Discovering exact, gauge-invariant, local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Peifeng FAN (范培鋒), Hong QIN (秦宏)and Jianyuan XIAO (肖建元)

1 Key Laboratory of Optoelectronic Devices and Systems, College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, People?s Republic of China

2 Advanced Energy Research Center,Shenzhen University,Shenzhen 518060,People?s Republic of China

3 Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, United States of America

4 School of Nuclear Science and Technology, University of Science and Technology of China, Hefei 230026, People?s Republic of China

Abstract Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas.However, exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort.Without such local conservation laws,energy and momentum can be instantaneously transported across spacetime,which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory.The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds.To overcome this difficulty, we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact, local conservation laws, in particular the energy–momentum conservation laws, for the electromagnetic gyrokinetic system.A weak Euler–Lagrange(EL)equation is established to replace the standard EL equation for the particles.It is discovered that an induced weak EL current enters the local conservation laws,and it is the new physics captured by the high-order field theory on heterogeneous manifolds.A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.

Keywords: electromagnetic gyrokinetic system, high-order field theory, heterogeneous manifolds, exact local energy–momentum conservation laws, weak Euler–Lagrange equation,gauge-invariant theory

1.Introduction

Gyrokinetic theory, gradually emerged since the 1960s [1–12],has become an indispensable tool for analytical and numerical studies [13–15] of instabilities and transport in magnetized plasmas, with applications to magnetic fusion and astrophysics.Present research on gyrokinetic theories focuses on endowing the models with more physical structures and conservation properties using modern geometric methods [16, 17], with the goal of achieving improved accuracy[18–21]and fidelity for describing magnetized plasmas.For example,the Euler–Poincare reduction procedure [22], Hamiltonian structure [23] and explicit gauge independence [24] have been constructed for gyrokinetic systems.These studies closely couple with the investigation of structure-preserving geometric algorithms of the guiding-center dynamics [25–27] for gyrokinetic simulations with long-term accuracy and fidelity.

One conservation property of fundamental importance for theoretical models in physics is the energy–momentum conservation.The gyrokinetic theory is no exception.However,exact local energy–momentum conservation laws for the gyrokinetic system with fully self-consistent time-dependent electromagnetic field are still unknown.It is worthwhile to emphasize that we are searching for local conservation laws instead of the weaker global ones.If a theoretical model does not admit local energy–momentum conservation law, energy and momentum can be instantaneously transported across spacetime,which is unphysical and detrimental for the purpose of studying energy and momentum transport in magnetized plasmas.

An effective approach is to start from variational principles,or field theories, and derive conservation laws by identifying first the underpinning symmetries admitted by the Lagrangians of the systems.This is the familiar Noether procedure[28,29].In principle, such a field theoretical methodology can also be adopted for gyrokinetic systems or the guiding-center drift kinetic system.However, as mentioned above, exact local energy–momentum conservation laws for the general gyrokinetic Vlasov–Maxwell system remain elusive despite continuous effort [22, 30–33].The technical difficulties involved can be viewed from two different angles.For the Eulerian formalism for gyrokinetic models, the Euler–Lagrange (EL)equation assumes a different form because the field variations are constrained [22, 34, 35], and the derivation of conservation laws from symmetries does not follow the standard Noether procedure for unconstrained variations.For Low’s type [36] of variational principles with mixed Lagrangian and Eulerian variations [17], particles (gyrocenters in this case) and the electromagnetic field reside on different manifolds.The electromagnetic field is defined on spacetime, but the particles are defined on the time axis only.This also differs from the standard Noether procedure.

Recently, this difficulty is overcome by the development of an alternative field theory for the classical particle-field system [37–39].This new field theory embraces the fact that different components, i.e.particles and electromagnetic field,reside on heterogeneous manifolds, and a weak EL equation was derived to replace the standard EL equation for particles.It was shown that under certain conditions the correspondence between symmetries and conservation laws is still valid, but with a significant modification.The weak EL equation introduces a new current in the corresponding conservation law.This new current,called weak EL current,represents the new physics captured by the field theory on heterogeneous manifolds [39].The field theory on heterogeneous manifolds has been successfully applied to find local conservation laws in the Vlasov-Poisson system and the Vlasov–Darwin system that were previously unknown [37, 39].

