ*b),
(8)

with π* now being the pullback for the map π(X, v) = X. The 2-form ω is non-degenerate iff B̃0 [to be introduced in (14)]. The equation iVω = dH for V=(Ẋ,v̇) implies

eẊ×B̃=μ|B|+mv̇b,
(9)
Ẋb=v,
(10)

with the modified field

B̃=B+mevc, where c=curl b.
(11)

These can be rearranged to give

Ẋ=1B̃vB̃+μeb×|B|,
(12)
v̇=μmB̃B̃|B|,
(13)

where

B̃=B̃b.
(14)

We call (12) and (13) first-order guiding-center motion (FGCM); “first-order” because, as shown by Littlejohn (1983), it is possible to derive higher-order approximations, but we will restrict attention to first-order in this paper.

The Hamiltonian formulation (7) and (8) and drift equations (12) and (13) hold for an arbitrary oriented 3D Riemannian manifold, with | |, ⋅, ×, ∇, div, and curl  interpreted appropriately. Note that the above system is defined for B̃0, which is a reasonable assumption because the zeroth-order term in (14) is |B| ≠ 0. In toroidal geometry, however, one can treat the degeneracy at B̃=0 to avoid any arising inconsistencies in gyrokinetics and derive at the same time a canonical Hamiltonian structure for the purpose of symplectic integration (Burby and Ellison, 2017).

The zeroth-order approximation to FGCM (using 1/e as a convenient smallness parameter) is

Ẋ=vb,
(15)
v̇=μmb|B|.
(16)

We call this zeroth-order guiding-center motion (ZGCM).

Both FGCM and ZGCM conserve H of (7). We write E for the value of H.

In ZGCM, the guiding center moves along a fieldline. It may be circulating, meaning v has constant sign, or bouncing, meaning it is confined to an interval where |B(X)| ≤ E/μ and v changes sign on reaching each end (the usual terminology for “bouncing” is “trapped,” but this is inappropriate in a context where the whole point is to determine whether the particles are confined).

In FGCM, there are drifts of the guiding center across the field. These come from the modification B̃, the ∇|B| term, and the B̃ denominator in (12). There are variants of FGCM, which agree to first order in 1/e, but we choose the one above because it has a natural Hamiltonian formulation, which we believe is important and, in particular, allows us to discuss its integrability.

Definition III.1.

A continuous symmetry of a Hamiltonian system (M, H, ω) on a manifoldMis aC1vector fieldUonMsuch that the Lie derivativesLUHandLUωare both zero.

It follows that there is a conserved quantity locally, and it can be extended globally under mild conditions. This is a Hamiltonian version of Noether’s theorem.

Theorem III.2.

Uis a continuous symmetry for a Hamiltonian system (M, H, ω) with vector fieldViff there is a conserved local functionKforV. If there are combinationsfU + gVofUandVwith closed or recurrent trajectories realizing a basis of first homologyH1(M), thenKis global.

Proof.
LUω = 0 and = 0 imply diUω = 0, so by Poincaré’s lemma, iUω = dK for some local function K, and then,
iVdK=iViUω=iUdH=LUH=0.
(17)
The converse is similar. If there is a combination w = fU + gV of U and V with a closed trajectory γ, then
γiUω=0T(fiU+giV)iUωdt,
(18)
where t is time along w and T is the period. The first term vanishes by antisymmetry of ω, and the second vanishes because of (17). For a recurrent trajectory, close it by a short arc and bound the error to obtain that the integral of iUω in its homology direction is zero [the concept of homology direction is described by Fried (1982)]. If ∫γiUω = 0 holds for γ representing a basis of H1(M), we deduce that K is global.□

Definition III.3.

A 2DoF Hamiltonian system with vector fieldVis integrable if it has a continuous symmetryUwith global conserved quantityK, andU, Vare linearly independent almost everywhere (a.e.) (equivalentlydK, dHare linearly independent a.e.).

Note that Definition III.1 implies that the symmetry U and the Hamiltonian vector field V commute because i[U,V]ω = LUiVωiVLUω = LUdH = dLUH = 0 and ω is non-degenerate.

For an integrable 2DoF system, the bounded regular components of level sets of (K, H) are 2-tori, and there is a coordinate system in which U, V are both constant vector fields on each of them. [A component C of a level set of a C1 function F : MN is regular if DF is surjective everywhere on C; in the present context, F = (K, H) and N=R2.] This is a special case of the Arnol’d-Liouville (AL) theorem (Arnol’d, 1978). Here is a statement and proof.

Theorem III.4.

IfU, Vare commuting vector fields on a bounded surfaceS, independent everywhere on it, thenSis a 2-torus, and there are coordinates (θ1, θ2) mod 2πon it in whichU, Vare constant.

Proof.

Let ϕU be the flow of U and ϕV be the flow of V. For t=(t1,t2)R2, let ϕt=ϕt1Uϕt2V. Because the two commute and are independent, ϕ is a transitive action of the group R2 on S. Choose a point 0 ∈ S, and let T be the set of tR2 such that ϕt(0) = 0. It is a discrete subgroup of R2 (as a group under addition). Then, S is diffeomorphic to R2/T. Since S is bounded, T must be isomorphic to Z2. Thus, T is generated by a pair (T1, T2) of independent vectors in R2. Let A be the matrix with columns (T1, T2). Then, we obtain an action of S1×S1 (with S1=R/2πZ) on S by (θ, x) ↦ ϕ/2π(x), where θ=(θ1,θ2)S1×S1 and xS. Keeping 0 ∈ S fixed, this action defines a diffeomorphism S1×S1S. In these coordinates, U is the first column of 2πA−1, and V is its second column, thus constant vector fields.□

Definition III.5.

Coordinates (θ1, θ2) mod 2πon a 2-torus in which commuting vector fieldsU, Von it are constant are called Arnol’d-Liouville (AL) coordinates.

The concepts of integrability and AL coordinates have higher dimensional analogs, but the 2DoF context suffices here.

Note that if K, H are C3, then dK, dH independent a.e. implies that the union of non-regular level sets of (K, H) has measure zero. We have not found this result in the literature but are grateful to Vassili Gelfreich for providing a proof, attached here as  Appendix B.

Definition IV.1.

Given a magnetic fieldBon an oriented 3D Riemannian manifoldQ, a vector fielduonQis a quasi-symmetry ofBifU = (u, 0) is a continuous symmetry for FGCM for all values of magnetic momentμ.

We assume B nowhere zero on Q in order for FGCM to make sense. Note that in contrast to most of the literature [e.g., Helander (2014)], we do not assume that B is MHS. Indeed, one might like to apply the concept of quasi-symmetry to magnetohydrodynamic equilibria with a mean flow [cf. Simakov and Helander (2011)] or with anisotropic pressure, for example. The concept of FGCM does not require an MHS field, so neither should quasi-symmetry.

A simple example of a quasi-symmetric magnetic field is any axisymmetric B in Euclidean space. Take u=ϕ=rϕ̂ in cylindrical coordinates (r, ϕ, z). Axisymmetry of B can be defined in various ways, e.g., LuB = 0 or Luβ = 0 for this u. They are equivalent because div u = 0 and

i[u,B]Ω=Luβ(div u)β.
(19)

Our first main theorem is as follows:

Theorem IV.2.
A vector fielduis a quasi-symmetry of a magnetic fieldBiff
Lu|B|=0,
(20)
Luβ=0,
(21)
Lub=0.
(22)

Proof.
Recall the Hamiltonian and symplectic form for FGCM,
H=12mv2+μ|B(X)|,
(23)
ω=eπ*βmd(vπ*b).
(24)
Then, LUH = μLu|B|, so LUH = 0 for all μ iff Lu|B| = 0. Next,
LUω=eLuβmd(vLub)=eLuβmdvLubmvdLub.
(25)
Apply this to an arbitrary pair of tangents to Q and set v = 0 to deduce that LUω = 0 implies Luβ = 0. Apply it to an arbitrary tangent ξ to Q and the vector (0, 1) tangent to Q×R to deduce that iξLub = 0, so Lub = 0.

In the other direction, if Luβ = 0 and Lub = 0, then LUω = 0.

So we have proved that u is a quasi-symmetry of B iff Lu|B| = 0, Luβ = 0, and Lub = 0.□

Remark IV.3.

Actually if (u, 0) is a continuous symmetry for FGCM for one set of non-zero e, m, μ then it is for all e, m, μ.

We write the three conditions of Theorem IV.2 in vector calculus for comparison (recall c = curl b):

u|B|=0,
(26)
curl (B×u)=0,
(27)
c×u+(ub)=0.
(28)

Quasi-symmetry has the following significant consequences:

Theorem IV.4.

Ifuis a quasi-symmetry of a magnetic fieldB, thenLuB = 0,LuΩ = 0, andLuB = 0.

