Variational Principle for Electrodynamics ofMoving Particles

Jerzy KijowskiCentrum Fizyki Teoretycznej PAN

Aleja Lotnikow 32/46, 02-668 Warsaw, Polandand

Dariusz ChruscinskiInstitute of Physics, Nicholas Copernicus University

ul. Grudziadzka 5/7, 87-100 Torun, Poland

Abstract

Consistent relativistic theory of the classical Maxwell field interacting with clas-sical, charged, point–like particles, proposed in [1], is now derived from a variationalprinciple. For this purpose a new electrodynamical Lagrangian based on fluxes isconstructed. As a result, we obtain the action principle where 1) field degrees offreedom and particle degrees of freedom are kept at the same footing, 2) contraryto the standard formulation, no infinities arise, 3) energy (Hamiltonian) is obtainedfrom the Lagrangian via the Legendre transformation, without any need of “addinga complete divergence”.

PACS: 03.50.De; 04.20.Fy; 41.10.-j; 41.70.+t

Contents

1 Introduction 2

2 Electrodynamics of moving particles: statement of results 5

3 Co-moving description of a relativistic field theory 7

4 Point particles and extended particles. Renormalization 12

5 Equations of motion from the variational principle 16

6 Variational principle based on fluxes 18

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7 The Lagrangian in the co-moving frame 25

8 Particle in an external potential 27

Appendixes 28

A Hamiltonian structure for a 2-nd order Lagrangian theory 28

B Maxwell equations in the co-moving frame 29

C Boundary momenta 32

D Proof of the conservation laws 35

1 Introduction

Recently, one of us (J.K) proposed a consistent relativistic theory of the classical Maxwellfield interacting with classical, charged, point-like particles (cf. [1]). For this purposean “already renormalized” formula for the total four-momentum of a system composedof both the moving particles and the surrounding electromagnetic field was proposed. Itwas proved, that the conservation of the total four-momentum defined by this formulais equivalent to a certain boundary condition for the behaviour of the Maxwell field inthe vicinity of the particle trajectories (in [1] this condition is called the fundamentalequation).

Field equations of such a theory are, therefore, precisely the linear, inhomogeneousMaxwell equations for the electromagnetic field surrounding the point-like sources. Thenew element introduced in [1], which completes this standard theory, is the above boundarycondition, with the particle trajectories playing the role of the moving boundary. Togetherwith this condition, the theory (called electrodynamics of moving particles) becomes causaland complete: initial data for both the field and the particles uniquely imply the evolutionof the system. This means e. g. that the particle trajectories may also be calculateduniquely from the initial data.

It was proved that the limit of this theory for e → 0 and m → 0 with their ratiobeing fixed, coincides with the Maxwell–Lorentz theory of test particles moving on thebackground described by the free field. However, for any finite value of e, the accelerationof the particle can not be equal to the Lorentz force, the latter being always ill defined,because of the field singularities implied by Maxwell equations.

Physically, the “already renormalized” formula for the total four-momentum (formula(33) in the present paper) was suggested by a suitable approximation procedure appliedto an extended-particle model. In such a model we suppose that the particle is a stable,soliton-like solution of a hypothetical fundamental theory of interacting electromagnetic

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and matter fields. Assuming, that for weak electromagnetic fields and vanishing matterfields (i. e. outside of the particles) the theory coincides with linear Maxwell electrodynam-ics, a formula was found, which gives in a good approximation the total four-momentumof a system composed of both the moving particles and the surrounding electromagneticfield. The formula uses only the “mechanical” information about the particle (position,velocity, mass m and the electric charge e) and the free electromagnetic field outside ofthe particle. It turns out, that the formula is meaningful also in the case of point particles.Hence, it can be taken as a starting point for a mathematically self-consistent theory ofpoint-like particles interacting with the linear Maxwell field.

In the present paper we give the variational formulation of the above theory. Thecorresponding, highly nontrivial canonical (Hamiltonian) structure, will be given in thenext paper.

The standard variational principle used in electrodynamics cannot be extended to thetheory containing also point-like particles interacting with the electromagnetic field. Sucha principle is based on the following Lagrangian, written usually in textbooks (see e. g.[2], [4]):

Ltotal = LMaxwell + Lparticle + Lint , (1)

with

LMaxwell = −1

4

√−gfµνfµν , (2)

Lparticle := −mδζ , (3)

and the interaction term given by

Lint := eAµuµδζ . (4)

Here by δζ we denote the Dirac delta distribution localized on the particle trajectory ζ.The above Lagrangian may be used to derive the trajectories of the test particles, whenthe field is given a priori. In a different context, it may also be used to derive Maxwellequations, if the particle trajectories are given a priori. Simultaneous variation withrespect to both fields and particles leads, however, to a contradiction, since the Lorentzforce will be always ill defined due to Maxwell equations.

But already in the context of the inhomogeneous Maxwell theory with given point-like sources, the variational principle based on Lagrangian (1) is of very limited use, sincethe interaction term Lint becomes infinite. As a consequence, the hamiltonian of such atheory will always be ill defined, although the theory displays a perfectly causal behaviour.

The main result of the present paper consists in removing this difficulty. Applyingan appropriate Legendre transformation to the field Lagrangian LMaxwell, we obtain anew, quasi-local variational principle for the Maxwell field. Already in the context of theinhomogeneous Maxwell theory with given (i. e. non-dynamical) point-like sources, ourLagrangian produces no infinities and enables us to describe the dynamics of the fieldinfluenced by moving particles as an infinite-dimensional Hamiltonian system. It turns

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out, that adding the particle Lagrangian (3) and varying it with respect to both fields andparticles is now possible and does not lead to any contradiction. As a result, we obtainprecisely the “electrodynamics of moving particles” proposed in [1].

It is not unusual that the same physical theory is described by different variationalprinciples. We derive our new variational principle, transforming the standard MaxwellLagrangian by an appropriately chosen Legendre transformation. Each of these two vari-ational principles is related to a specific way of controlling the boundary data of the field.Changing the variational principle means changing the physical quantities, which are keptfixed at the boundary during the variation. In the standard approach, the variation isperformed with values of the electromagnetic potentials Aµ being kept on the boundary.In our approach, we keep at the boundary the value of the electric and the magneticfluxes.

Passing from the lagrangian description to the hamiltonian one, different Lagrangianslead to different field Hamiltonians, describing the field dynamics with different boundaryconditions. It is worthwhile to notice that the Hamiltonian obtained from our Lagrangianis equal to the field energy: 1

2(D2 + B2). Other Hamiltonians, related to other boundary

conditions, which may be obtained from other Lagrangians (e. g. the standard one (2)),are not positive and even not bounded from below (see [8]).