In this paper,we extend the field theory for particle-field system on heterogeneous manifolds to systems with highorder field derivatives in non-canonical phase space coordinates,and apply it to systematically derive local conservation laws for the electromagnetic gyrokinetic system from the underpinning spacetime symmetries.In particular, the exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system are derived.For gyrokinetic systems, the finite-larmor-radius (FLR) effect is important,and the Lagrangian density must include derivatives of the field up to certain desired orders.Therefore, extending the field theory on heterogeneous manifolds to systems with high-order field derivatives is a necessary first step.We first extend the theory to include arbitrary high-order field derivatives, and then derive the energy–momentum conservation law for the electromagnetic gyrokinetic system.

Another difference between the present work and most previous studies [40, 41] is that we do not separate the electromagnetic field into perturbed and background parts.The field theory and conservation laws are expressed in terms of the total distribution functions and the 4-potential(φ(t,x) ,A(t,x)).This ensures that the Lagrangian density does not explicitly depends on the spacetime coordinatesxandt, and always admits exact 3D energy–momentum conservation laws.In most previous studies [40, 41], the magnetic field is separated into perturbed and background parts,and conservation laws were derived for the perturbed fields.However, such conservation laws exist only when the background field is symmetric with respect to certain spacetime coordinates.For example, in the tokamak geometry, if the background magnetic field is assumed to be symmetric in the toroidal direction by neglecting the ripple field, a toroidal angular moment conservation law in terms of the perturbed fields can be derived [41].But the general 3D momentum conservation cannot be established in these previous studies because the background magnetic field is inhomogeneous in the radial and poloidal directions.There is another advantage of not separating electromagnetic field into perturbed and background parts.Recently, Chenet al[42, 43] pointed out that for studying long-term dynamics it is desirable to express gyrokinetic equations in terms of the total fields, and such gyrokinetic equations were derived [42, 43].

In the present study,we also adopt a systematic approach to remove the electromagnetic gauge dependence from the electromagnetic gyrokinetic system using a gauge-symmetrization method recently developed for classical charged particle-electromagnetic field theories [44].The standard Belinfante–Rosenfeld method [45–47] symmetrizes the energy–momentum tensor (EMT) using a super-potential associated with the angular momentum but does not necessarily make the EMT gauge-invariant for a general field theory.The result reported in [44] shows that a third order tensor called electromagnetic displacement-potential tensor can be constructed to explicitly remove the gauge dependency of the EMT for high-order electromagnetic field theories.This method is applied here to render the exact, local energy–momentum conservation laws derived for the electromagnetic gyrokinetic system gauge-invariant.

This paper is organized as follows.In section 2, we extend the field theory for particle-field systems on heterogeneous manifolds to systems, such as the gyrokinetic system, with high-order field derivatives in non-canonical phase space coordinates.The weak EL equation is developed as necessitated by the fact that classical particles and fields live on different manifolds.Symmetries for the systems and the links between the symmetries and conservation laws are established.In section 3, the general theory developed is applied to derive the exact, gauge-invariant, local energy–momentum conservation laws induced by spacetime translation symmetries for the electromagnetic gyrokinetic system.

2.High-order field theory on heterogeneous manifolds

Before specializing to the electromagnetic gyrokinetic system,we develop a general high-order field theory on heterogeneous manifolds for particle-field systems using noncanonical phase space coordinates.A weak EL equation is derived.Exact local conservation laws are established from the underpinning symmetries.The weak EL current in the conservation laws induced by the weak EL equation is the new physics predicted by the field theory on heterogeneous manifolds.