Proof.
To prove LuB = 0, use B = |B|b, so
LuB=(Lu|B|)b+|B|Lub.
(29)
By Theorem IV.2, Lu|B| = 0 and Lub = 0. Thus, LuB = 0.
To prove LuΩ = 0, note that βb = |B|Ω. Applying Lu, we obtain
Luβb+βLub=(Lu|B|)Ω+|B|LuΩ.
(30)
According to Theorem IV.2, the first three terms of this are zero. As |B| ≠ 0, we obtain LuΩ = 0.
To prove that LuB = 0, note that it can alternatively be written as [u, B] = 0. Use the formula
i[u,B]Ω=LuiBΩiBLuΩ,
(31)
which holds for any pair of vector fields u, B and any differential form Ω, in particular, the volume-form. By Theorem IV.2, Luβ = 0, and we just proved that LuΩ = 0. So using Ω non-degenerate, we see that [u, B] = 0, cf. (19).□

Similarly to that for Lub in (28), the first result of Theorem IV.4 is written as

u×J=(uB),
(32)

where J = curl B, because

LuB=iudB+diuB=iuiJΩ+d(uB).
(33)

The second says

div u=0,
(34)

and for the third,

LuB=[u,B]=uBBu=curl (B×u)+(div B)u(div u)B.
(35)

Since div B = 0 and we already proved that div u = 0, then [u, B] = 0 can be written in this case as curl (B × u) = 0.

Noting that some steps in the above proof are reversible, we can derive various alternative necessary and sufficient conditions for quasi-symmetry. The following theorem gives some examples, from which we shall frequently use (i) or (ii). Case (i) is a slight generalization of the formulation by Burby and Qin (2013).

Theorem IV.5.

A vector fielduis a quasi-symmetry of a magnetic fieldBiff any of the following sets of conditions hold:

  • Lu|B| = 0,Luβ = 0,LuB = 0;

  • LuΩ = 0,Luβ = 0,LuB = 0;

  • LuΩ = 0,LuB = 0,LuB = 0.

Proof.

(i) To prove the first set, we use (29). Thus, under Lu|B| = 0 and B ≠ 0, we obtain LuB = 0 iff Lub = 0, which converts Theorem IV.2 to (i).

  • (ii) The second comes from the first and (30).

  • (iii) The third comes from the second and (31).

Here are some additional consequences of quasi-symmetry.

Theorem IV.6.

Ifuis a quasi-symmetry ofB, then

  • Lu(uB) = 0, Lu(ub) = 0,

  • [u, b] = 0, [u, B/|B|2] = 0, [u, u] = 0 (whereuis the component ofuperpendicular toB), and

  • [u, J] = 0 (whereJ = curl B), [u, [J, B]] = 0,Lu(JB) = 0, andLJ(uB) = 0.

Proof.

(i) uB = iuB so Lu(uB) = iuLuB + i[u,u]B, both of which are zero.

For ub, apply Lu to uB = |B|ub and use the above plus B ≠ 0 to deduce that Lu(ub) = 0.

(ii) For [u, b], use [u, B] = 0, B = |B|b, and Lu|B| = 0 to obtain [u, b] = 0.

Similarly, [u, B/|B|2] = 0 or indeed [u, f(|B|)B] = 0 for any function f.

u = u − (uB)B/|B|2, and Lu on each of these terms is zero, so Luu = 0.

(iii) J = curl B translates to iJΩ = dB. Apply Lu to each side. LuiJΩ = iJLuΩ + i[u,J]Ω and LudB = dLuB = 0. However, LuΩ = 0, so we deduce that i[u,J]Ω = 0. Ω is non-degenerate, so [u, J] = 0.

For [u, [J, B]], we use the Jacobi identity [u, [J, B]] + [J, [B, u]] + [B, [u, J]] = 0. We already proved that [B, u] = 0 and [u, J] = 0. So [u, [J, B]] = 0.

Lu(JB) = LuiJB = iJLuB = 0, using [u, J] = 0.

LJ(uB) = iJd(uB) = iJLuB = 0, using (33).□

If B is a vacuum field with quasi-symmetry u, note that Theorem IV.6(i) can be strengthened to uB = const., using (33) [or (32)].

The condition Luβ = 0 of Theorem IV.2 merits additional comment. We discuss it in a more general context than quasi-symmetry. Specifically, we require only Luβ = 0, div B = 0, and div u = 0.

Because = 0 and β = iBΩ, Luβ = 0 is equivalent to diuiBΩ = 0. Thus, by Poincaré’s lemma, iuiBΩ = for some function ψ locally (in vector calculus, B × u = ∇ψ), and both u and B are tangent to regular level sets of ψ.

An important question is whether ψ is global. It is global if there are combinations fu + gB with closed or recurrent trajectories realizing a basis of H1(Q). For the case of Q being a solid torus with a circulating magnetic field, B has a closed trajectory realizing H1(Q), so ψ is global. For more complicated domains, it might fail.

Definition V.1.

A flux function for a fieldBonQis a globally defined functionψ:QRwithiB = 0 and ≠ 0 a.e.

Note that the existence of a flux function ψ is an assumption of the standard approach to quasi-symmetry [e.g., Helander (2014)], whereas here, we derived it as a consequence, at least as a local function. For many purposes, however, we will need to assume that ψ is global and has non-zero derivative a.e. (see Definition VI.1). Note that in the paper [Helander (2014)], ψ is chosen to be the toroidal flux enclosed by the level set, and the term “flux function” is used for any function of ψ.

If ψ is global, it follows from the classification of surfaces that bounded regular components of level sets of ψ are 2-tori because they support a nowhere-zero vector field (u or B).

Definition V.2.

The bounded regular components of level sets of a flux function are called flux surfaces.

Furthermore, u and B are independent everywhere on such a 2-torus because iuiBΩ = . Luβ = 0 with div u = 0 implies that [u, B] = 0 [cf. (19)]. Using Theorem III.4, it follows that there are coordinates (θ1, θ2) mod 2π on the 2-torus in which both u and B are constant vector fields. There are many works on “flux coordinates,” including the book (D’haeseleeret al., 1991) and the recent paper (Kruger and Greene, 2019), but we are not aware of any of them using this very natural AL approach. The closest we have seen is the paper of Hamada (1962).

We now derive a formula for the winding ratios of X = u, B on a flux surface.

Definition V.3.

The winding ratioιXof a vector fieldXon a 2-torus with coordinates(θ1,θ2)S1×S1is the limit astof the ratio of the number of revolutions made inθ1along a trajectory of vector fieldXto that inθ2.ιXis considered as a point in the projective lineRP1to include the option ofand to ignore the sign ofX.

For vector fields X of “Poincaré type” (those having a cross section) on a 2-torus, the limit exists and is the same for all trajectories and for both signs of X. Since u and B are conjugate to non-zero constant vector fields on each flux surface, they are of Poincaré type.

As X conserves Ω and ψ, it also conserves an area-form on flux surfaces. Indeed, let

A=inΩ, where n=ψ|ψ|2.
(36)

In vector calculus, A(ξ,η)=n(ξ×η). Then, the restriction AC of A to a regular component C of a level set of ψ is non-degenerate and conserved by X. To prove the conservation, it is enough to work out LXA on (u, B), which form a basis of tangents to C. Using [u, B] = 0, we have

iuiBLXA=LXiuiBinΩ=LXindψ=LX1=0.
(37)

Theorem V.4.
Ifdivu = divB = 0 andiuiBΩ = , then forX = uorBon a bounded regular componentCof a level set ofψ,
ιX=γ1iXACγ2iXAC,
(38)
whereγjis any closed loop onCmaking one turn inθjand none in the other.

Proof.
As LXAC=0, we deduce that iXAC is closed, so its integral round a closed loop γ depends on only the homology class [γ] of the loop. Take a long piece of trajectory of X on C and close it by a short arc on C, making a closed loop γ. It has homology class close to N([γ1] + ιX[γ2]) for some large integer N. iXAC(γ̇) is zero except on the short arc. Taking the limit, we obtain
[γ1]+ιX[γ2]iXAC=0,
(39)
hence the formula of the theorem.□

The same formula applies to the current density J for an MHS field with p constant on flux surfaces.

Let us compute the conserved quantity K of FGCM resulting from quasi-symmetry. Recall from Theorem III.2 that K results from iUω = dK. Recall from (8) that ω = −*βmd(vπ*b) and from Definition IV.1 that U = (u, 0). So

iUω=eiuβmLU(vπ*b)+mdiu(vb)=dK,
(40)

with

K=eψ+mvub,
(41)

using LU(vπ*b) = vLub = 0. In particular, we see that K is global iff ψ is global.

Definition VI.1.

A magnetic fieldBis quasi-symmetric if it has a quasi-symmetryu, and the associated flux functionψis global with ≠ 0 a.e.

Theorem VI.2.

IfBis quasi-symmetric, then FGCM is integrable.

Proof.
By Definition IV.1, (u, 0) is a continuous symmetry for FGCM for all μ. From (41), the associated local conserved quantity K is global if ψ is global. To complete the verification of integrability (see Definition III.3), we must check that dH, dK are independent a.e. The derivatives dH and dK are independent at (X, v) iff rdH + sdK = 0, r,sR implies r = s = 0. Now,
rdH+sdK=r(mvdv+μd|B|)+s(edψ+mubdv+mvd(ub)).
(42)
If this is zero and (r, s) ≠ (0, 0), then the coefficient rmv + sm ub of dv is zero, and so
μubd|B|+v(edψmvd(ub))=0.
(43)
However, this is quadratic in v, so if ≠ 0 at X, it is zero for at most two values of v (typically for none because the coefficients are 1-forms in 3D). ≠ 0 a.e. in X, so dH, dK are independent a.e. in (X, v).□

K governs how far particles move from a flux surface. Using conservation of K, we see that

v=eψ+Kmub,
(44)

as long as ub ≠ 0. Hence, by conservation of H in (7), the tori for FGCM are given in projection to guiding-center position by

12(eψ+K)2m(ub)2+μ|B|=E,
(45)

with parallel velocity recovered by (44). This can be written as

ψ=Ke±ube2m(Eμ|B|).
(46)

We see the same division of motion into circulating and bouncing, as for ZGCM. Note that by Lu|B| = 0, the set of X where |B(X)| > E/μ is a set of u-lines.

