aa r X i v : . [ m a t h - ph ] D ec Gyrosymmetry: global considerations
J. W. Burby and H. Qin
1, 2 Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543,USA Dept. of Modern Physics, University of Science and Technology of China, Hefei,Anhui 230026, China (Dated: 17 June 2018)
In the guiding center theory, smooth unit vectors perpendicular to the magneticfield are required to define the gyrophase. The question of global existence of thesevectors is addressed using a general result from the theory of characteristic classes.It is found that there is, in certain cases, an obstruction to global existence. Inthese cases, the gyrophase cannot be defined globally. The implications of this facton the basic structure of the guiding center theory are discussed. In particular itis demonstrated that the guiding center asymptotic expansion of the equations ofmotion can still be performed in a globally consistent manner when a single globalconvention for measuring gyrophase is unavailable. The latter fact is demonstrateddirectly by deriving a new expression for the guiding-center Poincar´e-Cartan formexhibiting no dependence on the choice of perpendicular unit vectors.1 . INTRODUCTION
There is no doubt that the Hamiltonian formulation of guiding center theory is a founda-tional aspect of modern gyrokinetic theories. Simply put, it provides a means for deformingthe single-particle phase space so as to illuminate the approximate symmetry associated tothe magnetic moment, the gyrosymmetry, while keeping the Hamiltonian structure of theparticle dynamics in focus. However, in spite of its importance and the number of years ithas been studied , there are still poorly understood subtleties in the theory.In this paper, we study the subtleties associated with the so-called “perpendicular unitvectors” that make an appearance in virtually every version of the theory . Thesequantities, hereafter referred to as e and e , are smooth unit vector fields everywhereperpendicular to the magnetic field and to one another, meaning they form an orthonormaltriad together with b = B/ || B || in the velocity space. From one point of view, they appear inthe formalism for the sake of identifying an angular variable θ , the gyrophase, that evolves ona fast timescale with respect to the evolution timescale of the remaining dynamical variables,thereby putting the guiding center problem in the setting of the generalized method ofaveraging described in Ref. 13. In particular, when the equations of motion for a stronglymagnetized charged particle are expressed using a cylindrical parameterization of velocityspace such that the cylindrical axis points along the magnetic field, then it can be shown thatthe polar angle associated to this cylindrical coordinate system furnishes such a fast angle.This angle is measured with respect to a pair of mutually orthogonal normalized vectors e , e lying in the plane perpendicular to B . Because the magnetic field varies spatially, e , e must also vary in space so as to accommodate the constraint e · B = 0. Thereforethese e , e furnish an example of perpendicular unit vectors (see Fig. 1). From another,more geometric point of view, the perpendicular unit vectors usher themselves into theformalism so as to facilitate parameterizing the zero’th-order symmetry loops, or KruskalRings associated with the gyrosymmetry; one of the vectors, say e , distinguishes apoint on each Kruskal Ring which then serves as a reference or zero angle. Interestingly,nobody’s version of the theory ever provides a general, constructive definition of these e , e in terms of known quantities. This is the first hint that there is more to these vector fieldsthan meets the eye.Perhaps the reason nobody provides such a definition is that, in the most general setting2 IG. 1. A typical arrangement of the perpendicular unit vectors e , e for a uniform magnetic fieldthat points out of the page. The two sets of arrows represent e and e . While in this case, e and e are not required to vary in space, for a more general sort of magnetic field, they would be.Reprinted from Phys. Plasmas 19, 052106 (2012). Copyright 2012 American Institute of Physics. where the guiding center expansion applies, e , e simply cannot be defined globally, that is,there might not even be one vector field defined over the entire configuration space that isat once perpendicular to B and of unit length. While it is easy to see that smooth e , e canalways be defined locally in some, generally tiny, open neighborhood of any point p in theconfiguration space , this in no way implies that these locally defined perpendicular unitvectors extend to well-defined global quantities . So could there be an obstruction theglobal existence of smooth e , e in some cases?If we take this question seriously, a more important one arises immediately. Is the guidingcenter theory still valid without global perpendicular unit vectors? As the theory is carriedout to higher order, expressions involving the perpendicular unit vectors and their derivativesappear in the equations of motion; see Ref. 3 for instance. So it might seem plausible thatthe existence of global equations of motion is tied to the global properties of e , e .Here we will put both of these questions to rest. We will provide a complete mathemat-ical description