2.1.Weak EL equation

We start from the action of particle-field systems and revisit the field theory on heterogeneous manifolds developed in[37–39].We extend the theory to include high-order field derivatives and use non-canonical phase space coordinates(Xa,Ua)for particles.The action of gyrokinetic systems assumes the following form with the field derivatives up to thenth order

In this section, we will work out the field theory for this general form of action without specializing to gyrokinetic models.The subscriptalabels particles,(Xa(t),Ua(t))is the trajectory of theath particle in phase space over the time axis.Xa(t)takes value in the 3D laboratory space,andψ(t,x)is a vector(or 1-form)field defined on spacetime.For gyrokinetic system,ψwill be the 4-potentials of the electromagnetic field, i.e.ψ=(φ,A).Lais Lagrangian of theath particle,including the interaction between the particle and fields.LFis the Lagrangian density for the fieldψ.Here,pr(n)ψ(t,x)as a vector field on the jet space is the prolongation of the fieldψ(t,x)[29], which containsψand its derivatives up to thenth order, i.e.

where ?μi? {?t,?x1,?x2,?x3},(i=1, 2,…,n),represents a derivative with respect to one of the spacetime coordinates.

The difference in the domains of the field and particles is clear from equation (1).The fieldsψis defined on the 4D spacetime, whereas each particle’s trajectory as a field is just defined on the 1D time axis.The integral of the Lagrangian densityLFfor the fieldψis over spacetime, and the integral of LagrangianLafor theath particle is over the time axis only.Because of this fact, Noethers procedure of deriving conservation laws from symmetries is not applicable without modification to the particle-field system defined by the action A in equation (1).

To overcome this difficulty,we multiply the first part on the right-hand side of equation (1) by the identity

whereδa≡δ(x?Xa(t))is Dirac’s δ-function.The actionA in equation (1) is then transformed into an integral over spacetime

Note that the Lagrangian of theath particleLais transformed to the Lagrangian densityLaby multiplyingδa.Obviously, the variation of the action we constructed here will not have any constraints,which will make the variational process easier.We now calculate how the action given by equation(4)varies in response to the field variationsδXa,δUaandδψ,

where

are Euler operators with respect toXa,Uaandψ, respectively.In equation(6),the termsδXaandδUacan be taken out from the space integral because they are fields just defined on the time axis.Applying Hamilton?s principle to equation (6),we immediately obtain the equations of motion for particles and fields

by the arbitrariness ofδXa,δUaandδψ.Equation (10) is the EL equation for fieldsψ.Equations (11) and (12) are called submanifold EL equations forXaandUabecause they are defined only on the time axis after integrating over the spatial dimensions [37–39].We can easily prove that the submanifold EL equations(11)and(12)are equivalent to the standard EL equations ofLa,

by substituting the Lagrangian density (5).

Our next goal is to derive an explicit expression forEUa( L)andEXa( L).From the EL equation (13),

becauseδadoes not depend onUa.However,( )LEXais not zero but a total divergence [37–39]

To prove equation (15), we calculate

We will refer to equation (15) as weak EL equation.The qualifier weak here indicates that the spatial integral ofEXa( L), instead ofEXa( L)itself, is zero [37–39].The weak EL equation plays a crucial role in connecting symmetries and local conservation laws for the field theory on heterogeneous manifolds.The non-vanishing right-hand-side of the weak EL equation (15) will induce a new current in conservation laws[37–39].This new current is called the weak EL current, and it is the new physics associated with the field theory on heterogeneous manifolds.

2.2.General symmetries and conservation laws

We now discuss the symmetries and conservation laws.A symmetry of the actionA is a group of transformations

such that

for every subdomain.Here,?gconstitutes a continuous group of transformations parameterized by ?.Equation(17)is called symmetry condition.To derive a local conservation law, an infinitesimal version of the symmetry condition is required.For this purpose,we take the derivative of equation(17)with respect to ?at =?0,

Following the procedures in [29], the infinitesimal criterion derived from equation (18) is

Here,vis the infinitesimal generator of the group of transformations and the vector field(1,n)vpris the prolongation ofvdefined on the jet space,which can be explicitly expressed as

where

are the characteristics of the infinitesimal generatorv.The superscript α is the index of the fieldsφandψ.The formulations and proofs of equations (22)–(24) can be found in [29].