Remark VI.3.

The converse of Theorem VI.2 is not obvious. Perhaps there could be magnetic fields for which FGCM has a velocity-dependent symmetry. After all, gyro-rotation is a velocity-dependent symmetry for charged particle motion in a uniform field.

Next, we examine the relation of a quasi-symmetry u to the Riemannian metric g. The conjecture of Garren and Boozer (1991) suggests that u is a Killing field for Euclidean metric g because an isometry with bounded orbits for Euclidean metric has to be a rotation.

Definition VII.1.

A vector fielduis a Killing field for a Riemannian metricgifLug = 0.

In Euclidean space, Lug = 0 can be written as ∇u + (∇u)T = 0.

We have not managed to prove or disprove that a quasi-symmetry is a Killing field yet, but the following theorem gets two-thirds of the way (by showing that the subspace of possibilities for Lug at a point is constrained to a codimension-4 subspace of the 6D space of symmetric 3 × 3 matrices). It applies to an arbitrary Riemannian metric g.

Theorem VII.2.
Letube a quasi-symmetry for magnetic fieldB, with flux functionψ. Whereveru, Bare independent, then ≠ 0 and (B, u, n) is a basis, withn = ∇ψ/|∇ψ|2. With respect to this basis,Lughas matrix
0000Lu|u|2inLuu0u[n,u]Lu|n|2,
(47)
and the diagonal terms are related by
Lu|n|2=|B|2|n|4Lu|u|2.
(48)

Proof.

≠ 0 where u, B are independent because iuiBΩ = and Ω is non-degenerate. Ω(B, u, n) = iniuiBΩ = in = 1 ≠ 0, so (B, u, n) is a basis.

The calculation of Lug makes use of Lemma VII.5 to follow, which says that the standard commutation relation iXLu = LuiXi[u,X] applies not only to differential forms but also to any covariant 2-tensor, thus including the case of a metric tensor. We apply this to the metric tensor g for X = B, u, n in turn.

For X = B, we obtain iBLug = 0, so the first row of Lug is zero and also the first column by symmetry of g.

For X = u, we obtain iuLug = Luiug = Luu. The diagonal component is obtained by contracting this with u: iuLuu = Luiuu = Lu|u|2.

For X = n, we obtain
inLug=Luingi[u,n]g.
(49)
For the off-diagonal term, we contract (49) with u. First, iuLuing = Luiuing = Lu(un) but un = 0 from iu = 0. Second, iui[u,n]g = u ⋅ [u, n]. For the diagonal term, we contract (49) with n. First,
inLuing=inLun=inLu(dψ/|ψ|2)=|ψ|2inLudψ+(Lu|ψ|2)indψ=Lu|ψ|2,
(50)
using Luψ = 0 and in = 1. Second,
ini[u,n]g=i[u,n]n=LuinninLun=Lu|ψ|2|ψ|2inLudψ(Lu|ψ|2)indψ=0.
(51)
Finally, we prove the indicated relation between the diagonal terms. Using B × u = ∇ψ,
|ψ|2=|B|2|u|2(uB)2.
(52)
We proved Lu(uB) = 0 and Lu|B| = 0, so applying Lu to the above equation, we obtain Lu|∇ψ|2 = |B|2Lu|u|2, hence the result.□

Because g is symmetric, Lug is symmetric, but we give the two alternative expressions for the off-diagonal components in (47), and one can check that they are equal (u ⋅ [n, u] = i[n,u]u = inLuuLuinu but nu = 0).

One could choose other normalizations of ∇ψ, but an advantage of the chosen one is that n ⋅ [u, n] = 0, as proved in (51), besides det g = 1 for (B, u, n).

Relation (48) can alternatively be obtained by using div u = 0 and the local expression Ω=±detgdx1dx2dx3, with det g being the determinant of the matrix representing g in coordinate system (x1, x2, x3) [i.e., g(ξ, η) = gijξiηj]. A calculation shows that div u=12tr(g1Lug) and hence (48).

The previous theorem highlights the importance of Luu not just for the off-diagonal term but also because Lu|u|2 = iuLuu. So

Theorem VII.3.

A quasi-symmetryuis a Killing field iffLuu = 0.

In vector calculus, Luu = 0 can be written as w = 0, where

w=v×u+|u|2,
(53)

with v = curl u. The relation between the two is w = Luu. In terms of w, (47) combined with (48) can be written as

0000uwnw0nw|B|2|n|4uw.
(54)

Either from (47), using (48) and the aforementioned symmetry of Lug, or (54), we see that for |B|, |∇ψ| ≠ 0, the (1,1)-minor of Lug vanishes iff both the diagonal and off-diagonal terms vanish.

Corollary VII.4.

Ifuis a quasi-symmetry, thenLugis degenerate. IfLugis non-zero, it has rank 2.

We conclude this section with the required lemma.

Lemma VII.5.
For any covariant 2-tensorgand vector fieldsu, X,
i[u,X]g=LuiXgiXLug.
(55)

Proof.
For arbitrary vector fields u, X and Y, and covariant 2-tensor g,
(Lug)(X,Y)=Lu(g(X,Y))g(LuX,Y)g(X,LuY).
(56)
This says
iYiXLug=LuiYiXgiYi[u,X]gi[u,Y]iXg.
(57)
Now, iXg is a differential form, so the usual commutation relation
LuiYiXg=iYLuiXg+i[u,Y]iXg
(58)
can be employed for the first term on the right. It results that
iYiXLug=iYLuiXgiYi[u,X]g.
(59)
This is true for all Y, hence the result.□

In the case of axisymmetry, the trajectories of u are all closed and have a common period (single points on the axis of symmetry and circles elsewhere, of period 2π). We say the flow of u generates a circle action.

Definition VIII.1.

A circle action on a manifoldMis a differentiable mappingΦ:S1×MM, (θ, x) ↦ Φθ(x) such that Φ0(x) = xand Φθ+θ(x) = Φθθ(x)) for allθ,θS1andxM.

The orbit of a point x under a circle action is either a point or diffeomorphic to a circle.

Burby and Qin (2013) formulated quasi-symmetry in terms of a circle action preserving FGCM. Given a circle action, one can obtain a vector field u = θΦθ|θ=0. It is a quasi-symmetry if Φ preserves FGCM. In our treatment of quasi-symmetry here, we do not require the trajectories of a quasi-symmetry to be circles, but we prove now that under mild conditions any quasi-symmetry does generate a circle action locally.

Theorem VIII.2.

Ifuis a quasi-symmetry forB,ψis global,Sis a bounded regular component of a level set ofψ(flux surface), and S contains a regular componentTof a joint level set of either

  • (|B|, ψ) withuB ≠ 0 or

  • (uB, ψ),

thenTis a circle and a closedu-line, and allu-lines onSare closed and of the same period. Furthermore, all nearby flux surfaces have the same properties and the same period. The circles are non-contractible on the flux surfaces, and all have the same rational winding ratio on this interval of flux surfaces.

Proof.

The vector field u is nowhere zero on S because iuiBΩ = ≠ 0. A bounded regular component T of a level set of two functions in 3D is a circle. Luψ = 0. In case (i), Lu|B| = 0 and d|B|, are independent, so T is a u-line. In case (ii), Lu(uB) = 0 and d(uB), are independent, so T is a u-line.

Now, [u, B] = 0, so by Theorem III.4, u is conjugate to a constant vector field on S. Because one u-line on S is closed, it follows that all u-lines on S are closed and have the same period.

Independence of and d|B| [respectively, d(uB)] on T implies the same for all nearby components of level sets of (ψ, |B|) [respectively, (ψ, uB)]. So we obtain the same result for all nearby flux surfaces.

To prove that the period of the u-lines is the same for nearby flux surfaces, we treat the two cases separately.

In case (i), let the function f(x) = 2π/τ(x), where τ(x) is the period of the u-line through the given point x and define the vector field R = u/f. Then, R generates a circle action Φ. Define the circle-average ⟨ω⟩ of any differential form or vector field ω by
ω=12π02πΦθ*ωdθ.
(60)
Note that the circle-average of LRω is zero for any ω. From u = fR follows
0=LuB=(BR)df+fLRB.
(61)
Take the circle-average of this equation to obtain
0=(BR)df
(62)
because f and BR are constant along Φ-orbits. So if uB ≠ 0, then df = 0. This applies on a neighborhood of S, so f is locally constant.

In case (ii), LuB = 0 implies that iuiJΩ = −d(uB) [Eq. (33)] and [u, J] = 0 [Theorem IV.6(iii)]. So u, J are commuting vector fields on each level set of uB. Now, d(uB) ≠ 0, so the u-lines on the level set of uB are closed and have the same period (using Theorem III.4). By independence of d(uB) and , this gives us a u-line on each nearby flux surface, and it has the same period. Thus, they all have the same period.