The relation between different Hamiltonians, corresponding to different boundary con-ditions, is similar to the relation between the internal energy and the free (Helmholtz)energy in thermodynamics. The first one describes the evolution of the thermodynamicsystem, when insulated adiabatically from any external influence, whereas the latter de-scribes the (completely different) evolution of the same system, when put into a thermalbath. From this point of view, controlling the electric and the magnetic fluxes on theboundary of a 3-dimensional volume V means insulating it adiabatically from any exter-nal influence, whereas the standard control of potentials still leaves the possibility of theenergy exchange between the exterior and the interior of V .

The paper is organized as follows.Section 2 contains the main results of the theory proposed in [1].In Section 3 we develop a new technique, which enables us to describe at the same

footing the field and the particle degrees of freedom. For this purpose we formulate anyrelativistic, hyperbolic field theory with respect to a non-inertial reference frame definedas a rest-frame for an arbitrarily moving observer. Such a formulation will be used as astarting point for our renormalization procedure.

In Section 4 we show how to extend the above approach to the case of electrodynamicalfield interacting with point particles. This enables us to derive in subsequent Sections theElectrodynamics of Moving Particles from a variational principle.

Finally, in Section 8 we present the lagrangian formulation for the particle interactingnot only with the radiation field, but also with a fixed, external potential, produced bya heavy external device. This is a straightforward extension of our theory, where thehomogeneous boundary condition for the radiation field is replaced by an inhomogeneouscondition, the inhomogeneity being provided by the external field.

The Appendixes contain mainly calculations. However, in Appendix A we present

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the hamiltonian formulation of theories arising from second order Lagrangians. Thisformulation is rather difficult to find elsewhere, although it surely belongs to the “folklore”of classical mechanics.

In the present paper we do not prove the existence and the uniqueness of the solution ofequations derived here. Such a proof may be obtained using two different methods. Thefirst one consists in analyzing directly the boundary value problem for linear Maxwellequations in a co-moving reference frame, which we describe in Section 3. The othermethod consists in expressing the electromagnetic field explicitly in terms of the initialdata and the trajectory. This way we end up with an ordinary differential equation forthe trajectory, where the field initial data play the role of parameters given a priori. Bothproofs are relatively simple and will be published elsewhere.

2 Electrodynamics of moving particles: statement of

results

In the present Section we briefly sketch the electrodynamics of moving particles, presentedin [1].

Let y = q(t) with t = y0, be the coordinate description of a time-like world line ζ ofa charged particle with respect to a laboratory frame, i.e. to a system (yµ), µ = 0, 1, 2, 3;of Lorentzian space-time coordinates.

The theory contains as a main part the standard Maxwell equations with point-likesources:

∂[λfµν] = 0 ,

∂µfµν = euνδζ , (5)

where uν stands for the particle four-velocity and δζ denotes the δ-distribution concen-trated on the smooth world line ζ:

δζ(y0, yk) =

√1− (v(y0))2 δ(3)(yk − qk(y0)) . (6)

Here v = (vk) = (qk) is the corresponding 3-velocity and v2 denotes the square of its3-dimensional length (we use the Heaviside-Lorentz system of units with c = 1). In thecase of many particles the total current is a sum of contributions corresponding to manydisjoint world lines and the value of charge is assigned to each world line separately.

For a given particle trajectory, equations (5) define a deterministic theory. This meansthat initial data for the electromagnetic field uniquely determine its evolution. However,if we want to treat also the particle initial data (q,v) as dynamical variables, the theorybased on the Maxwell equations alone is no longer deterministic: the particle trajectorycan be arbitrarily modified in the future or in the past without changing the initial data.

This non-completeness of the theory is usually attributed to the fact that the parti-cle’s equations of motion are still missing. We stress, that such an interpretation is false.

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Indeed, it was proved in [1] that the field initial data fully determine the particle’s accel-eration and this is due to Maxwell equations only, without postulating any equations ofmotion. More precisely, there is a one-to-one correspondence between the (r−1)-term ofthe field in the vicinity of the particle and the acceleration of the particle. The easiest wayto describe this property of Maxwell theory is to use the particle’s rest-frame. For this pur-pose consider the 3-dimensional hyperplane Σt orthogonal to ζ at the point (t,q(t)) ∈ ζ.We shall call Σt the “rest frame hyperplane”. Choose on Σt any system (xi) of cartesiancoordinates centered at the particle’s position and denote by r the corresponding radialcoordinate. The initial data for the field on Σt are given by the electric induction fieldD = (Di) and the magnetic induction field B = (Bi) fulfilling the conditions div B = 0

and div D = e δ(3)0 . Maxwell equations can be solved for arbitrary data, fulfilling the

above constraints, but the solution will be usually non-regular, even far away from theparticles. To avoid singularities propagating over a light cone from (t,q(t)), the singularpart of the data in the vicinity of the particle has to be equal to

Dk =e

4π

[xk

r3− 1

2r

(ai

xixk

r2+ ak

)]+ O(1) , (7)

where a = (ak) is the acceleration of the particle (in the rest frame we have a0 = 0) andO(1) denotes the nonsingular part of the field (the magnetic field Bk(r) cannot containany singular part). This gives the one-to-one correspondence between the (r−1)-term ofthe field and the particle’s acceleration, which is implied by the regularity of the fieldoutside of the trajectory ζ.

Hence, for regular solutions, the time derivatives (D, B, q, v) of the Cauchy data(D,B,q,v) of the composed (fields + particles) system are uniquely determined by thedata themselves. Indeed, D and B are given by the Maxwell equations, q = v and v maybe uniquely calculated from equation (7). Nevertheless, the theory is not complete and itsevolution is not determined by the initial data. This non-completeness may be interpretedas follows. Field evolution takes place not in the entire Minkowski space M , but onlyoutside the particle, i. e. in a manifold with a non-trivial boundary Mζ := M − {ζ}. Theboundary conditions for the field are still missing!

To find this missing condition, the following method was used. Guided by an extendedparticle model, an “already renormalized” formula was proposed in [1], which assigns toeach point (t, qk(t)) of the trajectory a four-vector pλ(t). It is interpreted as the total four-momentum of the physical system composed of both the particle and the field (see Section4 for details). For a generic trajectory ζ and a generic solution of Maxwell equations (5)this quantity is not conserved, i. e. it depends upon t. The conservation law

d

dtpλ(t) = 0 (8)

is proposed as an additional equation, which completes the theory. It was shown that,due to Maxwell equations, only 3 among the 4 equations (8) are independent. Given alaboratory reference frame, we may take e. g. the conservation of the momentum p = (pk):

d

dtp = 0 (9)

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as independent equations. They already imply the energy conservation

d

dtp0(t) = 0 . (10)

In Section (5) we will show, that the above momentum p may be obtained from thevariational principle as the momentum canonically conjugate to the position of the particle,whereas p0 is equal to the total Hamiltonian of the composed (particle + field) system.