Having derived the weak EL equation (15) and infinitesimal symmetry criterion (19), we now can establish the conservation law.We cast the infinitesimal criterion(19)into an equivalent form

where the 4-vector fieldsPυaandPFvcontain high-order derivatives of the fieldψ.They are the boundary terms[28, 29] calculated by integration by parts

The last two terms in equation (25) vanish due to the EL equations(10)and(14),while the third term is not zero because of the weak EL equation (15) and induces a new current for system.If the characteristicqais independent ofx, the local conservation law of the symmetry is finally established as

where

2.3.Statistical form of the conservation laws

The local conservation law (31) is written in terms of particle’s phase space coordinates(Xa(t) ,Ua(t))and fieldψ(t,x).To express it in the statistical form in terms of distribution functions of particles and field, we classify the particles into several species by their invariants such as mass and charge.A particle indexed by the subscriptacan be regarded as thepth particle of thes-species,i.e.ais equivalent to a pair of indices

For each species, the Klimontovich distribution function is defined to be

FunctionsLa,qaandin equation(31)distinguished by the indexa～spare same functions in phase space for the same species.For such a functionga(x,u),the labela～spcan be replaced just bys, i.e.

In the conservation law (31), the summations in the form ofcan be expressed in terms of the distribution functionsFs(t,x,u),

Using equation (36), the conservation law (31) can be equivalently written in the statistical form in terms of the distribution functionsFs(t,x,u)and fieldψ(t,x)as

Note that in equation (37), the index for individual particlesahas been absorbed by the Klimontovich distribution functionFs(t,x,u),which serves as the bridge between particle representation using(Xa(t),Ua(t))and distribution function representation.In section 3, local conservation laws for the electromagnetic gyrokinetic system will be first established using the particle representation in the form of equation (31).They are then transformed to the statistical form in the form of equation (37) using this technique.

3.Exact, gauge-invariant, local energy–momentum conservation laws for the electromagnetic gyrokinetic system

In this section, we apply the field theory on heterogeneous manifolds for particle-field systems developed in section 2 to the electromagnetic gyrokinetic system, and derive the exact,gauge-invariant, local energy–momentum conservation laws of the system from the underpinning spacetime translation symmetries.For the general electromagnetic gyrokinetic system specified by the Lagrangian density in equation (38),the final conservation laws are given by equations (98) and(126).The derivation is explicitly illustrated using the firstorder system specified by the Lagrangian density in equation (58).

We emphasize here that the present study starts from the general field theory of the gyrokinetics.Once the action or Lagrangian of the gyrokinetic system is given, the conservation(energy and momentum)laws can be systematically derived using the method we developed in this work.We take the validity of the gyrokinetic field theory(or the Lagrangian of the system) as a prerequisite.

3.1.The electromagnetic gyrokinetic system

When the field theory on heterogeneous manifolds developed in section 2 is specialized to the electromagnetic gyrokinetic theory,Xais the gyrocenter position,Ua=(ua,μa,θa)consists of parallel velocity, magnetic moment and gyrophase,and the fieldψ(t,x)=(φ(t,x) ,A(t,x))is the 4-potential.As in the general case,the Lagrangian density of the systemL is composed of the field Lagrangian densityLFand particle LagrangianLa,

For the general electromagnetic gyrokinetic system,LFis the standard Lagrangian density of the Maxwell field theory

For particles

whereL0ais the leading order of the LagrangianLaof theath particle,L1ais the first-order,etc.AndδLarepresents all highorder terms of ofLa.The expressions ofL0aandL1aare give by equations (59) and (60), respectively.The expansion parameter is the small parameter of the gyrokinetic ordering,i.e.

Here,kand ω measure the spacetime scales of the electromagnetic fieldEandBassociated with the total total 4-potential(φ,A) , and ρ and Ω are the typical gyro-radius and gyro-frequency of the particles.