The u-lines are non-contractible on the flux surfaces because of the conjugacy to a constant vector field. The winding ratio of u as a function of ψ is continuous and rational, so it is constant.□

Remark VIII.3.

There is also a partial converse to the above theorem. Namely, if u is a quasi-symmetry that is equal to the infinitesimal generator of a circle action Φθ, then there is a globally defined ψ such that iuβ = . This may be seen by directly computing the exterior derivative of ψ = iuA⟩, where A is the vector potential.

Note that case (ii) can not occur for an MHS field with p constant on flux surfaces because of Theorem IX.2 to follow, but it might be useful in other situations.

The rational winding ratio m : n of the u-lines is called the type of the quasi-symmetry. With θ1 poloidal and θ2 toroidal, m : n = 0 : 1 is called quasi-axisymmetric (QA), 1 : 0 is called quasi-poloidal (QP), and anything else is called quasi-helical (QH).

Note that both quasi-symmetry and circle action allow the possibility of short fibres, for example, a region of m : n QH may shrink onto a closed u-line (which will be also a B-line) around which the rest have the winding ratio m : n (a “Seifert fibration”). If the u-period of the main u-lines is 2π, then the period of the short fibre is 2π/n.

Note also that the construction in case (ii) of tori with uB constant supporting commuting vector fields u, J applies for a general quasi-symmetry u, without requiring the derivatives of uB and ψ to be independent. The formula of Theorem V.4 for winding ratios extends to those for u and J on these tori, with n replaced by ∇(uB)/|∇(uB)|2.

The standard approach to quasi-symmetry, as exemplified by Helander (2014), assumes a magnetohydrostatic field B from the start.

Definition IX.1.

A magnetic fieldBis magnetohydrostatic (MHS) ifJ × B = ∇pfor some functionp, whereJ = curlB.

It then assumes a flux function ψ, i.e., a function such that B ⋅∇ψ = 0 and ∇ψ ≠ 0 a.e. Given the MHS assumption, this is not a great restriction because p satisfies B ⋅∇p = 0; the only catch is that dp might not be non-zero a.e. [in particular, in the surrounding vacuum region but also because dp must vanish at all rational surfaces where the resonant Fourier harmonic of 1/|B|2 does not vanish (Boozer, 1981)]. Then, the level sets of ψ are assumed to be bounded in the region of interest and hence 2-tori. Actually, Helander (2014) takes ψ to be the toroidal flux bounded by the level set of ψ, but that is not essential. The standard approach also assumes that p is constant on flux surfaces, which is automatic if the field has density of irrational flux surfaces but might otherwise be a restriction. Then, it is proved that there are “Boozer” coordinates (Boozer, 1981), which, in particular, make the magnetic field lines straight. Guiding-center motion is formulated in the Boozer angles as a Lagrangian system and seen to have an ignorable linear combination of the angles if the field strength is constant along a family of straight lines on each flux surface (not, in general, the same straight lines as the field lines), and so guiding-center motion is integrable. The field is said to be quasi-symmetric if this is the case.

An alternative approach is due to Hamada (1962) but requires dp ≠ 0 a.e. It constructs a different coordinate system on flux surfaces but with similar properties, and quasi-symmetry is identified as the result of an ignorable coordinate again. Helander (2014) identifies a whole class of coordinate systems that will do as well.

Here, we explain how our approach connects to these. In our definition and analysis of quasi-symmetry, we have not required the field to be MHS, but if it is MHS and has quasi-symmetry u in our sense with a global flux function ψ and if p is constant on flux surfaces, it turns out that uB is also. The latter in this case will be denoted by C. We showed that quasi-symmetry implies that [u, B] = 0 and [u, B/|B|2] = 0. We claim that the resulting AL coordinates on flux surfaces augmented by ψ give Hamada coordinates in the first case and Boozer coordinates in the second one. Because of Lu|B| = 0, it follows that |B| is constant along the u-lines, which are straight in either case. So under the assumptions of MHS with p constant on flux surfaces, quasi-symmetry in our sense implies quasi-symmetric in the standard sense.

We will now prove these statements. Afterward, we will address the converse question.

Theorem IX.2.

Ifuis a quasi-symmetry for an MHS fieldBandpis constant on flux surfaces, thenuBis constant on flux surfaces.

Proof.

We already have Lu(uB) = 0 (Theorem IV.6). Now, LB(uB) = LBiuB = iuLBB because [u, B] = 0. Translated to differential forms, the MHS equation says iBdB = dp. So LBB = dp + d|B|2. Thus, LB(uB) = iudp + iud|B|2. The first term is zero because iu = 0 and p is constant on flux surfaces. The second is zero from Lu|B| = 0. u and B are independent and tangent to flux surfaces. Combining these two results, uB is constant on flux surfaces.□

Thus, in the MHS case, uB is a function C(ψ), and so together with B × u = ∇ψ, we can describe the magnetic field in terms of u and ψ. [Technically, uB = C(ψ) is not quite correct because if a level set of ψ has more than one regular component, then uB could take different values on the different components, but we use the formulation uB = C(ψ) with this understanding. Similarly, we will write p being constant on flux surfaces as p = p(ψ).] We can do the same for J. The results are stated in the following theorem:

Theorem IX.3.
Ifuis a quasi-symmetry for an MHS fieldBwithpconstant on flux surfaces, thenBhas the form
B=1|u|2(C(ψ)u+u×ψ).
(63)
AssumeCandpare absolutely continuous functions ofψ. Then,
J=p(ψ)uC(ψ)B.
(64)

Proof.

Equation (63) comes straight from B × u = ∇ψ if we cross with u and use uB = C(ψ). Absolute continuity of a function is enough for its derivative to exist a.e. and for the fundamental theorem of calculus to hold. The MHS condition implies that J is tangent to flux surfaces, so it is a linear combination of u and B, say J = κu + λB for some functions κ, λ. Then, from ∇p = J × B = κu × B = −κψ, we find κ = −p′. Likewise, from (32), we have ∇C = u × J = λu × B = −λψ, and so λ = −C′.□

Remark IX.4.

(a) The function C has an interpretation as a current. More precisely, if γ1, γ2 are u-lines with values ψ1, ψ2 of flux function and their (common) period is denoted τ (Sec. VIII), then the current I across any surface S spanning γ1 and γ2 is I = ∫SJ = ∫γ1γ2B = ∫γ1γ2Buds = (C(ψ1)−(ψ2)) τ, where s is time along u.

(b) For future reference, it is also useful to express the quasi-symmetry u in terms of B and ψ. So now, if we cross B × u = ∇ψ with B, we have
u=1|B|2(C(ψ)BB×ψ),
(65)
using uB = C(ψ). An alternative expression can be obtained by crossing B × u = ∇ψ with ∇|B| and using u ⋅ ∇|B| = 0. In this way, the quasi-symmetry can be written as
u=ψ×|B|B|B|.
(66)
We emphasize that the two expressions are not equivalent as the former assumes MHS fields in using uB = C(ψ), while the latter does not but uses Lu|B| = 0 instead.

Definition IX.5.

Given a magnetic fieldBwith a flux functionψ, coordinates (θ1, θ2, ψ),θjS1=R/2πZ, are magnetic coordinates if theB-lines are straight in them.

Note that this remains true under any linear transformation on θj in SL(2,Z), so it is conventional to take θ1 in the poloidal direction on a flux surface and θ2 in the toroidal direction, defined (up to orientation) by its embedding in R3. For topologists, a poloidal loop is a “meridian,” and a toroidal one is a “longitude.”

Definition IX.6.

A set of magnetic coordinates is called Hamada coordinates if the current density takes the formJ = ∇I ×∇θ1 + ∇G ×∇θ2for some functionsI, Gofψonly (in differential forms, the current flux-formj = iJΩ = dI1 + dG2).

Definition IX.7.

A set of magnetic coordinates is called Boozer coordinates if the magnetic field takes the formB = Iθ1 + Gθ2 + KψwithI, Gfunctions ofψonly (in differential forms,B = Idθ1 + Gdθ2 + Kdψ).

Note that in each case, there is freedom to choose where to put the origin of (θ1, θ2) on each flux surface.

Theorem IX.8.

Ifuis a quasi-symmetry for an MHS fieldB, then AL coordinates for [u, B] = 0 give Hamada coordinates, and |B| is constant along a system of straight lines in these coordinates.

Proof.

Quasi-symmetry implies [u, B] = 0 (Theorem IV.4). Construct AL coordinates (θ1, θ2) for this commutation relation and augment to 3D by ψ. Then, both u and B are constant vector fields on each flux surface in these coordinates, i.e., u = u11 + u22 and B = B11 + B22, with the coefficients being functions of ψ only, where i is short for θi. In particular, the B-lines are straight, so (θ1, θ2) are magnetic coordinates.