The four-momentum pλ(t) was defined in terms of an integral over any hypersurface Σ,which intersects the trajectory at the point (t, qk(t)). It has been proved in [1] that, due toMaxwell equations, the integral (global) condition (9) is equivalent to a (local) boundarycondition for the behaviour of the Maxwell field in the vicinity of the trajectory. Thecondition was called the fundamental equation of the electrodynamics of moving particles.In particle’s reference frame it may be formulated as a relation between the (r−1) and the(r0) terms in the expansion of the radial component of the electric field in the vicinity ofthe particle:

Dr(r) =1

4π

(e

r2+

α

r

)+ β + O(r) , (11)

where by O(r) we denote terms vanishing for r → 0 like r or faster. For a given value ofr both sides of (11) are functions of the angles (only the r−2 term is angle–independent).The relation between the acceleration and the (r−1) – term of the electric field given in(7) may be rewritten in terms of the component α of this expansion:

α = −eaixi

r(12)

(it implies that the quadrupole and the higher harmonics of α must vanish for regularsolutions). In this notation the fundamental equation (equivalent to the conservation law(8)) reads:

DP(mα + e2β) = 0 , (13)

where by DP(f) we denote the dipole part of the function f on the sphere S2. Togetherwith this condition, Maxwell theory becomes complete, causal and fully deterministic:initial data for particles and fields uniquely determine the entire history of the system.

3 Co-moving description of a relativistic field theory

To construct the variational formulation of the above theory, we will need a descriptionof electrodynamics with respect to the particle’s rest-frame. This is not an inertial frame.In the present section we show how to extend the standard variational formulation of fieldtheory to the case of non-inertial frames.

Consider any field theory based on a first-order relativistically-invariant Lagrangiandensity

L = L(ψ, ∂ψ) , (14)

7

where ψ is a (possibly multi-index) field variable, which we do not need to specify at themoment. As an example, ψ could denote a scalar, a spinor or a tensor field.

We will describe the above field theory with respect to accelerated reference frames,related with observers moving along arbitrary space-time trajectories. Let ζ be such a(time-like) trajectory, describing the motion of our observer. Let y = q(t), or yk = qk(t),k = 1, 2, 3; be the description of ζ with respect to a laboratory reference frame, i. e. to asystem (yλ), λ = 0, 1, 2, 3; of Lorentzian space-time coordinates.

We will construct an accelerated reference frame, co-moving with ζ. For this purposelet us consider at each point (t,q(t)) ∈ ζ the 3-dimensional hyperplane Σt orthogonal toζ, i.e. orthogonal to the four-velocity vector U(t) = (uµ(t)):

(uµ) = (u0, uk) :=1√

1− v2(1, vk) , (15)

where vk := qk. We shall call Σt the “rest frame surface”. Choose on Σt any system (xi)of cartesian coordinates, such that the particle is located at its origin (i. e. at the pointxi = 0).

Let us consider space-time as a disjoint sum of rest frame surfaces Σt, each of themcorresponding to a specific value of the coordinate x0 := t and parametrized by thecoordinates (xi). This way we obtain a system (xα) = (x0, xk) of “co-moving” coordinatesin a neighbourhood of ζ. Unfortunately, it is not always a global system because differentΣ’s may intersect. Nevertheless, we will use it globally to describe the evolution of thefield ψ from one Σt to another. For a hyperbolic field theory, initial data on one Σt

imply the entire field evolution. We are allowed, therefore, to describe this evolution as a1-parameter family of field initial data over subsequent Σ’s.

Formally, we will proceed as follows. We consider an abstract space-time M := T ×Σdefined as the product of an abstract time axis T = R1 with an abstract, three dimensionalEuclidean space Σ = R3. Given a world-line ζ, we will need an identification of points ofM with points of physical space-time M . Such an identification is not unique because oneach Σt we have still the freedom of an O(3)-rotation.

Suppose, therefore, that an identification F has been chosen, which is local with respectto the observer’s trajectory. By locality we mean that, given the position and the velocityof the observer at the time t, the isometry

F(q(t),v(t)) : Σ 7→ Σt (16)

is already defined, which maps 0 ∈ Σ into the particle position (t,q(t)) ∈ Σt.As an example of such an isometry which is local with respect to the trajectory we

could take the one obtained as follows. Choose the unique boost transformation relatingthe laboratory time axis ∂/∂y0 with the observer’s proper time axis U . Next, define theposition of the ∂/∂xk - axis on Σt by transforming the corresponding ∂/∂yk – axis of thelaboratory frame by the same boost. It is easy to check, that the resulting formula for Freads:

y0(t, xl) := t +1√

1− v2(t)xlvl(t) ,

8

yk(t, xl) := qk(t) +(δkl + ϕ(v2)vkvl

)xl . (17)

Here, the following function of a real variable has been used:

ϕ(τ) :=1

τ

(1√

1− τ− 1

)=

1√1− τ(1 +

√1− τ)

. (18)

The function is well defined and regular (even analytic) for v2 = τ < 1. The operator(δk

l + ϕ(v2)vkvl) acting on rest-frame variables xl comes from the boost transformation.Suppose, therefore, that for a given trajectory ζ a local isometry (16) has been chosen,

which defines Fζ : M 7→ M . This mapping is usually not invertible: different pointsof M may correspond to the same point of space-time M because different Σt’s mayintersect. It enables us, however, to define the metric tensor on M as the pull-backF ∗

ζ g of the Minkowski metric. The components gαβ of the above metric are defined bythe derivatives of Fζ , i. e. they depend upon the first and the second derivatives of theposition q(t) of our observer.