Before carrying out the detailed derivation of the energy–momentum conservation laws, we shall point out a few features of the electromagnetic gyrokinetic system defined by equation (38).In the gyrokinetic formalism adopted by most researchers, the electromagnetic potentials (fields) are separated into perturbed and background parts

where subscript ‘0’ indicates the background part, and subscript ‘1’ the perturbed part.Here,A1～?A0andφ1～?φ0.Letk1andω1denote the typical wave number and frequency of the electromagnetic field associated with the perturbed 4-potential(φ1,A1).While gyrokinetic theory requires equation (43), it does allow

The energy conservation law derived in [40, 41] is for the perturbed field(φ1,A1)when the background field(φ0,A0)does not depend on time explicitly.Because the background magnetic fieldB0(x)= ?×A0depends onx, the momentum conservation law in terms of(φ1,A1)cannot be established in general, except for the case where x(B0)is symmetric with respect to specific spatial coordinates.

In the present study, we do not separate the electromagnetic potentials (fields) into perturbed and background parts,and the theory and the energy–momentum conservation laws are developed for the total field(φ,A) .Therefore, it is guaranteed that the Lagrangian density L defined in equation (38) does not break the space translation symmetry,and that the exact local energy–momentum conservation laws always exist.

To ensure the validity of the gyrokinetic model used in the present study,we assume that the amplitudes of perturbed fields(φ1,A1) are much smaller, while the frequency and wave number can be much lager than the background parts.It is also important to observe that condition (46) is consistent with the gyrokinetic ordering (43), because the amplitude of the perturbed field is smaller by one order of ?.Since our theory is developed for the total field(φ,A), only the gyrokinetic ordering(43)is required,and it is valid for cases with condition(46).To express the FLR effects of the gyrokinetic systems using the total field(φ,A), it is necessary and sufficient to include high-order field derivatives in the Lagrangian densityL, which is the approach we adopted.The general theory developed include field derivatives to all orders, and we explicitly work out the first-order theory,which includes field derivatives up to the second-order.

Without specifying the explicit form ofLFandLa,the equations of motion for φ andAderived directly from the equation (10) are

where

The following equations

are used in the last steps of equations (47) and (48), andεin equation (55) is the Levi-Civita symbol in the Cartesian coordinates.In equation(49),ρgandjgare charge and current densities of gyrocenter, andPandMin equations (50) and(51) are polarization and magnetization, which contain field derivatives up to thenth order.Using equations(47)and(48),the equations of motion for fields(φ,A),are then transformed into

We will derive the exact, gauge-invariant, local energy–momentum conservation laws for the general electromagnetic gyrokinetic system specified by the Lagrangian density in equation (38).The final conservation laws are given by equations (98) and (126).To simplify the presentation, we only give the detailed derivation for the following first-order electromagnetic gyrokinetic theory which only keepsL1ainδLa[17]

wheremaandqaare mass and charge of theath particle, andwais the perpendicular velocity.The Routh reduction has been used to decouple the gyrophase dynamics.Note that the first-order LagrangianL1acontains second-order spacetime derivatives of the electromagnetic 4-potential(φ,A) .The prolongation field involved is thus pr(2)ψ(t,x).

From equations (50) and (51), we can obtain the polarizationPand magnetizationMfor the first-order theory as

The detailed derivations of equation (67) and (70) are shown in appendix A.

3.2.Time translation symmetry and local energy conservation law

First, we look at the local energy conservation.It is straightforward to verify that the action for the gyrokinetic system specified by the Lagrangian density in equation(38)is invariant under the time translation

because the Lagrangian density does not contain the time variables explicitly.Using equations (20) and (22), the infinitesimal generator and its prolongation of the group transformation are calculated as

whereξ= 1t,ξ= 0 and(see equations (20)–(23)).The infinitesimal criterion (19) is reduced to

which is indeed satisfied as the Lagrangian density does not depend on time explicitly.Because the characteristic of the infinitesimal generatoris independent ofx,the infinitesimal criterion(74)will induce a conservation law by calculating terms in equation (31).Using equations (24) and (26)–(30), these terms for the first-order theory specified by equation (58) are

The detailed derivations of equations (77)–(80) are shown in appendix B.The velocityX.a,as a function of(Xa(t),Ua(t)),is determined by the equation of motion of theath particle[17],which can be obtained by the EL equation (13).Substituting equations (75)–(80) into (31), we obtain the local energy conservation law

where

In this case, the leading order weak EL current is

which is the canonical energy flux of particles.Here, p1aand m1ain equation (84) are first-order polarization and magnetization for theath particle, and pa1and ma1 are obviously gauge-invariant.