Under the MHS condition with p constant on flux surfaces, J is given by (64). Thus, the current 2-form is j = iJΩ = −piuΩ − CiBΩ. Now, Ω can be written as Ω=Jdψdθ1dθ2, where J is the Jacobian of the coordinate system. The Jacobian in these coordinates is a function of ψ only because div u = div B = 0 implies div i = 0 for i = 1, 2, so 0=di1Ω=dJdψdθ2 and 0=di2Ω=dJdψdθ1. Thus, dJdψ=0, and hence, J is a function of ψ. Finally,
j=J(p(u2dψdθ1u1dψdθ2)C(B2dψdθ1B1dψdθ2)),
(67)
and so has the form j11 + j22 for some ji(ψ), which (together with being magnetic coordinates) is the defining condition of Hamada coordinates. Because u is constant in these coordinates on a flux surface, it follows from Lu|B| = 0 that |B| is constant along a system of straight lines.□

Note that J is also constant in Hamada coordinates on each flux surface because we proved in (64) that it is a linear combination (by functions of ψ) of u and B, which are constant on each flux surface. Thus, instead of making AL coordinates for [u, B] = 0, we could equally well make AL coordinates for [J, B] = 0 or for [J, u] = 0, the first of which holds for any MHS field and the second of which holds for any quasi-symmetric field. To prove that [J, B] = 0 for any MHS field, note simply that

i[J,B]Ω=iJLBΩLBiJΩ=LBdB=dLBB=0
(68)

because LBB = d(p + |B|2). [J, B] = 0 gives Hamada coordinates, and |B| is constant along a system of straight lines if the field is quasi-symmetric.

Theorem IX.9.

Ifuis a quasi-symmetry for an MHS fieldB, then AL coordinates for [u, B/|B|2] = 0 give Boozer coordinates, and |B| is constant along a system of straight lines in these coordinates.

Proof.

Quasi-symmetry implies [u, B/|B|2] = 0 (Theorem IV.6). Construct AL coordinates (θ1, θ2) for this commutation relation and augment by ψ. Then, both u and B/|B|2 are constant vector fields on each flux surface in these coordinates. In particular, the B-lines are straight, even if |B| might vary along them. So these are magnetic coordinates again.

Under the MHS condition with p constant on flux surfaces, we see that the AL coordinates have the additional property that lines of ∇ψ × B are straight too. This is because uB is constant on flux surfaces, so ∇ψ × B/|B|2 = uCB/|B|2 [coming from (65)] is a constant vector field on each flux surface. This is a property of Boozer coordinates, but the defining property of Boozer coordinates (beyond being magnetic coordinates) is that B = Idθ1 + Gdθ2 + Kdψ for some functions I, G, K with I, G functions of ψ only. Now, we prove that this holds for our AL coordinates. Without the constraints on I and G, B has the above form because the coordinate 1-forms form a basis for the cotangent space. As before, we proved that C = uB is constant on flux surfaces, so C = iuB = Iu1 + Gu2. In addition, 1=iB/|B|2B=Ih1+Gh2, where hj = Bj/|B|2. The coefficients (uj, hj) in these two equations are functions of ψ only, so it follows that I and G are too. Thus, the AL coordinates for [u, B/|B|2] = 0 in an MHS quasi-symmetric field are Boozer coordinates. Because u is constant in these coordinates and |B| is constant along u-lines, it follows that |B| is constant along a system of straight lines on each flux surface.□

Again, we note that the same coordinates are obtained if we start from the commutation relation [u, ∇ψ × B/|B|2] = 0, which holds in quasi-symmetry, or from [B/|B|2, ∇ψ × B/|B|2] = 0, whose significance comes from the next result.

Theorem IX.10.

IfBis MHS with flux functionψandpconstant on flux surfaces, then [B/|B|2, ∇ψ × B/|B|2] = 0.

Proof.
Write h = B/|B|2 and k = ∇ψ × h. We apply
i[h,k]=LhikikLh
(69)
to , B, k, in turn, using the operator
Lh=|B|2LB+d|B|2iB,
(70)
which reduces to |B|−2LB on 0-forms. In the first case, we get immediately i[h,k] = 0, since Lh = 0 and ik = 0. Next, inserting the MHS condition in the form LBB = d(p + |B|2), we have
i[h,k]B=ikLhB=ik(|B|2d|B|2+|B|2d|B|2)=0,
(71)
as well. For the last case, note first that
i[h,k]k=Lh|k|2ikLhk=|B|2LBikk|B|2ikLBk=|B|2i[B,k]k
(72)
since iBk = 0. However, using ikΩ = h, we also see that
iBi[B,k]Ω=iB(LBikΩikLBΩ)=iB(dψLBh)=dψLB1=0,
(73)
since LBψ = 0, LBΩ = 0, and iB = 0. Therefore, the vector field [B, k] is parallel to B, and so i[h,k]k = 0 too. Since (∇ψ, B, k) form a basis for the tangent space, we deduce that [h, k] = 0.□

Thus, starting with an MHS field B with a flux function ψ and p constant on flux surfaces, one can construct Boozer coordinates as AL coordinates for the commutation relation of Theorem IX.10. If B is in addition quasi-symmetric, then |B| is constant along a system of straight lines, namely, the u-lines, because Lu|B| = 0 and u, given by (65), is a constant vector field on each flux surface.

The standard approaches to quasi-symmetry are based on a symmetry (ignorable coordinate) of the gyro-averaged Lagrangian in Boozer or Hamada coordinates. There are two guiding-center Lagrangians that appear in the quasi-symmetry literature. One of these is given by

L(X,Ẋ)=12m(bẊ)2+eAẊμ|B(X)|,
(74)

with μ being a positive real parameter, which bears a resemblance to a well-known Lagrangian for the (non-gyro-averaged) Lorentz force, namely, LLL(q,q̇)=m|q̇|2/2+eAq̇. However, the resemblance is misleading. In contrast to LLL, the Lagrangian (74) is degenerate, which means that its associated Legendre transformation is not invertible. A consequence of this degeneracy is that the Euler–Lagrange equations derived from (74) do not uniquely specify a second-order system of ordinary differential equations on TQ. Nevertheless, the Lagrangian (74) is still a valid Lagrangian for FGCM in a weaker sense. Indeed, if X is the Q-component of a solution of the system (12) and (13), then X must be a critical point of the action functional defined by integrating (74) with respect to time. That said, the fundamental role of (74) in the theory of charged particle dynamics in strong magnetic fields is unclear. While (74) is a good Lagrangian for FGCM, it is not a good Lagrangian for higher-order guiding-center dynamics. In fact, there are no known extensions of (74) to higher-order guiding-center theory.

The other guiding-center Lagrangian that appears in the quasi-symmetry literature is

L(X,v,Ẋ,v̇)=(eA+mvb)Ẋ(12mv2+μ|B|),
(75)

which is due to Littlejohn (1983). This Lagrangian bears resemblance to another Lagrangian for the Lorentz force, LPP(q,v,q̇,v̇)=(eA+mv)q̇m|v|2/2. In this case, the resemblance is more than superficial. After first writing the variational principle for the Lorentz force in Poincaré–Cartan form in the extended phase space (q, v, t) as δαPP = 0, with αPP = (eA + mv) ⋅ dq − (m|v|2/2)dt [see Arnol’d (1978)], Littlejohn used sequences of near-identity transformations in phase space and asymptotic symplectic reduction to systematically derive (75). In fact, Littlejohn’s derivation naturally produces generalizations of (75) that account for all higher-order effects in guiding-center theory. In this sense, it is clear that the Lagrangian (75) plays a fundamental role in the theory of strongly magnetized particle dynamics, in contrast to the Lagrangian (74).

In a similar way, the variational principle for (75) in extended state space (X, v, t) is expressed as δγα = 0 over variations of compact support of paths γ with

α=eA+mvb(12mv2+μ|B|)dt.
(76)

The vector field V resulting from such a variational principle is given by the unique choice (V, 1) in the kernel of (which is one-dimensional because the extended state space is odd-dimensional and B is assumed non-zero). Then, a continuous symmetry is defined to be a vector field U such that LUα = dZ for some function Z. This is because flowing with such a vector field does not change the variational principle (actually, one could replace dZ by any closed 1-form, but it is conventional to take it exact). It follows, as in the standard Noether theorem, that iU = d(ZiUα) and so i(V,1)d(ZiUα) = −iUi(V,1) = 0, so K = ZiUα is conserved by (V, 1). If U = (u, 0), then

iUα=eiuA+mviub=euA+mvub.
(77)

Finally, we address the converse question: given a quasi-symmetric system in the standard sense, identify the quasi-symmetry in our sense.

Theorem IX.11.

If the magnetic fieldBhas a flux functionψand is MHS withpconstant on flux surfaces, density of irrational surfaces, p′(ψ) ≠ 0 a.e. and |B| is constant along a family of straight lines in Hamada coordinates, then it is quasi-symmetric withu = −(J + CB)/pfor a functionCofψsuch thatC = uB.

Proof.
For an MHS field, [J, B] = 0. If dp ≠ 0, then J, B are independent. Take AL coordinates for this commutation relation. By the discussion after Theorem IX.8, they are Hamada coordinates. If |B| is constant along a family of straight lines, then that implies |B| is constant along
u=κJ+λB
(78)
for some functions κ, λ of p or equivalently of ψ if a different flux function has been chosen with the same level sets. Their ratio is determined by the lines of constant |B|, but their magnitudes are otherwise free.

Now, iuiBΩ = κiJiBΩ = −κdp = −κp. Therefore, Luβ = 0. Choose, in particular, κ = −1/p′ to obtain iuiBΩ = .