of the obstruction to global perpendicular unit vectors and show that this3bstruction does not always vanish. However, we demonstrate that the obstruction doesindeed vanish if the physical domain is an open solid torus. Then we will show that theguiding center theory does provide consistent global equations of motion in the absence ofglobal e , e owing to the fact that the symmetry associated with the magnetic moment isalways globally defined. To illustrate this second point, we provide an expression for theguiding center Poincar´e-Cartan one-form in terms of globally defined physical quantities like B ; neither the perpendicular unit vectors nor the gyrophase appear in the expression.The paper is structured as follows. In II we provide a simple example of a magneticfield that does not admit global e , e . Then in III, we provide a complete mathematicaldescription of the obstruction to global perpendicular unit vectors in the most general case.As an example illustrating the theory, we prove in IV that if the physical domain , D ,particles are tracked through is an open, solid torus, then it is always possible to find global e , e . This is even true when the magnetic field lines are chaotic! Then we give a non-trivialexample of a magnetic field that does not admit global e , e . Finally in V, we show thatthe guiding center theory does provide consistent global equations of motion in the absenceof global e , e . II. A MOTIVATING EXAMPLE: THE FIELD DUE TO A MAGNETICMONOPOLE
The field due to a magnetic monopole provides probably the simplest illustration of theobstruction to the existence of global e , e . Perhaps the simplicity comes at the cost ofphysical relevance, but the latter will be reclaimed later after developing some machinery.Amusingly, the possibility that this example is physically relevant has never been ruled out.See Ref. 22 for an interesting discussion of the current status of magnetic monopoles intheoretical physics.The monopole field is given by B ( x ) = 1 || x || e r ( x ) , (1)where e r is the radial unit vector from a spherical coordinate system about the origin. It isdepicted in Fig. 2. Sufficiently far from the singularity at the origin, we could in principledevelop the guiding center approximation. So let the physical domain D where particles4 IG. 2. The magnetic field due to a magnetic monopole. Note that ∇ · B = 0 except at theorigin, which is depicted as a large central dot. Reprinted from Phys. Plasmas 19, 052106 (2012).Copyright 2012 American Institute of Physics. would move consist of the open region exterior to some sphere of radius r o centered on theorigin. Now we will check if there is a perpendicular unit vector defined on all of D .If there were such a vector field, e , then it could be restricted to a sphere centered onthe origin with radius r a > r o , S r a . Because B | S r a is parallel to the vector normal to S r a , e | S r a would have to be everywhere tangent to S r a . Thus, e | S r a : S r a → T S r a , where T S r a denotes the tangent bundle of S r a , would furnish an example of a smoothnon-vanishing tangent vector field on the sphere. But this situation is impossible by thefamous “hairy ball theorem”. It follows that no such e exists.There are two essential features of this example. First of all, notice that D has a “hole”due to excluding the region with r < r o , thus giving D the shape of a peach without the pit.If instead D were chosen to be some solid spherical region separated from the singularity atthe origin, then it would be possible to find an e (we won’t prove this now). But then D would be hole free. So we see that the obstruction to the existence of e is somehow relatedto the topology of D , in particular the presence of holes (or lack thereof) is important.5econd, notice that the utility of the hairy ball theorem derives entirely from the fact thatthe planes perpendicular to the magnetic field are arranged to be tangent to the spheres S r a .Thus this distribution of perpendicular planes impacts the existence of global perpendicularunit vectors. In particular, note that even if D had holes, were the monopole field replacedwith a uniform field, then global perpendicular unit vectors could be found. III. THE GENERAL OBSTRUCTION TO GLOBAL PERPENDICULARUNIT VECTORS
Now consider the general problem of finding a perpendicular unit vector defined on theentirety of an arbitrary physical domain D . D can have any number of holes, but we willinsist that it be an open subset of R and that the magnetic field in this region is neverzero. Thus D might have the appearance of a block (not just a slice) of swiss cheese. Inpractice, D would be determined by first choosing a domain where particles will move, andthen removing those regions where the guiding-center ordering breaks down.We claim that the key ingredients in the solution to this existence problem are the holestructure of D and the divergence-free vector field N discussed in depth by Littlejohn , N = 12 b (cid:18) Tr( ∇ b · ∇ b ) − ( ∇ · b ) (cid:19) (2)+ ( ∇ · b ) b · ∇ b − b · ∇ b · ∇ b. In particular, in order for a global perpendicular unit vector to exist, it is both necessary andsufficient that there be zero net flux of N through each boundary-free surface encapsulatinga hole in D .Using just Stoke’s theorem, it