Because (xk) are cartesian coordinates on Σ, the space-space components of g aretrivial: gij = δij. The only non-trivial components of g are, therefore, the lapse functionand the (purely rotational) shift vector:

N =1√−g00

=√

1− v2 (1 + aixi) ,

Nm = g0m =√

1− v2 εmklωkxl , (19)

where ai is the observer’s acceleration vector in the co-moving frame. The quantity ωm

is a rotation, which depends upon the coordination of isometries (16) between differentΣt’s. Because ωm depends locally upon the trajectory, it may also be calculated in termsof the velocity and the acceleration of the observer, once the identification (16) has beenchosen. In the case of example (17), it is easy to check that

ai =1

1− v2

(δik + ϕ(v2)vivk

)vk , (20)

ωm =1√

1− v2ϕ(v2)vkvlεklm , (21)

where vk is the observer’s acceleration in the laboratory frame.The metric F ∗

ζ g is degenerate at singular points of the identification map (i.e. where theidentification is locally non-invertible because adjacent Σ’s intersect, i. e. where N = 0),but this degeneration does not produce any difficulties in what follows.

The simplest O(3)-coordination of the isometries (16) would consist in Fermi-propa-gating the xk – axis along ζ, i. e. in putting ωm ≡ 0. Such a coordination is, however,non-local with respect to the trajectory. Indeed, the identification Ft between Σ and Σt

would be, in this case, a result of the Fermi – propagation of a given mapping Fto fromthe initial time t0 to the actual time t. Such a mapping cannot be described by a local

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formula (16). We stress, however, that for our construction we do not need to specify anycoordination F , provided it is local.

Once we know the metric (19) on M, we may rewrite the invariant Lagrangian densityL of the field theory under consideration, just as in any other curvilinear system ofcoordinates. The Lagrangian obtained this way depends on the field ψ, its first derivatives,but also on the observer’s position, velocity and acceleration. Variation with respect to ψproduces field equations in the co-moving system (xα). Due to the relativistic invarianceof the theory, variation of the Lagrangian with respect to the observer’s position q shouldnot produce independent equations but only conservation laws, implied already by thefield equations.

For our purposes we will keep, however, at the same footing the field degrees of freedomψ and (at the moment, physically irrelevant) observer’s degrees of freedom qk. For such aLagrangian theory, we perform a partial Legendre transformation, and pass to the Hamil-tonian description of the field degrees of freedom, keeping the Lagrangian description ofthe “mechanical” degrees of freedom. For this purpose we define

LH := L− Πψ , (22)

where Π is the momentum canonically conjugate to ψ:

Π :=∂L

∂ψ. (23)

The function LH plays the role of a Hamiltonian (with negative sign) for the fields anda Lagrangian for the observer’s position q. It is an analog of the Routhian function inanalytical mechanics. For a mechanical system with n degrees of freedom the Routhianfunction obtained from the Lagrangian by a partial Legendre transformation

R = R(q1, q2, ..., qn; q1, ..., ql, pl+1, ..., pn) = L−n∑

k=l+1

pkqk (24)

generates the following symplectic relation

dR =l∑

k=1

(pkdqk + pkdqk) +n∑

k=l+1

(−pkdqk + qkdpk) . (25)

This means that R is a Lagrangian in variables (q1, ..., ql; q1, ..., ql) and a Hamiltonian(with negative sign) in variables (ql+1, ..., qn; pl+1, ..., pn). We can choose the lagrangianor the hamiltonian mode for any degree of freedom in a completely independent way.For l = 0, R becomes the complete Hamiltonian and for l = n it becomes the completeLagrangian.

In case of formula (22), the Lagrango-Hamiltonian LH generates the hamiltonian fieldevolution with respect to the accelerated frame, when the “mechanical degrees of free-dom” qk are fixed. Due to (19), this evolution is a superposition of the following threetransformations:

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• time-translation in the direction of the local time-axis of the observer,

• boost in the direction of the acceleration ak of the observer,

• purely spatial O(3)-rotation ωm.

It is, therefore, obvious that the numerical value of the generator LH of such an evolutionis equal to

LH = −√

1− v2(H + akRk − ωmSm

), (26)

where H is the rest-frame field energy, Rk is the rest-frame static moment and Sm is therest-frame angular momentum, all of them calculated on Σ. The factor

√1− v2 in front

of the generator is necessary, because the time t = x0, which we used to parameterize theobserver’s trajectory, is not the proper time along ζ but the laboratory time.

Now, it is easy to prove that the Euler-Lagrange equations obtained when varying LH

with respect to the observer’s position q(t) are satisfied identically if the field equationsare satisfied. The proof follows directly from the conservation laws of the total four-mo-mentum Pα and the total angular momentum Mαβ of the field, implied by Noether’stheorem. Indeed, the four-momentum conservation reads:

∇0Pα = Pα + Γα0βPβ = 0 , (27)

where Γαβγ are the Christoffel symbols of the metric gαβ, calculated on the trajectory, i.e.

at xk = 0. Putting P0 = H and calculating Γ from (19), one immediately obtains thefollowing “accelerated-frame version” of Noether conservation laws:

H = −√

1− v2 akPk , (28)

Pk =√

1− v2(−akH− ε ml

k ωmPl

). (29)

Putting Mk0 = Rk and Mij = εijkSk, the angular momentum conservation ∇0Mαβ = 0may be rewritten in a similar same way:

Rk =√

1− v2(Pk − εkimaiSm − εkilω

iRl)

, (30)

Sm =√

1− v2(εmilaiRl − εmijωiSj

). (31)

The above conservation laws are implied by field equations only. It is a matter of simplecalculations (see Section 5), that the Euler-Lagrange equations obtained by varying (26)with respect to the observer’s position q(t) are satisfied identically if equations (28) – (31)are satisfied. To perform such a calculation, an explicit formula for the mapping Fζ isnecessary, which implies an explicit formula for ak and ωk in terms of (v, v). In Section 5we use for this purpose formulae (20) and (21), corresponding to the embedding (17).

The fact, that the Euler-Lagrange equations of the theory are not independent, istypical for a gauge theory. This property may be nicely described in the hamiltonianpicture. Considering LH as the generator of the evolution of both the field degrees of free-dom and the observer’s degrees of freedom, we may perform the Legendre transformation

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also with respect to the latter, and find this way the complete Hamiltonian of the entire(observer + field) system (see again Section 5 for details). It may be proved, that q playsthe role of a gauge parameter: momenta canonically conjugate to observer’s position arenot independent but subject to constraints. Reducing the theory with respect to theseconstraints we end up with the “true” degrees of freedom, namely those describing thefield. Fixing the trajectory plays the role of “gauge fixing” and the “evolution equations”of the observer are automatically satisfied if the field equations are satisfied.