It remains to prove that LuB = 0. From (78), iuiJΩ = λiBiJΩ = λdp = λp, by the MHS equation. Also [u, B] = 0 since [J, B] = 0. From the MHS condition again LBB = d(p + |B|2), and [u, B] = 0, we have LB(uB) = iuLBB = iud(p + |B|2) = 0 because both p and |B| are constant along u. Then, density of irrational surfaces implies, assuming continuity, that uB is constant on flux surfaces. In other words, uB is a function C of ψ. Therefore,
LuB=iudB+diuB=iuiJΩ+d(uB)=(λp+C)dψ.
(79)
Thus, choosing λ = −C′/p′, we satisfy the last condition for quasi-symmetry.□

Remark IX.12.

In the previous proof, we can show that LuB = 0 using circle averaging instead, as follows. First, we note that if |B| is not constant on a flux surface, then the u-lines are closed. If, exceptionally, |B| is constant on a flux surface, then the ratio is undetermined, but we can choose it to make the u-lines closed. The period τ of the u-lines is constant on flux surfaces but, in general, varies with ψ. We are free to simultaneously scale κ and λ by any function of ψ, however. Thus, we can scale them to make τ = 2π. With this choice, we can now apply circle averaging [defined by (60)] to (79). The average of the left-hand side is zero, being Lu of something. All of λ, p, and C are constant along u. Thus, the average of the right-hand side is just itself. Consequently, 0 = (λp′ + C′). It follows that LuB = 0.

Theorem IX.13.

If magnetic fieldBhas a flux functionψand is MHS withpconstant on flux surfaces and density of irrational surfaces, and |B| is constant along a family of straight lines in Boozer coordinates, then it is quasi-symmetric withu = (CB + ∇ψ × B)/|B|2for a functionCofψsuch that(C2/2)=|u|2puJ.

Proof.
For an MHS field with a flux function ψ, we proved that [B/|B|2, ∇ψ × B/|B|2] = 0 (Theorem IX.10), and the corresponding AL coordinates are Boozer (discussion after Theorem IX.10). If |B| is constant along a family of straight lines in these coordinates, then there are functions C, λ of ψ only such that |B| is constant along
u=(CB+λψ×B)/|B|2.
(80)
The ratio of C, λ is determined by the lines of constant |B|, but their magnitudes are otherwise free. We see that C = uB; hence, uB is constant on flux surfaces.

Now, B × u = λψ, so we obtain iuiBΩ = λdψ and Luβ = 0 accordingly. Let us choose λ = 1 to obtain iuiBΩ = .

It remains to prove that LuB = 0. The MHS equation can be written as iBiJΩ = dp. Thus, from the above expression for u, we deduce that
iuiJΩ=1|B|2CiBiJΩiJ(dψB)=1|B|2(Cdp+(iJB)dψ)=κdψ
(81)
since iJ = 0, with
κ=(Cp+JB)/|B|2=(|u|2p+uJ)/C.
(82)
Therefore,
LuB=iuiJΩ+d(uB)=(κ+C)dψ.
(83)
Next, note that [u, B/|B|2] = 0 because u is given by (80), B ⋅ ∇C = 0, and [B/|B|2, ∇ψ × B/|B|2] = 0. However, |B| is constant along u, so [u, B] = 0. Then, apply LB to (81). We have [u, B] = 0 and [J, B] = 0 from the MHS condition, and LBΩ = 0. Hence, LBκ = 0. Using density of irrational surfaces, it follows, assuming continuity, that κ is constant on flux surfaces, i.e., κ = κ(ψ). Then, choose κ = −C′ to obtain LuB = 0. Inserting this in (82) proves the result for (C2/2).□

Remark IX.14.

Alternatively, in the previous proof, we can show that LuB = 0 using circle-averaging, as follows. If |B| is not constant on a flux surface, then the u-lines are closed. If it is constant, then we can choose the ratio C : λ to make the u-lines closed. In either case, the period of the u-lines is constant on a flux surface. We can scale C, λ simultaneously by a function of ψ to make the period 2π. Apply circle averaging to (83) and use κ, C constant along u to obtain 0 = (κ + C′). Thus, LuB = 0.

Another common treatment of quasi-symmetry for an MHS field with flux function and p constant on flux surfaces [e.g., Simakov and Helander (2011)] is based on the relation

B×ψ=Eψ×|B|+FB,
(84)

where E = −|B|2/B ⋅ ∇|B| and F = B ×∇ψ ⋅ ∇|B|/(B ⋅ ∇|B|). Quasi-symmetry in this approach is then formulated as F being constant on flux surfaces,

F=F(ψ).
(85)

This setup fits in our framework because Eq. (84) is none other than iuiBΩ = and Lu|B| = 0 combined together, and condition (85) says that uB is constant on flux surfaces in the MHS case. To see the first one, cross B × u = ∇ψ with B to obtain B × ∇ψ = (uB)B − |B|2u and insert (66) for u. The second one follows from Theorem IX.2 since the function F is precisely uB, i.e., F = C for MHS fields.

Finally, we show that Lub = 0 implies that ∫cdl is constant when the curve c is drawn from any continuous family of field line segments within a given flux surface and with fixed endpoint values of |B|. Let γ:[s0,s1]R3 be the restriction of a field line to an interval [s0, s1] such that |B|(γ(s0)) = k0 and |B|(γ(s1)) = k1, where k0,k1R+. If ϕλ is the u-flow, then γλ = ϕλγ is a field line segment contained in the same flux surface as γ for each λ. In addition, the integral Iλ=γλdl is independent of λ because

ddλϕλγdl=ddλϕλγb=ddλγϕλ*b=γLub=0.
(86)

Therefore, the arc lengths of the field line segments γλ are all the same. Moreover, because Lu|B| = 0, the endpoint values of |B| for γλ are independent of λ, i.e., |B|(γλ(s0)) = k0 and |B|(γλ(s1)) = k1 for each λ. The desired result now follows upon noting that any continuous family of field line segments in a given flux surface with fixed endpoint values of |B| can be generated by flowing some field line segment along u.

In the axisymmetric case, magnetohydrostatics is reduced to a nonlinear elliptic partial differential equation called the Grad–Shafranov (GS) equation (Grad and Rubin, 1958; Shafranov, 1958) [but previous versions were published by Lüst and Schlüter in 1957 and by Chandrasekhar and Prendergast in 1956, and in the fluids context, the analogous equation was published by Hicks in 1899]. It takes as input two functions p(ψ) and C(ψ) and is an equation for ψ(r, z) in cylindrical polar coordinates. The GS equation has a nice variational principle (Berestycki and Brézis, 1980) (but probably there are earlier references), reasonable existence theory for solutions (Berestycki and Brézis, 1980; Ambrosetti and Mancini, 1980), and well developed codes for its numerical solution, e.g, Jardin (2010), Chap. 4.

Here, we generalize the GS equation to magnetohydrostatics with a general quasi-symmetry u.

First, we derive a pre-GS equation, which does not assume magnetohydrostatics. Furthermore, the only part of quasi-symmetry that it uses is iuiBΩ = .

Theorem X.1.
IfiuiBΩ = , then
Δψu×v|u|2ψ+uv|u|2uBuJ=0,
(87)
where Δ = div ∇,v = curlu, andJ = curl B.

Proof.
We have B × u = ∇ψ, and so
Bu=iB×uΩ=iψΩ.
(88)
Applying d to the above equation, we obtain
ΔψΩ=d(Bu)=dBuBdu=iJΩuBivΩ=(uJBv)Ω.
(89)
Hence, by non-degeneracy of Ω,
Δψ=uJBv.
(90)
For the last term of (90), contract (88) with v and u to find
v×uψ=iuiv(Bu)=iu((Bv)u(uv)B)=(Bv)|u|2(uv)(uB).
(91)
Hence,
Bv=v×uψ+(uv)(uB)/|u|2.
(92)
Substituting this into (90), we deduce Eq. (87).□

An immediate consequence of iuiBΩ = is also the additional equation u ⋅ ∇ψ = 0, which comes by contracting with u.

Next, we derive two further conditions, which follow from div B = 0 and LuB = 0, assuming, in addition, the properties div u = 0 and Lu(uB) = 0 of a quasi-symmetry.

Theorem X.2.

IfiuiBΩ = , div u = 0, andLu(uB) = 0, thendivB = 0 iffBw = 0, wherew = Luu.

Proof.
Recalling β = iBΩ, we have
|u|2dβ=(iudβ)u=(Luβdiuβ)u=(i[u,B]Ω+iBLuΩ)u=(i[u,B]u)Ω
(93)
=(LuiBuiBLuu)Ω=(Bw)Ω
(94)
because diuβ, LuΩ, and LuiBu are zero. = (div B) Ω, hence the result.□

Theorem X.3.

Under the assumptions of the previous theorem,LuB = 0 iffB × w = [u, ∇ψ].

Proof.
Note first that since iuLuB = LuiuB = 0, we have iu(LuBu) = −|u|2LuB. Thus, LuB vanishes iff LuBu does. Applying then Lu to (88), we obtain
(LuB)u=LuiψΩBLuu=i[u,ψ]ΩiB×wΩ,
(95)
since LuΩ = 0. Ω is non-degenerate, hence the result.□

Thus, the pre-GS equation requires extra conditions to guarantee that div B = 0 and LuB = 0. Both of them are automatic in the case of isometries, that is, if u is a Killing field. To see this, write w = Luu = iuLug and [u,∇ψ] = Lu() − iψLug, recalling Lemma VII.5, and take into account the first condition, Luψ = 0. In the general case, however, they appear to be non-trivial additional conditions.