is easy to see that the latter condition is indeed necessary forglobal existence. If there were a globally defined e , then the vector R = ( ∇ e ) · ( b × e ) wouldbe globally defined. It is straightforward to show that this implies N has a globally definedvector potential N = ∇ × R . Therefore, if S were a boundary-free surface encapsulating ahole in D , Z S N · dA = Z S ∇ × R · dA = Z ∂S R · dl = 0 . (3)To show sufficiency is not nearly as simple. Unfortunately, a properly rigorous demon-stration would require a lengthy digression into the theory of principal bundles and charac-teristic classes, topics that are discussed by a master of these subjects in Ref. 24. While6e will make no attempt to provide the complete digression, we have included an appendixdescribing how the theorem on page 118 of the last reference can be applied to our existenceproblem to prove sufficiency of our flux condition. More curious readers will want to readRef. 24 in detail.Regardless of how the flux condition is proved, however, it is helpful to understand thefollowing physical argument for why it is feasible . As already discussed by Littlejohn in Ref.25, N can be interpreted as a kind of magnetic field whose coupling constant is the magneticmoment (instead of the electric charge). If the flux condition on N is not satisfied, thenbecause ∇ · N = 0 on D , then there must be monopole sources for the field N lurking in D ’s holes, Z S N · dA = 2 πQ gyro . (4)Here Q gyro we term the gyrokinetic monopole charge contained in the hole encapsulatedby S . A striking fact, which pushes the N -magnetic field analogy even further, is that Q gyro must be an integer. The latter can of course be identified with Dirac’s quantizationcondition on the magnetic charge, a point also discussed in Ref. 24. Now recall that whentracking the evolution of the quantum phase of an electron outside of a Dirac monopole, asingle global convention for measuring this phase is impossible; there must be at least twodistinct measurement conventions, corresponding to the domains where the vector potentialcan be defined without singularities. In gyrokinetics, the gyrophase can be considered ananalogue of the quantum phase and R an analogue of the vector potential. To make thisanalogy precise again requires discussing principal bundles. However, because gyrophaseand quantum phase represent redundant physical information, albeit in different contexts,it is perhaps reasonable on physical grounds. Thus it should not be surprising that a singleconvention for measuring gyrophase, corresponding to a choice of perpendicular unit vector,is unavailable when D encapsulates gyrokinetic monopoles. Likewise, because there are noissues defining the quantum phase when an electron’s physical domain does not encapsulateDirac monopoles, it should not be surprising that there are not issues defining the gyrophasewhen D does not encapsulate gyrokinetic monopoles.7 V. SOME EXAMPLE ASSESSMENTS OF THE EXISTENCE OFGLOBAL PERPENDICULAR UNIT VECTORS
Now we will apply the machinery developed in the previous section to assess the existenceof global perpendicular unit vectors for a few example choices of D and B . Because of theirrelevance to magnetic confinement, we will first treat the broad class of examples where D is an open solid torus and B is only constrained to be non-vanishing on D . We will showthat, in these examples, global perpendicular unit vectors can always be found. Then wewill consider a more exotic example where B is linear and vanishes at a single point and D is taken to be the region surrounding the field null. In this case global perpendicular unitvectors do not exist.When D is an open solid torus, for instance the region contained within the vacuumvessel of a tokamak device, then it is intuitively clear that every boundary-less 2-dimensionalsurface contained in D arises as the boundary of some 3-dimensional region. This statementcan of course be demonstrated rigorously using some basic results from algebraic topology .Stoke’s theorem then implies that, because ∇· N = 0, the flux of N through any such surfacemust vanish. Therefore we arrive at the following conclusion: when D is an open solid torus,global perpendicular unit vectors always exist.It is worth mentioning that this conclusion holds even when there are chaotic magneticfield lines. To see that this is reasonable , consider a typical tokamak field that has beensubjected to a resonant magnetic perturbation. Often, for instance in Ref. 27, these per-turbations are not large enough to completely kill the toroidal component of the magneticfield at any point within the last closed flux surface (assume this region is D ). However, itis will known that they may nonetheless create regions of chaotic field lines. Therefore, inspite of the presence of chaotic field lines, the vector E = e R × B = B φ e z − B z e φ , (5)where e R , e φ are the cylindrical radial and azimuthal unit vectors, vanishes nowhere in D and so defines a global perpendicular unit vector e = E / || E || . Similarly “X-points” and“O-points” lead to no obstruction to a global e , e .Now consider the magnetic field given by B ( x, y, z ) = ye x + ze y + xe z . (6)8et D = R \ S r o , where S r o is a solid sphere centered on the origin whose radius is muchlarger than any gyroradius of interest. Thus we exclude from D the only region where thegyrocenter coordinate system cannot be treated perturbatively. Note that there is nothingsingular about B at 0 even though b is. Also note that the current density ∇ × B is uniform.It is straightforward to compute the flux of N through a sphere of any radius centeredon the origin, which turns out to be − π . This implies there is a gyrokinetic monopolecharge Q gyro = − D ’s hole. This rules out the possibility of the existence of aperpendicular unit vector defined on all of D . Note that we could not have proven this lastresult by appealing directly to the hairy ball theorem; instead we had to utilize the moregeneral flux condition. V. HOW THE GUIDING CENTER THEORY WORKS WITHOUTGLOBAL PERPENDICULAR UNIT VECTORS