Because electromagnetic field in the field theory is represented by the 4-potential(φ,A), the conservation laws depends on gauge explicitly.To remove the explicit gauge dependency from the Noether procedure, we can add the identity to equation (81), and rewrite the two terms on the left-hand side of equation (86) as follows

The details of the derivation of equations(87)and(88)can be found in [44].The resulting energy conservation is

In equations (89),is drift velocity of the guiding-center, and it is a function of(Xa(t),Ua(t))determined by the EL equation(13).The detailed expression ofcan be found in[17].

Following the procedure in section 2.3,equation(89)can be expressed in terms of the Klimontovich distribution functionFs(t,x,u)and the electromagnetic field

where

are the zeroth-order polarization and magnetization for particles of thes-species.The polarizationP1and magnetizationM1are contained in the first-order terms of equation (90).In the limit of guiding-center drift kinetics, the first-order terms in equation (90) are neglected, and we have

In the limit of guiding-center drift kinetics, if the ×E BtermDinLais also ignored, namely

then the polarization vector fieldP0and magnetization vector fieldM0reduce to

Thus, the energy conservation law is further reduced to

which, in terms of the distribution function and the electromagnetic field, is

Equation (97) agrees with the result of Brizardet al[48] for guiding-center drift kinetics.Note that before the present study, local energy conservation law was not known for the high-order electromagnetic gyrokinetic systems.Our local energy conservation law for the electromagnetic gyrokinetic systems (90) and(98) recover the previous known results for the first-order guiding-center Vlasov–Maxwell system and the drift kinetic system as special cases.

The above derivation of local energy conservation law is for the first-order theory specified by equation (58).For the general electromagnetic gyrokinetic system of arbitrary highorder specified by equation (38), an exact, gauge-invariant,local energy conservation law can be derived using the same method.It is listed here without detailed derivation

where

3.3.Space translation symmetry and momentum conservation law

We now discuss the space translation symmetry and momentum conservation.It is straightforward to verify that the action of the gyrokinetic system specified by equation(38)is unchanged under the space translation

wherehis an arbitrary constant vector.Note that this symmetry group transforms bothxandXa.

It is worthwhile to emphasize again that in order for the system to admit spacetime translation symmetry and thus local energy–momentum conservation laws, we do not separate the electromagnetic field into background and perturbed components.This is different from other existing studies in gyrokinetic theory,which separate the background magnetic field from the perturbed magnetic field, and as a result no momentum conservation law can be established in these studies for the plasma dynamics in tokamaks or devices with inhomogeneous background magnetic fields.

The infinitesimal generator corresponding to equation (103) is

Becauseξ= 0t,ξ θ= =haand(see equations (20)–(23)), the prolongation ofvis the same asv,

The infinitesimal criterion (19) is then satisfied since

where used is made of the fact that?δa?x= ??δa?Xa.The characteristics of the infinitesimal generator (104) are

The infinitesimal criterion (105) thus implies a conservation law becauseqais a constant vector field independent ofx.

We now calculate each term in equation(31)for the firstorder theory specified by equation (58) to obtain the conservation law.Using the definitions ofPυaandPFv(see equations (26)–(30)), the most complicated termsin the conservation law can be explicitly written as

The detailed derivations of equations (107)–(110) are shown in appendix B.Substituting equations (107)–(110) into (31),we obtain the momentum conservation laws as

where

In this situation, the leading order weak EL current is

which is the canonical momentum flux of particles along the directionh.Akin to the situation of equation (81) in section 3.2,equation(111)is gauge dependent.We can add in the following identity

to remove the explicit gauge dependency (see [44]).The two terms in equation (115) can be rewritten as