The condition of Theorem X.2 is also automatic if w = 0. For a quasi-symmetry, the latter was precisely the condition for u to be a Killing field (Theorem VII.3). The next result shows that this is also true if instead u satisfies the first and third supplementary conditions.

Theorem X.4.

Ifdivu = 0,Luψ = 0, andB × w = [u, ∇ψ], thenw = 0 iffLug = 0.

Proof.

If Lug = 0, then straightforwardly w = Luu = iuLug = 0.

Let w = 0. Then, [u, ∇ψ] = 0. Using LuΩ = 0 and Lu = dLuψ = 0,
i[u,u×ψ]Ω=Luiu×ψΩiu×ψLuΩ=Lu(udψ)=Luudψ=0.
(96)
Thus, [u, u ×∇ψ] = 0 too, since Ω is non-degenerate.
Consider then the basis (u, ∇ψ, u ×∇ψ). iuLug = Luu = 0. Furthermore, since u commutes with ∇ψ and u × ∇ψ,
iψLug=Luiψg=Ludψ=dLuψ=0,
(97)
iu×ψLug=Luiu×ψg=Lu(u×ψ)=LuiψiuΩ=iψiuLuΩ=0,
(98)
using Luψ = 0 and LuΩ = 0 again. Hence, Lug = 0.□

We now turn to combining quasi-symmetry with magnetohydrostatics, obtaining another of our main results.

Theorem X.5.
If MHS fieldBis quasi-symmetric with quasi-symmetryu, flux functionψ, andpa function ofψ, then
Δψu×v|u|2ψ+uv|u|2C(ψ)+CC(ψ)+|u|2p(ψ)=0,
(99)
wherev = curluandC = uB.

Proof.

In the MHS case with p constant on flux surfaces, uB is a function C(ψ) from Theorem IX.2, and uJ = −CC′ −|u|2p′ from (64). Substitute these into the pre-GS equation (87) to obtain (99).□

Equation (99) is our quasi-symmetric Grad–Shafranov equation. For given u, C(ψ), and p(ψ), it comprises a semilinear elliptic PDE for the dependent variable ψ. Solutions of (99), however, do not necessarily give MHS fields. There are several additional conditions that are required.

First of all, Eq. (99) needs supplementing by the condition Luψ = 0, equivalently

uψ=0,
(100)

which restricts (99) to u-invariant solutions, reducing it effectively to 2D. For the special case of axisymmetry, u=rϕ̂, then |u| = r, v = 2, uv = 0, and u×v=2rr̂, so the usual GS equation is recovered in cylindrical polar coordinates. Similarly, for the case of helical symmetry, u=rϕ̂+lẑ, where l is a constant, then |u|=r2+l2, v = 2, uv = 2l, and u×v=2rr̂, so the helical GS equation (Johnson,et al., 1958) is obtained.

Recalling Theorem IX.3, the magnetic field B can be obtained by formula (63). If w = 0, as for axisymmetry and helical symmetry, no further conditions beyond (99) and (100) are required for MHS fields.

If w ≠ 0, however, it is not automatic from (63) that div B = 0 nor that LuB = 0. Thus, in general, one must add the conditions of Theorems X.2 and X.3 to ensure them. The first one reads

(u×w)ψ(uw)C(ψ)=0
(101)

as we can see by replacing B from (63) into Bw = 0. For the non-isometry case, the second one can be reduced to

v×w2(w)uψ=0,
(102)
(uv)w+2((u×w))uψ+|w|2C(ψ)=0,
(103)

as the next result shows. It is worth noting that, owing to [u, ∇ψ], the original condition in this case involves second-order partial differential equations, but they can be reduced to first-order ones by making use of (100), as described in (104) below. Thus, in the end, the second-order quasi-symmetric Grad–Shafranov equation is augmented by four first-order quasilinear partial differential equations given by (100)–(103).

Theorem X.6.

LetBbe of the form(63)withC ≠ 0. Ifuis not locally a Killing field, thenB × w = [u, ∇ψ] reduces to(102) and (103)under(100) and (101).

Proof.
First of all, for any vector field X, we have
[u,ψ]X=i[u,ψ]X=(LuiψiψLu)X=LuiXdψiψLuX=(i[u,X]+iXLu)dψiψLuX=iψ([u,X]LuX),
(104)
using Lu = dLuψ = 0 from (100). Moreover, switching to vector calculus,
[u,X]LuX=[(u)X(X)u]iudXd(iuX)=[(u)X(X)u+u×curl X(uX)]=[v×X2(X)u].
(105)

Now, if u × w = 0, then uw = 0 from (101) for C ≠ 0, and so w = 0; hence, Lug = 0 from Theorem X.4. Therefore, if u is not locally a Killing field, (u, w, u × w) is a basis. Then, project B × w = [u, ∇ψ] to the directions X = u, w, u × w and use (104) and (105).

For X = u, we see directly from (104) that the projection of B × w = [u, ∇ψ] to u is trivially satisfied,
0=u(B×w[u,ψ])=w(u×B)+iψLuu=wψ+wψ.
(106)
For X = w, we obtain (102),
0=w(B×w[u,ψ])=iψ[v×w2(w)u].
(107)
For X = u × w, using (100) and (101) in its original form Bw = 0, we arrive at (103),
0=u×w(B×w[u,ψ])=[(B×w)×u]wiψ[v×(u×w)2((u×w))u]=[Cw(wu)B]wiψ[(vw)u(vu)w2((u×w))u]=C|w|2+iψ[(vu)w+2((u×w))u].
(108)

The four first-order partial differential equations (100)–(103) in 3D imply a linear dependence among them. Without going into the tedious details, we comment that the latter though is equivalent to the degeneracy of Lug, which we saw earlier in Theorem VII.2.

Finally, the current density J can be found from (64). However, J = curl B is not automatic either. Still for solutions of (87) and therefore (99), this amounts to LuB = 0 again as the next result shows.

Theorem X.7.

LetB, Jbe of the form (63) and (64). On the set of solutions of the pre-GS equation,LuB = 0 iffJ = curl B.

Proof.
Write LuB = iudB + d(iuB) and J = curl B as s = 0, where s = dBiJΩ. Using uB = C and iuiJΩ = −C, we see that
LuB=ius.
(109)
For the converse, note that |u|2s = iu(us) + uius and
us=d(uB)+ivΩB(iJu)Ω=(Δψ+BvJu)Ω.
(110)
Thus, in light of (90), we deduce that on solutions of (87),
s=|u|2uLuB,
(111)
which completes the proof.□

In conclusion, the system of Eqs. (99)–(103) describes the conditions that a quasi-symmetry u and the corresponding flux function ψ must satisfy in magnetohydrostatics.

Two questions arise: (i) does (99) has solutions, and (ii) how do we incorporate the supplementary conditions (100)–(103)?

In this section, we principally address the first question.

There are a number of relevant results on the existence theory for semilinear elliptic PDEs, for example, Theorem 15.12 in the book of Gilbarg and Trudinger (2001) and Theorem 9.12 in the book of Amann (1976). However, even for the axisymmetric GS equation, there are regimes with no solutions (Ambrosetti and Mancini, 1980) and regimes with more than one solution. We have to do more work to reach definitive conclusions.

In the meantime, however, we address here the question whether (99) has a variational principle because it would be one useful route to prove existence of a minimizer or some other critical point by variational means, cf. Berestycki and Brézis (1980), and to understand the set of solutions. To that end, we resort to the Helmholtz conditions, as formulated in Olver (1993).

For a variational problem of the form DLψ=0 (often written δL[ψ]=0) with L[ψ]=QL(ψ,ψ)dV (where L is called the Lagrangian and may involve more derivatives) on smooth functions ψ:QR, the Euler–Lagrange operatorE on smooth functions ψ is defined by writing

DLψv=(E[ψ],v)
(112)

for all v:QR (often written δψ) satisfying suitable boundary conditions, where (f, g) = ∫Qfg dV is the standard inner product on L2(Q,R). So the Euler–Lagrange equations are E[ψ] = 0.

The Helmholtz conditions say that a differential operator E on functions ψ:QR is the Euler–Lagrange operator for some variational problem iff (f, DEg) = (DEf, g) for all functions f, g for which both sides are defined. We write this as DE* = DE, where DE* is the adjoint operator defined wherever (DEf, g) = (f, DE*g) makes sense, and refer to such operators as self-adjoint, but ignoring the question of equality of domains that is part of the standard definition.

A catch with applying the Helmholtz conditions is that the variational property is not preserved between equivalent equations, not even, for example, λE[ψ] = 0 and E[ψ] = 0, where λ is some non-zero function on Q. In fact, the Helmholtz criterion for the left-hand side of (99) as it stands implies the highly restrictive case u × v = 0 since Δ is self-adjoint. However, the axisymmetric and helical cases suggest the use of the factor λ = |u|−2, so let us consider

E[ψ]=|u|2Δψ|u|4u×vψ+|u|4uvC(ψ)+|u|2CC(ψ)+p(ψ).
(113)

Theorem XI.1.
Egiven by (113) is the Euler–Lagrange operator for some variational problemLiffLuu = 0. In this case,
L[ψ]=12|u|2(|ψ|2C(ψ)2)+C(ψ)Yψp(ψ)dV,
(114)
whereYis a vector field such thatdivY = uv/|u|4.