When a perpendicular unit vector cannot be defined globally, the usual notion of gy-rophase looses its global meaning as well. So what happens to the guiding center pertur-bation expansion? Because D can always be covered by (perhaps tiny) open regions U α inwhich local e , e are defined, the perturbation procedure can certainly be carried out ineach of these patches. The result of each of these local calculations would then consist offormal phase space coordinate changes given as formal one-to-one maps φ α : U α × R → R that lead to simpler equations of motion in the new coordinates. However, these coordinatechanges will not necessarily fit together to define a global coordinate change, i.e. an invert-ible mapping of the entire phase space into itself. Therefore, when calculating the motionof a particle as it moves from one U α to the next, it becomes necessary to occasionally passthe mechanical state from one φ α to another in order to continue using the simplified equa-tions of motion provided by the perturbation theory. While this can be done formally bydeveloping expressions for φ α ◦ φ − β , practically it would involve truncating asymptotic serieseach time the particle crossed from one U α to the next. This could lead to coherently accu-mulating error in a simulation, and, in general, would destroy the Hamiltonian propertiesof the simplified equations of motion.A far better approach is to look for a global change of coordinates to accomplish theperturbation theory from the outset . This way the difficulties associated with truncating9he expansions of the φ α ◦ φ − β could be avoided altogether. We have found that such aglobal coordinate change can be found for the guiding center problem owing essentially tothe fact that the zero’th order symmetry is globally defined. We arrived at this conclusion byapplying a version of Lie perturbation theory to the guiding center problem that synthesizesLittlejohn’s Poincar´e-Cartan one-form approach developed in Ref. 28 (also see Ref. 29) withthe group-theoretic structure provided by a zero’th order symmetry. Littlejohn’s formalismprovided the means for performing the perturbation expansion in each of the regions ofphase space where the perpendicular unit vectors can be defined, while the globally definedsymmetry served as the needle that sews these local calculations into a global result.Because the mathematical formalism we used to arrive at this conclusion draws heavilyon fiber bundle theory, we will not reproduce our method of proof here. The key point, how-ever, is simple. Because the coordinate change used in the perturbation theory is defined interms of the flow map of a Lie generator, i.e. a vector field, the coordinate change will beglobally defined if and only if the Lie generator is globally defined. A Lie generator will beglobally defined if and only if its local expressions transform as a vector should upon chang-ing from one local coordinate system to another. If the coordinate systems we use on phasespace consist of locally defined cylindrical velocity space parameterizations, correspondingto different local conventions for measuring the gyrophase, then the vector transformationlaw simplifies to the condition for gyrogauge invariant Lie generators . Thus, provided gy-rogauge invariant Lie generators are used, the coordinate change derived in the perturbationtheory will be globally defined as desired. Readers interested in a more detailed discussioncan contact one of us via email.This fact has the happy consequence that, provided gyrogauge invariant Lie generatorsare employed, the guiding- or gyro- center Poincar´e-Cartan one-form must be a globallydefined quantity even when the perpendicular unit vectors are not.Therefore, if we work in aglobally defined coordinate system, such as the obvious cartesian position and velocity coor-dinates, the Poincar´e-Cartan one-form will be manifestly independent of the perpendicularunit vectors. We will demonstrate this explicitly to drive home the point that the guidingcenter theory will work even without global perpendicular unit vectors.For simplicity we will only consider the time-independent case. Let A denote the magneticvector potential vector field and B = ∇ × A denote the magnetic field. Then the Poincar´e-10artan one-form, ordered in one of the standard ways , is given by ϑ ǫ ( x, v, t ) = A ( x ) · dx + ǫv · dx − ǫ v · vdt (7)= ϑ + ǫϑ + ǫ ϑ . One can consider all variables dimensionless or not. In the latter case, A should be consideredto be normalized by