The formalism introduced in this Section cannot be a priori used for the descriptionof the electromagnetic field, since the naive Legendre transformation (22) from the La-grangian to the Hamiltonian picture does not lead to the correct local expression for thefield energy (we obtain the “canonical Hamiltonian” which differs from the field energy bya complete divergence). However, we may take the electrodynamical Routhian (Lagrango- Hamiltonian) (26), where H, Rk and Sm are the conventional energy, static momentand the angular momentum of the electromagnetic field. They are defined as appropriateintegrals of the components of the Maxwell energy-momentum tensor

T µν = fµλfνλ − 1

4δµν fκλfκλ . (32)

Of course, Maxwell equations derived from such an LH imply conservation laws (28) -(31). Hence, variation of LH with respect to the observers position q(t) produces Euler-Lagrange equations which are automatically satisfied by virtue of the field equations.

We will show in the sequel, that the renormalized version of the above Lagrangian(26) really implies the Electrodynamics of Moving Bodies. For this purpose no furtherjustification of LH is necessary. However, in Section 6, we will construct a new Lagrangianfor electrodynamics, which is directly related with LH via the simple Legendre transfor-mation (22). This way, finally, the entire content of the present Section may be appliedto electrodynamics.

4 Point particles and extended particles. Renormal-

ization

In this Section we briefly sketch the renormalization procedure proposed in [1]. It enablesus to extend the definition of “Lagrango-Hamiltonian” (26) to the case of electromagneticfield interacting with point-like particles.

Given a solution of inhomogeneous Maxwell equations (5), we define the total energy-momentum of the composed (particle + field) system by the following “already renormal-ized” formula:

pλ(t) := muλ(t) + P∫

Σ

(T µ

λ −T(t)µλ

)nµdΣ , (33)

where Σ is any hypersurface which intersects the trajectory at the point (t,q(t)), andT(t)

µν

is the energy-momentum tensor of the “uniformly moving particle”, i. e. of theCoulomb field, boosted in such a way that the position and the velocity of its singularity

12

match the velocity of our particle at (t,q(t)). By “P” we denote the principal value ofthe singular integral, defined by removing from Σ a sphere K(0, r) around the particleand then passing to the limit r → 0. We assume that Σ fulfills standard asymptoticconditions at infinity (this means that all the surface integrals at infinity, appearing inthe energy-momentum conservation laws, vanish – see [7]). The parameter m denotes thedressed (i. e. already renormalized) mass of the particle.

It was proved in [1] that the above integral is well defined. Moreover, it is invariantwith respect to changes of Σ, provided the intersection point with the trajectory does notchange. Hence, the total four-momentum of the composed (particle + field) system maybe defined this way, at each point (t,q(t)) of the trajectory separately.

Similarly, we define the total angular-momentum Mµν of the composed (field + par-ticle) system as the sum composed of 1) an integral containing appropriate componentsof T and 2) an integral containing the difference T − T. The renormalization consistsin replacing the first integral by the particle’s angular-momentum. In the present paperwe assume the particle to be of scalar type, i.e. that its angular momentum vanishes (ageneralization to the case of particles with non-vanishing internal angular-momentum isrelatively straightforward and will be given in the next paper).

The above definition is a result of the following physical idea, concerning the particle’sstructure. Consider a general field theory describing the electromagnetic field interactingwith a hypothetical multi-component matter field φ = (φK). We assume that the dynam-ical equations of this “super theory” may be derived from a gauge-invariant variationalprinciple

δL

δA= 0 (34)

δL

δφ= 0 , (35)

where A = (Aµ) is the electromagnetic potential, i.e. fµν := ∂µAν−∂νAµ. As an exampleone may consider the complex (charged) scalar field or the classical spinorial Dirac field,interacting with electromagnetism. For our purposes, however, no further assumptionsabout the geometric character of the field φ are necessary. Moreover, we assume Maxwellequations as a limiting case of the above field equations, corresponding to sufficientlyweak electromagnetic fields and vanishing matter fields.

We will suppose that the particles, whose interaction with the electromagnetic field weare going to analyze, are simply global solutions of the above field theory. Each solutionof this type is characterized by a tiny “strong field region”, concentrated in the vicinityof a time-like trajectory ζ, which we may call approximate trajectory of an extendedparticle. Outside of the strong field region the matter fields vanish (or almost vanish inthe sense, that the following approximation remains valid) and the electromagnetic fieldis sufficiently weak to be described by Maxwell equations.

To be more precise, we imagine the “particle at rest” as a stable, static, soliton-like solution of our hypothetical “super theory”. The solution is characterized by two

13

parameters: its total charge e and its total energy m. We stress that m is an alreadyrenormalized mass, (or dressed mass), including the energy of the field surrounding theparticle. Within this framework questions like “how big the bare mass of the particleis and which part of the mass is provided by the purely electromagnetic energy?” aremeaningless. In the strong field region (i. e. inside the particle) the energy density maybe highly non-linear and there is probably no way to divide it consistently into such twocomponents.

Due to relativistic invariance of the theory, there is a 6 parameter family of the “uni-formly moving particle” solutions obtained from our soliton via Poincare transformations.

An arbitrarily moving particle is understood as a “perturbed soliton”. This meansthat it is again an exact solution of the same “super theory”, with its strong-field-regionconcentrated in the vicinity of a time-like world line ζ, which is no longer a straight line,as it was for “uniformly moving particles”.

Let us calculate the total four-momentum pλ of such a solution. For this purpose wechoose any Σ and integrate the total energy-momentum tensor T µ

λ of our “super theory”over Σ. It is usefull to decompose T µ

λ:

T µλ = Tµ

λ + (T µλ −Tµ

λ) . (36)

Here, by Tµλ we denote the total energy-momentum of the “super theory”, corresponding

to the “uniformly moving particle” solution, which matches on Σ the position and thevelocity of our particle.

Integrating the first term we obtain obviously muµ, where uµ is the four-velocity ofour particle on Σ. Consequently, pλ may be decomposed as a sum of two terms: 1) thetotal four-momentum of the uniformly moving particle (and its surrounding field) and 2)the difference between the two, given by the integral of the last term in (36) over Σ.

The exact value of this integral cannot be found without knowing exactly the internalstructure of both solutions. But it is easy to find a good approximation, which is basedon the following observation: the contribution of the interior of the particles is smallbecause of the stability of the soliton. Indeed, stability means that the soliton is a localminimum of energy in the space of the field initial data. Hence, the variation of T µ

λ insidethe particle (corresponding to the perturbation of the soliton) is small. This means thatthe purely electromagnetic contribution, corresponding to the surrounding Maxwell fields,approximates with a good accuracy the above quantity.