Proof.
Note first that the last three terms of E are functions of ψ only. Therefore, their derivative is a multiplication operator, which is always self-adjoint. Thus, the problem is reduced to just E[ψ] = |u|−2ΔψF ⋅ ∇ψ, where F = u × v/|u|4. Now E is a linear differential polynomial, and so DE = E, i.e.,
DE=|u|2ΔF.
(115)
Integration by parts shows that DE*g = Δ(g|u|−2) + div (gF) for the formal adjoint of DE. Using the identities Δ(g|u|−2) = div ∇(g|u|−2) = |u|−2Δg + 2∇|u|−2 ⋅ ∇g + gΔ|u|−2 and div (gF) = g div F + F ⋅ ∇g, we arrive at
DE*=|u|2Δ(2G+F)div G,
(116)
where G = −∇|u|−2F. In other words, G = (d|u|2 + iudu)/|u|4 = Luu/|u|4. Hence, DE* = DE iff G = 0, i.e., Luu = 0.

The first term of (114) introduces −|u|−2Δψ −∇|u|−2 ⋅ ∇ψ into the Euler–Lagrange operator. For Luu = 0, ∇|u|−2 reduces to −F, and so we recover the first two terms of (113). The third term of the Lagrangian yields −Cdiv Y = −Cuv/|u|4, and the remaining terms easily restore the rest of (113).□

Note that existence of a vector field Y such that div Y = uv/|u|4 may look difficult to satisfy, but it can be expressed equivalently as saying that a = udu/|u|4 is exact, since du = ivΩ and so udu = (ivu)Ω and a = diYΩ accordingly. In this way, we see that it is not much of a restriction because a is automatically closed, being a top-form.

In the case of axisymmetry where uv = 0, the variational functional (114) for Y = 0 recovers the Lagrangian (Berestycki and Brézis, 1980) for the usual GS equation in cylindrical polar coordinates.

In the case of helical symmetry, uv/|u|4=2l/(r2+l2)2, then Y=l/(r(r2+l2))r̂, so we derive a Lagrangian for the helical GS equation.

Recall, however, Theorem X.4 for u satisfying the supplementary conditions. In this case, unfortunately, Luu = 0 is the condition for u to be a Killing field and thus in Euclidean space holds only if u generates an orbit of SE(3). Then, we are back to the axisymmetric GS equation (rejecting translations and the helical case because they do not have bounded flux surfaces). This is where the second question comes in. Can the extra conditions be incorporated as constraints in a variational principle? Perhaps, as in Sec. X, one should view the problem as a simultaneous system of equations for ψ and u.

An alternative approach to extending the GS equation to general quasi-symmetry is to circle-average the MHS equation in the form iBdB = dp and derive an equation for ψ with respect to the circle-averaged Riemannian metric. Strangely, the resulting GS equation always has a variational principle. However, as in our analysis here, there are extra conditions that must be satisfied, and it is not clear that they can. This will be written in a separate publication.

Is every quasi-symmetry a Killing field? At least in the Euclidean case? Or at least if one requires magnetohydrostatics? Or might there be some “Kovalevskaya” examples? (Recall that Kovalevskaya found non-axisymmetric integrable cases for the dynamics of a top.) It is not even clear whether these questions are global or local in nature. The isometry condition Lug = 0 is certainly a local one, and this at least hints that the questions may be local. If this is indeed the case, a prolongation analysis based on the Cartan–Kuranishi theorem may be sufficient to provide definitive answers. We will report on such an analysis in a future publication.

The main point of stellarators is to achieve confined guiding-center trajectories without significant toroidal current. Is quasi-symmetry compatible with this goal? The toroidal current enclosed by a flux surface is just ∫γB round any poloidal loop on the flux surface. It is conventionally written as 2πI(ψ). Not surprisingly, ∫polB = 0 iff the winding ratio for the current is ιJ = 0. We see no incompatibility between this and quasi-symmetry, but it depends on there being some non-axisymmetric quasi-symmetries.

Quasi-symmetry may be too strong an ideal to aim for. Weaker conditions would suffice for single-particle confinement. Omnigenity (Helander, 2014) is one such concept, which just requires flux surfaces and the average drift in flux function for a guiding center to be zero to leading approximation. This is automatic for circulating particles of ZGCM on irrational flux surfaces but requires a condition for all bouncing particles (Landreman and Catto, 2012) and for circulating particles on rational surfaces (this is not usually recognized but follows by the same arguments as for bouncing particles). Quasi-symmetry implies omnigenity, but perhaps not vice versa, so omnigenity would allow a bit more scope (Landreman and Catto, 2012).

More generally, one invariant torus for FGCM at each value of energy and magnetic moment will confine all those inside it. This might be too weak an approach, however, because particle interactions would lead to exchange of energy and magnetic moment and drive them across the confining tori.

Alternatively, approximate quasi-symmetry may be enough; after all first-order guiding-center motion is only approximate. This can be achieved to some order by near-axis expansions (Landreman ,et al., 2019), and we intend to address it in more detail.

Quasi-symmetry may also be too weak an ideal to aim for. Even if quasi-symmetric field configurations do exist, they may do a poor job of confining the hot alpha particles generated by thermonuclear burn. The issue is such particles have much larger gyro-radii than bulk plasma particles. In the best case, confinement properties of alpha particles might be well-captured by guiding-center theory with higher-order corrections, in which case it would be interesting to study possible hidden symmetries of these high-order terms. In the worst case, guiding-center theory is useless for describing alpha particle orbits, and other approaches, perhaps based on a more brute-force optimization approach, should be pursued.

Finally, we have restricted attention to symmetries of FGCM of the form U = (u, 0), with u being a vector field on guiding-center position, not involving the parallel velocity. Might there be parallel-velocity-dependent symmetries that render FGCM integrable? Likewise, we have concentrated here entirely on the context of Hamiltonian symmetries. Might there be relevant non-Hamiltonian symmetries? This work is in progress.

This work was supported mainly by a grant from the Simons Foundation (601970, RSM) and partly by the National Science Foundation under Grant No. DMS-1440140 while the first and third authors were in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during the Fall 2018 semester. Research presented in this article was also supported by the Los Alamos National Laboratory LDRD program under Project No. 20180756PRD4.

We are grateful to other members of the MSRI semester for useful comments and of the Simons collaboration team, in particular, to Adam Golab for researching the literature on existence results for solutions of the GS equation and discussion on the supplementary conditions for its quasi-symmetric generalization. We would also like to thank Vassili Gelfreich for  Appendix B and Miles Wheeler for helpful suggestions about existence of solutions of the quasi-symmetric GS equation.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

To add the effect of an electrostatic potential Φ to the theory of this paper, add eΦ(q) to H in both equations (2) and (7). The drift equations (12) and (13) gain additional terms b×Φ/B̃ and eΦ/B̃, respectively. The conditions of Theorems IV.2 and IV.5 need augmenting by LuΦ = 0.

To add relativistic effects, one simply replaces p = mv by p = γmv with Lorentz factor γ=(1|v|2/c2)1/2 and the kinetic energy in H by c(m2c2+|p|2)1/2. Likewise, the magnetic moment changes to μ=p2/(2m|B|), and the kinetic part of the guiding-center Hamiltonian changes to c(m2c2+p2+2mμ|B|)1/2. The conditions for quasi-symmetry are unchanged.

Alternatively, taking a fully space–time view and allowing general time-dependent electric and magnetic fields, the equation of motion is

dpdτ=eiUF,
(A1)

where F is the Faraday 2-form, U is the contravariant 4-velocity, p = mU is the covariant 4-momentum, and τ is proper time for the particle. In a time–space coordinate system (t, x, y, z) with locally Minkowski metric ds2 = −c2dt2 + dx2 + dy2 + dz2,

F=Bxdydz+Bydzdx+Bzdxdy+Exdxdt+Eydydt+Ezdzdt
(A2)

and U = γ(c, vx, vy, vz). Noting that two of Maxwell’s equations are equivalent to saying F is closed, the motion of the charged particle can be written in Hamiltonian form with

H=|p|*22m,
(A3)
ω=dϑeF
(A4)

on the level set H = mc2/2, where ϑ is the tautological 1-form on T*Q for space–time Q and |p|* is the induced metric on cotangents. The number of degrees of freedom is now 4. Guiding-center reduction can still be performed resulting in a 3DoF system. Integrability would now require two further integrals beyond the Hamiltonian. It would be interesting to find out whether time-translation symmetry can be replaced.

Theorem B.1.

(V. Gelfreich) If (i)F : MnNkbetween manifolds of dimensionsnkisCnk+1and (ii) the setXMnwheredFis not of rankkhas measureμn(X) = 0, thenμn(Z) = 0, whereZ = F−1(Y) andY = F(X).

Proof.

Under the Cnk+1 condition, Sard’s theorem implies μn(Y) = 0. Every point x in the complement of X is regular for F so has a neighborhood Vx where F can be used for k components of a smooth coordinate system. Then, μn(ZVx) = 0. Since μn(X) = 0, for every ϵ > 0, there exists an open set UM such that XU and μn(X) < ϵ. The complement of U is closed and so can be covered by a countable number of the neighborhoods Vx. Therefore, μn(Z) < ϵ. Hence, μn(Z) = 0.□

Corollary B.2.

IfF:M4R2,F = (H, K) isC3, then the union of the non-regular sets has measure zero.

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