the charge-to-mass ratio of the particle in question so that ∇ × A hasthe units of frequency. The coordinates used in this expression are cartesian position andvelocity, ( x, v ).This one-form defines the dynamical vector field X ǫ ( x, v, t ) through the formula X ǫ y dϑ ǫ = 0 . (8)It is straightforward to verify that this implies˙ x ( x, v ) = ǫv (9)˙ v ( x, v ) = v × B ( x ) . (10)Now, applying Littlejohn’s gyrogauge invariant Poincar´e-Cartan perturbation theory in adomain of phase space where we have a locally defined perpendicular unit vector as in Ref.25, the truncated Poincar´e-Cartan one-form becomesˆ ϑ ( x, v k , v ⊥ , t ) = (cid:18) A ( x ) + v k b ( x ) (cid:19) · dx (11)+ 12 v ⊥ || B ( x ) || (cid:18) dθ − R ( x ) · dx (cid:19) − (cid:18) v k + 12 v ⊥ (cid:19) dt, which involves the unphysical e , e through R = ( ∇ e ) · e . The coordinates used in thisexpression are cartesian position x and cylindrical velocity coordinates ( v ⊥ , v k , θ ), where θ is measured with respect to the local perpendicular unit vector e .Now we simply change back to cartesian position and velocity coordinates according tothe mapping v = v k b ( x ) + v ⊥ cos( θ ) e ( x ) − v ⊥ sin( θ ) b ( x ) × e ( x ) x = x. ϑ ( x, v, t ) = (cid:18) A ( x ) + v · b ( x ) b ( x ) (cid:19) · dx (12)+ 12 (Π( x ) · v ) || B ( x ) || (cid:20)(cid:18) ∇ b · b ( x ) × vv · b ( x ) || b ( x ) × v || (cid:19) · dx − (cid:18) b ( x ) × v || b ( x ) × v || (cid:19) · dv (cid:21) − v · vdt, where Π( x ) = 1 − b ( x ) b ( x ) is the perpendicular projection tensor. Clearly the perpendic-ular unit vectors appear nowhere in the expression. Furthermore, it has exactly the samesymmetry properties as Eq. (11) because no approximations were made passing from thatexpression to this one. In particular, the dynamical equations implied by the new expressionmust conserve the magnetic moment exactly.Boghosian has achieved a similar result previously in the relativistic context. However,he decided to introduce extra variables with compensatory Lagrange multipliers presumablyin order to continue to work with the parallel and perpendicular velocity as coordinates.Thus the above expression is indeed a distinct and, to our knowledge, new result. One pointregarding its derivation is especially important: if R were to be neglected in Eq. (11), thedependence of the one-form on the gyrophase convention would not disappear upon passingto cartesian position and velocity coordinates. We would also like to mention that we haverecently been informed of a Lie perturbation method that succeeds in attaining 12 directly,without ever resorting to the cylindrical velocity space parameterization. VI. CONCLUSION AND DISCUSSION
To summarize, we have identified the obstruction to the global existence of perpendicularunit vectors in terms of a flux condition on the nameless vector N . Through examples,we showed that this obstruction does not trivially vanish in all cases. In particular, wehave given two simple examples where the guiding center ordering is applicable, but globalperpendicular unit vectors fail to exist. However, we demonstrated that when the physicaldomain particles move through is an open solid torus, global perpendicular unit vectorsalways exist. We have also provided a physically plausible explanation for the flux conditionin terms the new concept of gyrokinetic monopoles.Then we proceeded to explain how the guiding center theory works when global per-12endicular unit vectors are unavailable. In particular, we derived an expression for theguiding-center Poincar´e-Cartan one-form in a coordinate system rectilinear in both positionand velocity only involving globally defined quantities.Looking at what we have done from a practical point of view, we have identified somedifficulties researchers will face when trying to simulate gyrophase-dependent dynamics in configurations where global perpendicular unit vectors cannot be defined. When dealingwith such deviant cases numerically, for instance in a particle-in-cell simulation, it will benecessary to either define a number of gyrophase conventions that cover the phase space andkeep track of which of these “patches” particles live in, or resort to the global expressionfor the Poincar´e-Cartan one-form given at the end of the previous section. In the formercase, care must be taken to avoid spending too much time keeping track of a particle’s“patch”, while in the latter case this could be avoided. However, the cost incurred by usingthe global version of the one-form comes in the form of complicated equations of motion.While simulations of the interior of tokamaks should be able to avoid multiple gyrophaseconventions by finding global perpendicular unit vectors (which must exist), this may notbe the case in configurations such as the polywell that involve field nulls in the region ofinterest. Around each of these field nulls, bubble-like regions must be excluded from thephysical domain to ensure the validity of treating the gyrocenter coordinate system pertur-batively. If gyrokinetic monopole charge resides in any of these cavities, then perpendicularunit vectors will be unavailable in the “safe” region exterior to these cavities. ACKNOWLEDGMENTS
This work was supported by the U.S. Department of Energy under contract numberDE-AC02-09CH11466.