We take the above observation, which is true for extended particles, as a starting pointfor our definition of the total four-momentum in the case of point particles. We decomposeenergy-momentum tensor of the field surrounding a point particle as a sum of two terms:1) the energy-momentum tensor corresponding to the uniformly moving particle and 2)the difference between the two. Our renormalization procedure consists, therefore, inreplacing the integral of the first (non-integrable) term by the dressed quantity muµ

whereas, miraculously, the second term is already integrable. This way we obtain theformula (33).

Let us calculate the renormalized total four-momentum and the total angular-momen-tum components in the particle’s rest frame. Decomposing the electric induction field on

14

the rest-frame surface Σt into the sum

D = D0 + D (37)

of the Coulomb field

D0 =er

4πr3. (38)

and the remaining part D, we obtain the following formulae for the renormalized rest-frame quantities:

H = m +1

2P

∫

Σ(D2 + B2 −D2

0) d3x = m +1

2

∫

Σ(D

2+ B2) d3x , (39)

Pl = P∫

Σ(D×B)l d

3x =∫

Σ(D×B)l d

3x +∫

Σ(D0 ×B)l d

3x , (40)

Rk =1

2P

∫

Σxk(D

2 + B2) d3x =1

2

∫

Σxk(D

2+ B2) d3x +

∫

ΣxkDD0 d3x , (41)

Sm =∫

Σεmklx

k(D×B)l d3x =∫

Σεmklx

k(D×B)l d3x . (42)

Other contributions of the Coulomb field were killed by the principal value operator P.Let us come back to our “super theory”. For a “moving particle solution” choose

the observer, which moves along the approximate particle trajectory and take the corre-sponding co-moving Lagrango-Hamiltonian (26). Its value may be well approximated ifwe replace the exact value of the quantities H,Rk and Sm by the above “renormalized”quantities (39), (41) and (42), containing only the external Maxwell field.

We know that the variation with respect to the observer’s trajectory vanishes auto-matically when the complete “super theory” is taken into account. But this is no longertrue if we approximate the extended particles by point-like particles, surrounded by theMaxwell field. We conclude that only those solutions of (5) may approximate the dynam-ics of the true extended particles, governed by the field equations of the “super theory”,for which the variation of the above renormalized Lagrangian LH with respect to theparticle’s trajectory does vanish.

The reason why the variational principle obtained this way gives now a non-trivialequation is that the conservation laws (28) - (31) are not necessarily satisfied for thesolutions of the inhomogeneous Maxwell equations. Indeed, they were implied by theNoether invariance of an autonomous, Lagrangian field theory. Noether theorem does notapply to the inhomogeneous Maxwell theory with given sources.

But, after all, the situation is not so bad. We prove in Appendix D that, for anyregular solution of (5), the renormalized quantities (39) - (42) satisfy necessarily threeamong the conservation laws, namely (28), (30) and (31). This implies (see Section 5) thatEuler-Lagrange equations obtained from Lagrangian LH are equivalent to the remainingmomentum conservation law (29), which in turn is equivalent to the fundamental equation(13) of the electrodynamics of moving particles.

This proves that the renormalized LH is really a correct Lagrangian for Electrody-namics of Moving Particles.

15

5 Equations of motion from the variational principle

In this Section we will prove explicitly the equivalence between the Euler-Lagrange equa-tions derived from LH and the momentum conservation laws.

Let us take LH given by (26), with H, Rk and Sm given by formulae (39), (41) and(42), as the Lagrango-Hamiltonian of our theory of point-like particles interacting withthe Maxwell field. Being the Hamiltonian (with opposite sign) for the fields, it obviouslygenerates the inhomogeneous Maxwell equations (5) (written in the particle’s co-movingframe). The remaining equations of the theory are obtained from the variation withrespect to the particle trajectory.

Observe that LH is a 2-nd order Lagrangian in the particle variables:

LH = LH(q,v, v; fields) .

Varying LH with respect to q we obtain the following Euler-Lagrange equations (seeAppendix A):

pk =∂LH

∂qk, (43)

where the momentum pk canonically conjugate to the particle’s position qk is defined as:

pk :=∂LH

∂vk− d

dt

(∂LH

∂vk

). (44)

Using formula (26) we obtain:

∂LH

∂vk=

vk√1− v2

H− ∂(√

1− v2al)

∂vkRl +

∂(√

1− v2ωm)

∂vkSm . (45)

The momentum canonically conjugate to the velocity vk equals (see Appendix A):

πk :=∂LH

∂vk= −

√1− v2

(∂al

∂vkRl − ∂ωm

∂vkSm

). (46)

To calculate the time derivative of πk we use conservation laws (30) and (31) which areproved in Appendix D. This way we obtain the following formula for pk:

pk =vk√

1− v2H +

(δl

k + ϕ(v2)vlvk

)Pl + Al

kRl + BmkSm , (47)

where Alk and Bm

k are given by the following expressions:

Alk =

d

dt

(√1− v2

∂al

∂vk

)− ∂(

√1− v2al)

∂vk−

− (1− v2)ε ilm

(∂am

∂vkωi +

∂ωm

∂vkai

),

Bmk = − d

dt

(√1− v2

∂ωm

∂vk

)+

∂(√

1− v2ωm)

∂vk+

+ (1− v2)ε iml

(∂al

∂vkai +

∂ωl

∂vkωi

). (48)

16

The quantities ai and ωl have to be expressed in terms of v and v via formulae (20)and (21). Using the following properties of the function ϕ(τ)

2ϕ(τ)− (1− τ)−1 + τϕ2(τ) = 0 ,

2ϕ′(τ)− (1− τ)−1ϕ(τ)− ϕ2(τ) = 0 , (49)

and the identity

vi(εiklvm + εilmvk + εimkvl) = v2εklm , (50)

one easily shows that Alk ≡ Bl

k ≡ 0 . Thus, we finally obtain the following formula forthe momentum pk canonically conjugate to the particle’s position:

pk =vk√

1− v2H +

(δl

k + ϕ(v2)vlvk

)Pl , (51)

which we immediately recognize as the space-like component of the total four-momentum(H,Pk), Lorentz-transformed from the particle’s co-moving frame to the laboratory frame.Since the Lagrangian LH does not depend explicitly on the particle position q we con-clude that the Euler-Lagrange equation (43) is equivalent to the conservation law of totalmomentum in the laboratory frame:

d

dtpk = 0 . (52)

To calculate the time derivative of pk

pk = H d

dt

(vk√

1− v2

)+ Pl

d

dt

(ϕ(v2)vlvk

)+

+vk√

1− v2H +

(δl

k + ϕ(v2)vlvk

)Pl , (53)

we need to know time derivatives ofH and Pl. In Appendix D we show that the rest-frameenergy H fulfills the conservation law (28). However, as we mentioned in the previousSection, the conservation law (29) for the rest-frame momentum Pl is not necessary sat-isfied for the solutions of inhomogeneous Maxwell equations. Let us denote by Xl thedeviation from this law. This means that we define Xl by the formula:

Pl =√

1− v2(−alH− ε mk

l ωmPk + Xl

). (54)

The following identities are easy to prove:

d

dt

(vk√

1− v2

)=√

1− v2(δl

k + ϕ(v2)vlvk

)al , (55)

and

d

dt

(ϕ(v2)vlvk

)= alvk −

√1− v2

(δm

k + ϕ(v2)vmvk

)εlj

mωj . (56)

17

Inserting them, together with (28) and (54), into (53) we finally obtain:

pk =√

1− v2(δl

k + ϕ(v2)vlvk

)Xl . (57)

It is easy to see that the matrix (δmk + ϕ(v2)vmvk) is non-singular, its inverse being equal

to (δkl −

√1− v2 ϕ(v2)vkvl). Hence, Euler-Lagrange equations pk = 0, derived from the

Lagrangian LH , are equivalent to Xl = 0, i. e. to the conservation law (29). This provesthat the electrodynamics of moving particles may indeed be derived from the LagrangianLH .

It is worthwhile to notice that (29) implies also the conservation of the remainingcomponent p0 of the total four-momentum of the system. Observe, that p0 is numericallyequal to the complete Hamiltonian of the composed system, obtained from the completeLegendre transformation (formula (46) is used to calculate πk):

H := pkqk + πkv

k − LH = pkvk +

√1− v2 H

=1√

1− v2

(H + vlPl

). (58)

Indeed, the last expression is equal to the p0 – component of the total four-momentum(H,Pk), Lorentz-transformed to the laboratory frame. We have, therefore, H = p0. Usingthe same methods as before we easily obtain

p0 = vlXl (59)

which ends the proof. We conclude, that in the case, when conservation laws (28), (30)and (31) are satisfied, the Euler-Lagrange equation derived from the Lagrangian (26) isequivalent to the remaining conservation law (29).

In Appendix D we show that for any regular solution of (5) the non-conservation vectorXk equals

Xk = mak − eβk , (60)

where βk is the dipole part of the function β defined by expansion (11), i. e.

DP(β) = βkxk

r, (61)

Due to relation (12), vanishing of Xk is therefore equivalent to the fundamental equation(13) of the electrodynamics of moving particles imposed on the solutions of (5). It impliesthe total four-momentum conservation (8). In particular, it guarantees that the totalHamiltonian H = p0 remains constant during the evolution.

6 Variational principle based on fluxes

In this Section we finally derive a quasi-local Lagrangian for electrodynamics, related tofield energy via the Legendre transformation (22).

18

The discrepancy between the “canonical” and the “symmetric” energy-momentumtensors was often interpreted as an argument against defining the field Hamiltonian viathe Legendre transformation. Such a conclusion is false. It was shown (see [8] and [9]) thatthis problem (also the problems of defining gravitational energy in General Relativity),is related to the fact, that there is no unique way to represent the field evolution as an(infinite dimensional) Hamiltonian system. Each such representation is based on a specificchoice of controlling the boundary value of the field, and corresponds to a specific choice ofthe Hamiltonian. This non-uniqueness is implied by the non-uniqueness of the evolutionof the portion of the field, contained in a finite laboratory V . Indeed, the evolution is notunique because external devices may influence the field through the open windows of ourlaboratory. To choose the Hamiltonian uniquely, we have to insulate the laboratory or,at least, to specify the influence of the external world on it. One may easily imagine anunsuccessful insulation, which does not prevent the external field from penetrating thelaboratory. From our point of view, an insulation is sufficient if it keeps under controla complete set of field data on the boundary ∂V in such a way, that the field evolutionbecomes mathematically unique.

For relatively simple theories (e.g. scalar field theory) the Dirichlet problem may betreated as a privileged one among all possible mixed (initial value + boundary value)problems which are well posed. This means that there is a natural way to insulatethe laboratory V adiabatically from the external world. But already in electrodynamics(and even more in General Relativity) any attempt to define the field Hamiltonian leadsimmediately to the question: how do we really define our Hamiltonian system?

It was shown in [9] that the “canonical energy” obtained from the Maxwell Lagrangianvia the ordinary Legendre transformation is a legitimate field Hamiltonian, describing theevolution of the field closed in a metal shell in such a way, that the potential A0 on theshell is controlled (e. g. the shell is grounded). On the contrary, the “symmetric” energy12(D2 +B2) is related to the control of the electric and the magnetic flux on the boundary.

Below, we construct a Lagrangian which is directly related to the latter energy. Thisway, starting from the variational picture, we obtain the Hamiltonian picture by an ordi-nary Legendre transformation.

Field equations of any (linear or nonlinear) electrodynamics may be written as follows:

δL(Aν , Aνµ) = ∂µ(FµνδAν) = (∂µFµν)δAν + FµνδAνµ , (62)

where Aνµ := ∂µAν and L is the Lagrangian density of the theory. The above formula(see [5]) is a convenient way to write down the Euler-Lagrange equations

∂µFνµ =∂L

∂Aν

, (63)

together with the relation between the electromagnetic field fµν = Aνµ − Aµν and theelectromagnetic induction density Fνµ describing the momenta canonically conjugate tothe potential:

Fνµ =∂L

∂Aνµ

= −2∂L

∂fνµ

. (64)

19

For the linear Maxwell theory, the Lagrangian density is given by the standard formula(2) and relation (64) reduces to Fµν :=

√−ggµαgνβfαβ.We will integrate both sides of (62) over a 3–dimensional volume V belonging to the

hyperplane Σt and consisting of the exterior of the sphere S(r0):

δ∫

VL =

∫

V∂0(Fk0δAk) +

∫

∂VFν⊥δAν (65)

(by ⊥ we denote the component orthogonal to the boundary). To describe the boundaryterm it is convenient to use spherical coordinates (ξa), a = 1, 2, 3; adapted to ∂V . Wechoose ξ3 = r as the radial coordinate and (ξA), A = 1, 2; as angular coordinates: ξ1 = Θ,ξ2 = ϕ. The Euclidean metric gab is diagonal:

g33 = 1 , g11 = r2 , g22 = r2 sin2 Θ , (66)

and the volume element λ = (det gab)1/2 is equal to r2 sin Θ. With this notation we have:

δ∫

VL =

∫

V∂0(FB0δAB + F30δA3) +

∫

∂V(FB3δAB + F03δA0) . (67)