Details of the derivation are shown in [44].Substituting equations (115)–(117) into (111), we obtain

where used is made of the following equations

Here, the drift velocity .Xaof the guiding-center in equation (118) is determined by the EL equation (13), which is regarded as a function of(Xa(t),Ua(t)).Using the procedure in section 2.3 the momentum conservation can be expressed in terms of the the Klimontovich distribution functionFs(t,x,u)and the electromagnetic field

For the special case of guiding-center drift kinetics, the first-order Lagrangian densityLa1 is neglected, and we have

In the limit of guiding-center drift kinetics, if the ×E BtermDinLais also ignored (see equation (94)), then the momentum conservation is further reduced to

Substituting the polarization vector fieldPand magnetization vector fieldMof the drift kinetic system (see equation (95))into (123), we have

In terms of the distribution functionFs(t,x,u)and the electromagnetic field(E(t,x) ,B(t,x)), equation (124) is

Equation (125), as a special case of the gyrokinetic momentum conservation law (121), is consistent with the result shown by Brizardet al[48] for the drift kinetics.

This completes our derivation and discussion of the momentum conservation law for the first-order theory.

For the general electromagnetic gyrokinetic system defined by equation (38), the following exact, gauge-invariant, local momentum conservation law can be derived using a similar method

where pδ sand mδ sare defined in equations(99)and(100),and

4.Conclusion

We have established the exact, gauge-invariant, local energy–momentum conservation laws for the electromagnetic gyrokinetic system from the underpinning spacetime translation symmetries of the system.Because the gyrocenter and electromagnetic field are defined on different manifolds, the standard Noether procedure for deriving conservation laws from symmetries does not apply to the gyrokinetic system without modification.

To establish the connection between energy–momentum conservation and spacetime translation symmetry for the electromagnetic gyrokinetic system,we first extended the field theory for classical particle-field system on heterogeneous manifolds [37–39] to include high-order field derivatives and using non-canonical phase space coordinates in a general setting without specializing to the gyrokinetic system.The field theory on heterogeneous manifolds embraces the fact that for classical particle-field systems, particles and fields reside on different manifolds,and a weak EL equation was developed to replace the standard EL equation for particles.The weak EL current,induced by the weak EL equation, is the new physics associated with the field theory on heterogeneous manifolds,and it plays a crucial role in the connection between symmetries and conservation laws when different components of the system are defined on different manifolds.

The high-order field theory on heterogeneous manifolds developed was then applied to the electromagnetic gyrokinetic system to derive the exact, local energy–momentum conservation laws from the spacetime translation symmetries admitted by the Lagrangian density of the system.And,finally, the recently developed gauge-symmetrization procedure [44] using the electromagnetic displacement-potential tensor was applied to render the conservation laws electromagnetic gauge-invariant.

Acknowledgments

P Fan was supported by the Chinese Scholarship Council(CSC) (No.201806340074), Shenzhen Clean Energy Research Institute and National Natural Science Foundation of China (No.12005141).H Qin was supported by the US Department of Energy (No.DE-AC02-09CH11466).J Xiao was supported by the National MC Energy R&D Program(No.2018YFE0304100), National Key Research and Development Program(Nos.2016YFA0400600,2016YFA0400601 and 2016YFA0400602), and the National Natural Science Foundation of China (Nos.11905220 and 11805273).

Appendix A.Derivations of polarization and magnetization in equations (67) and (70)

In this appendix, we give the derivations of zeroth-order polarizationP0and magnetizationM0.From the definition ofP0,M0and Lagrangian density of theath particle (see equations(67),(70),(58)and(5)),they are derived as follows

and

In obtaining equations(A1)and(A2),the following equations were used

Appendix B.Derivations of equations (77)–(80) and(107)–(110)

In this appendix, we show the detailed derivations of equations (77)–(80) and (107)–(110), which are boundary terms induced by time and space translation symmetries.For the time translation symmetry,using equations(24)and(26)–(30), equations (77)–(80) can be proved as follows

Similarly, for the space translation symmetry, using the definitions ofPυaand PFv(see equations (26)–(30)),equations (107)–(110) are demonstrated as follows

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