Appendix A: Principal circle bundles
Here we define and discuss the notion of principal circle bundle. A more complete ex-position can be found in Ref. 36. First some terminology. Let P be a manifold andΦ : S × P → P a smooth map, where S = R mod 2 π denotes the circle. If θ , θ ∈ S ,then we take the symbol θ + θ to mean addition modulo 2 π . For a fixed θ ∈ S define13he map Φ θ : P → P by the formula Φ θ ( p ) = Φ( θ, p ), where p is any point in P . Φ is saidto be a left circle action when Φ θ ◦ Φ θ = Φ θ + θ and Φ is the identity on P . Given apoint p ∈ P , the set O p = { p ′ ∈ P |∃ θ ∈ S s.t. Φ θ ( p ) = p ′ } is called the orbit of Φ through p . A left circle action is said to be free if Φ θ ( p ) = p if and only if θ = 0. Intuitively, aleft circle action is free if when the second argument of Φ is held fixed at p o , the resultingmap establishes a one-to-one correspondence between the orbit through p o and the circle. A principal circle bundle is a manifold P together with a free left circle action Φ : S × P → P .If there is a manifold B and a smooth map π : P → B such that π is surjective, its Jacobianmatrix has full rank at each point p ∈ P , and π − ( b ) is an entire orbit for each b ∈ B , then P/S ≡ B is referred to as the base of the principal circle bundle P and π is referred toas the bundle projection map. Because it can be shown such a B and π can always befound for a principal circle bundle, the following intuitive picture of such bundles emerges.A principal circle bundle is nothing more than a collection of circles (the orbits) smoothlyparameterized by the base P/S .There is a subtle aspect of this picture however. Notice that while it is possible to fixa point p o ∈ P as the second argument in Φ and establish a correspondence between theorbit through p o and S , if Φ θ ( p o ) were used in place of p o , the result would be a different correspondence between the same two objects O p o and S . This is because O p o = O Φ θ ( p o ) .Therefore, while the orbits O p “look” like distorted copies of the circle, they lack a naturalchoice for the 0, or reference angle.On the other hand, it is often convenient take a bunch of nearby orbits and smoothlyassign to each of them a reference point so that each point on this bunch of orbits can beassigned an angle in an unambiguous way. Such an assignment of reference points is calleda local section. Formally, given an open subset U α ⊂ P/S of the base, a local section s α : U α → π − ( U α ) is a mapping from U α into the collection of orbits that project onto U α that satisfies the equation π ◦ s α = id U α , which simply says that s α assigns a single point toeach of the orbits “attached” to U α . Local sections can always be found. However, a globalsection s : P/S → P , which would smoothly assign a reference point to all of the orbitsthat make up P , may not exist. If a global section does exist, then the principal bundle isreferred to as being trivial .In the presence of a local section, the process of assigning an angle to each point in thebunch of orbits attached to U α can be formalized as a special coordinate system on π − ( U α )14nown as a bundle chart. If p ∈ π − ( U α ), then, because the action is free, there is a unique g α ( p ) ∈ S such that p = Φ g α ( p ) s α ( π ( p )) . This defines the functions g α : π − ( U α ) → S . Thebundle charts φ α : π − ( U α ) → U α × S are then given by the formula φ α ( p ) = ( π ( p ) , g α ( p )) . By this definition, when looking at a principal circle bundle locally in a bundle chart, itlooks like a bunch of bike tires hanging on a multi-dimensional horizontal rod. The orbitsare the tires while the base is the rod. It is also useful to think of the bundle charts as“symmetry-aligned” coordinate systems, where the symmetry is defined by Φ.
Appendix B: Principal connections
This appendix gives the definition of a principal connection and briefly explores someof the basic properties of these objects relevant to this article. A much more thoroughdiscussion can be found in Ref. 37.Given a principal circle bundle ( P, Φ) and a real number ξ , the infinitesimal generator ξ P associated to ξ is the vector field on P given by ξ P ( p ) = ddθ (cid:12)(cid:12) θ =0 Φ ξθ ( p ). So ξ P points in thedirection of the symmetry associated with Φ. A principal connection , or connection form on P is a one-form, A , with the following two properties:1) ∀ ξ ∈ R , A ( ξ P ) = ξ ∀ θ ∈ S , Φ ∗ θ A = A . Connection forms have a useful local structure when viewed in the bundle charts definedin the previous section. Let s α : U α → π − ( U α ) be a local section and φ α its associatedbundle chart. Define the gauge field A α : T ( U α ) → R and the Maurer-Cartan one-form θ L : T S → R by A α = s ∗ α A (B1) θ L ( θ, ξ ) = ξ, (B2)where we have made the identification T S = S × R . Note that θ L is nothing more thanthe coordinate differential on S . It is not difficult to show that on π − ( U α ) A is made upof these two quantities according to A = π ∗ A α + g ∗ α θ L . (B3)15his formula has two important consequences. First of all, if A β is another gauge fielddefined on an overlapping patch of P/S , U α ∩ U β = ∅ , then it must be related to A α on theoverlap: A α = A β + g ∗ αβ θ L , (B4)where g αβ : U α ∩ U β → S is the circle-valued function defined by the relation g αβ ( π ( p )) = g β ( p ) − g α ( p ). Second, it implies that the gauge field strengths F α = dA α , apparently onlylocally defined quantities, actually define a global two-form, the curvature form F , overthe entire base P/S . This result follows from applying the exterior derivative to (B4) andrecalling that dθ L = 0. On any of the U α , F = F α . As discussed in Ref. 24, the curvaturetwo-form encodes the basic topological properties of the principal circle bundle it comesfrom.Connection forms also provide a convenient structure for expressing the transformationlaw for the bundle chart representatives of globally defined vector fields on P . If X : P → T P is a smooth vector field on P , then given a bundle chart φ α , its bundle chartrepresentative is X α ≡ φ α ∗ X : U α × S → T U α × S × R ; the bundle chart representativesare just the vector field expressed in the coordinates provided by the bundle charts. Set X α ( u, θ ) = ( w α ( u, θ ) , θ, ξ α ( u, θ )), where w α ( u, θ ) ∈ T u ( P/S ) and ξ α ( u, θ ) ∈ R . Using thefact that φ ∗ α X α = φ ∗ β X β on π − ( U α ∩ U β ), it is straightforward to show that the bundle chartrepresentatives are related by w α ( u, θ ) = w β ( u, θ ′ ) (B5) ξ α ( u, θ ) = ξ β ( u, θ ′ ) + g ∗ βα θ L ( w β ( u, θ ′ )) , (B6)where θ ′ = θ + g αβ ( u ). Using the transformation law for the gauge fields, this can be recastas η α ( u, θ ) ≡ ξ α ( u, θ ) + A α ( w α ( u, θ )) (B7) w α ( u, θ ) = w β ( u, θ ′ ) (B8) η α ( u, θ ) = η β ( u, θ ′ ) . (B9)So we see that the w α and η α are local representatives of globally defined maps. To beprecise, w α = w ◦ φ − α and η α = η ◦ φ − α , where w : P → T ( P/S ) and η : P → R are16lobally defined maps only constrained to satisfy τ P/S ◦ w = π ( τ P/S is the tangent bundleprojection map associated to T ( P/S )).Conversely, if there is an assignment of a local vector field X α to each of the bundle charts φ α whose components satisfy (B8) and (B9), then this collection of locally defined vectorfields will define a global vector field X : P → T P that agrees with each of the X α in thebundle charts.Why is expressing the vector transformation law in terms of the gauge fields useful?Because of the organization it brings to the process of stitching together local vector fieldsinto a global one. The vector transformation law for passing from one arbitrary (non-bundle)coordinate chart to another would be quite messy to work with for this purpose. By workingwith the bundle charts and finding expressions for the gauge fields, the process is streamlinedto finding the two functions w and η . Appendix C: Sufficiency of the flux condition
The appropriate way to tackle this problem is to recognize that SD is actually a prin-cipal circle bundle and that the existence of a globally defined perpendicular unit vectoris equivalent to the existence of a global section of SD (see appendix A for the necessarybackground on principal circle bundles). Because a principal circle bundle admits a globalsection if and only if it is a trivial bundle, the existence problem can be solved by appealingto the well-established topological classification of principal circle bundles . This classifi-cation theorem tells us that if we can find any so-called principal connection on SD (seeappendix B for the necessary background on principal connections), which is a special sortof one-form over SD , then the curvature of this connection, a closed two-form over D in-duced by the principal connection, will be exact if and only if SD is a trivial bundle. Thus,given the curvature form, existence of global perpendicular unit vectors can be tested byintegrating the curvature form over a collection of cycles that generate D ’s second homologygroup H ( D, Z ) . If all of these integrals vanish, then the curvature form must be exactand a global section of SD must exist.So in order to furnish a solution to the existence problem, all that we must still do is1) prove that SD is a principal circle bundle whose global sections, if they exist, coincidewith global perpendicular unit vectors and 2) derive an expression for the curvature form17ssociated to some principal connection on SD . Then existence can be determined in anyparticular case after finding the “holes” in D .First notice that SD is indeed a manifold. Actually it is a submanifold of T D = D × R defined by the algebraic equations v · v = 1 (C1) v · b ( x ) = 0 , where ( x, v ) ∈ T D . Next, consider the following circle action on SD :Φ θ ( x, v ) = ( x, exp (cid:16) θ ˆ b ( x ) (cid:17) v ) , (C2)where ˆ b ( x ) is the 3 × b ( x ) w = b ( x ) × w , and exp denotesthe matrix exponential. Hence this circle action simply rotates all of the circles that comprise SD by θ radians. Furthermore, the action is free. Therefore ( SD,
Φ) forms a principal circlebundle.To see that the sections of this circle bundle are equivalent to the perpendicular unitvectors, we first show that the base space of the bundle can be identified with D . Definethe map π : SD → D by π ( x, v ) = x. (C3) π is a surjective submersion and π − ( x ) is equal to the circle in SD over x , which is an entireorbit of the action Φ. It follows that SD/S = D with π serving as the bundle projectionmap. Thus a global section of SD would consist of a smooth map of the form s : D → SD with the property π ( s ( x )) = x , that is, s ( x ) must lie in the circle over x . Because all ofthe points on the circle over x are by definition perpendicular to b ( x ) and of unit length, s would be a global perpendicular unit vector. Conversely, any global perpendicular unitvector would define such an s .Now we move on to define a principal connection on SD . Because it will be necessary towork with the space T SD ⊆ T T D , we make the following identification:
T T D = T ( D × R ) = T D × T R = ( D × R ) × ( R × R ) . Accordingly, a typical element of the 12 dimensional space
T T D will be denoted ( x, u, v, a ),where ( u, a ) forms the tangent vector over the point ( x, v ) ∈ T D . Clearly, each element of18 SD can also be written in this way (of course u and a will be constrained in this case). Itwill also be helpful to define a metric on T D . Recall that such a metric on
T D defines aninner product on each of the tangent spaces in
T T D . The useful metric in this case assignsan inner product to each ( x, v ) ∈ T D given by (cid:28) ( x, u, v, a ) , ( x, u ′ , v, a ′ ) (cid:29) = u · u ′ + a · a ′ . (C4)Note the distinction between this inner product denoted by square brackets and the usualdot product between vectors in R . Finally, a principal connection A : T SD → R can bedefined by A ( x, u, v, a ) = (cid:28) ( x, u, v, a ) , ( x, , v, b ( x ) × v ) (cid:29) = a · b ( x ) × v. (C5)The two defining properties of a principal connection (appendix B) are straightforward tocheck.Next we derive an expression for the curvature form associated to A . Because a localsection s α : U α ⊆ D → π − ( U α ) must be of the form s α ( x ) = ( x, e ( x )) , (C6)where e is a locally defined perpendicular unit vector, the gauge fields must be of the form A α ( x, w ) = s ∗ α A ( x, w ) = (cid:18) w · ∇ e ( x ) (cid:19) · b ( x ) × e ( x ) (C7) ≡ w · R ( x ) . As the notation suggests, R = ( ∇ e ) · b × e = ( ∇ e ) · e is the well-known quantity thatappears elsewhere in the guiding center formalism. Therefore, the curvature form F = dA α is given by the equation ∗ F = N · dx, (C8)where ∗ is the hodge star and N = ∇ × R . By the transformation law for curvature formsgiven in appendix B, N must be a globally defined quantity even when e , and therefore R ,is not. In fact, there is an expression giving N in terms of b : N = 12 b (cid:18) Tr( ∇ b · ∇ b ) − ( ∇ · b ) (cid:19) (C9)+ ( ∇ · b ) b · ∇ b − b · ∇ b · ∇ b. F should be integrated over a collection of cycles that generate D ’s second homologygroup H ( D, Z ). Intuitively, this amounts to calculating the flux of N through a collectionof closed, bounded, boundary-less surfaces that encapsulate the “holes” in D . If all of theseintegrals vanish, then there will be global perpendicular unit vectors. Otherwise, owing tothe ensuing non-trivial topology of SD , global perpendicular unit cannot be defined, evenin principle. REFERENCES M. Kruskal, “The gyration of a charged particle,” Project Matterhorn Report PM-S-33(NYO-7903) (Princeton University, 1958). T. G. Northrop,
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