On each sphere S(r) = {r = const} the 2-dimensional covector field AB splits into a sumof the “longitudinal” and the “transversal” part:

AB = u,B +ε CB v,C , (68)

where the coma denotes partial differentiation and εAB is a sqew-symmetric tensor, suchthat λεAB is equal to the Levi-Civita tensor-density (i. e. λε12 = −λε21 = 1). Thefunctions u and v are uniquely given by the field AB up to additive constants on eachsphere separately. Inserting this decomposition into (67) and integrating by parts weobtain:

δ∫

VL =

∫

V∂0(−FB0,B δu + F30δA3 −FB0,C ε C

B δv) +

+∫

∂V(−FB3,B δu + F03δA0 −FB3,C ε C

B δv) . (69)

Using identities ∂BFB0 + ∂3F30 = 0 and ∂BFB3 + ∂0F03 = 0, implied by equations (63)in the case of a gauge-invariant theory, and integrating again by parts we finally obtain:

δ∫

VL =

∫

V∂0

[F30δ(A3 − u,3 )− (FB0||CεBC)δv

]+

+∫

∂V

[F03δ(A0 − u,0 )− (FB3||CεBC)δv

]. (70)

Here, by “||”we denote the 2-dimensional covariant derivative on each sphere S(r). Thequantities (A0 − u,0 ) and (A3 − u,3 ) are “almost” gauge invariant: only their monopolepart (mean-value) on each sphere may be affected if we change the additive constant inthe definition of u (the choice of an additive constant in the definition of v is irrelevant,because it is always multiplied by quantities which vanish when integrated over a sphere).

20

The sum of the volume and surface integrals in (70) is however gauge invariant. It maybe easily checked that

4(A3 − u,3) = r2BC||DεCD , (71)

where 4 denotes the 2-dimensional Laplace-Beltrami operator on S(r) multiplied by r2

(the operator 4 does not depend on r and is equal to the Laplace-Beltrami operatoron the unit sphere S(1)). The operator 4 is invertible on the space of monopole–freefunctions (functions with vanishing mean value on each S(r)). This functional space willplay an important role in further considerations and all the dynamical quantities of thetheory will belong to this space. To fix both terms in (70) uniquely we choose u in sucha way that the mean value of (A3 − u,3) vanishes on each sphere. Hence, with the abovechoice of the additive constants the quantity A3 − u,3 becomes gauge invariant:

A3 − u,3 = r24−1(BC||DεCD) . (72)

Let us observe that the function v is also gauge invariant (up to an additive constant,which does not play any role and may also be chosen in such a way that its mean valuevanishes on each sphere). Indeed, we have:

B3 = (curlA)3 = AC||DεDC = −r−24v . (73)

Due to the Maxwell equation divB = 0, the function B3 is monopole–free and theLaplasian 4 may again be inverted:

v = −r24−1B3 . (74)

The formula (70) could be also obtained directly from (67) by imposing the followinggauge conditions:

AB||B = 0 , (75)

∫

S(r)λA3 = 0 . (76)

The above condition does not fix the gauge uniquely: we still may add to Aµ the gradientof a function of time f = f(t). This residual gauge changes only the monopole part ofA0, but both the volume and the surface integrals in (70) remain invariant with respectto such a transformation.

Assuming the above gauge, we have u ≡ 0 and AB = ε CB v,C . To simplify the notation

we will, therefore, replace our invariants (A0 − u,0 ) and (A3 − u,3 ) by the values of A0

and A3, calculated in this particular gauge.Formula (70) is analogous to the lagrangian formula in classical mechanics

dL(q, q) =d

dt

(pkdqk

)= pkdqk + pkdqk . (77)

In the case of electrodynamics, we have a system with infinitely many degrees of freedomdescribed by four functions: F30, A3, FB0||CεBC and v. Two of them will play the role of

21

field configurations and the remaining two will be the conjugate momenta. But there isalso a boundary term in (70), typical for field theory. Killing this term by an appropriatechoice of boundary conditions is necessary for transforming the field theory into an (infinitedimensional) dynamical system (see [5], [8]). From this point of view, the quantity v (or,equivalently B3) is a good candidate for the field configuration, since controlling it at theboundary will kill the term δv in the boundary integral. On the contrary, δA0 can not bekilled by any simple boundary condition imposed on A3. We conclude, that it is ratherF03 = λD3 than A3, which has to be chosen as another field configuration.

Hence, we perform the Legendre transformation in formula (70), both in the volume

F30δA3 = −F03δA3 = −δ(F03A3) + A3δF03 , (78)

and on the boundary ∂V :

F03δA0 = δ(F03A0)− A0δF03 . (79)

This way, using (72) and (74), we obtain from (70) the following result:

δ∫

V

[L + ∂0(λD3A3)− ∂3(λD3A0)

]=

=∫

Vλr2∂0

[4−1(BA||BεAB)δD3 −4−1(DA||BεAB)δB3

]

+∫

∂V

[−λA0δD

3 − r24−1(F3A||BεAB)δB3]

. (80)

In the above formula, the function v has been replaced by the right-hand side of (74).Moreover, the operator 4−1, which is self-adjoint on the functional space of monopole-free functions on a sphere, was moved from δB3 to functions which multiply it under theintegral sign.

We see that (D3, B3) play the role of field configurations, whereas the remainingfunctions (BA||BεAB, DA||BεAB) describe the conjugate momenta. Controlling the config-urations at the boundary we kill the surface integral over ∂V and obtain this way aninfinite dimensional dynamical system describing the field evolution. There is, however, aproblem with such a control, because the electric induction D3 cannot be controlled freelyon the boundary. The reason is that the total electric flux through both components of∂V (i. e. through S(r0) and through the sphere at infinity) must be the same:

∫

S(r0)F03 =

∫

S(r∞)F03 = e , (81)

where e is the electric charge contained in S(r0). Hence, we have to separate the monopole-free (“radiative”) part of D3 (which can be independently controlled on both ends of V )from the information about the electric charge. For this purpose we split the electricinduction D3 into

D3 =e

4πr2+ D

3, (82)

22

where D3

is again a monopole-free function. It follows from (81) that the monopole partof D3 (equal to e/4πr2) is nondynamical and drops out from the volume integral of (80)

because it is multiplied by a monopole-free function BA||BεAB. The remaining part D3

(which does not carry any information about the charge e), together with B3, can betaken as the true, unconstrained degrees of freedom of the electromagnetic field.

In the same way we split the scalar potential A0:

A0 = φ + A0 , (83)

where φ(r) is the mean value of A0 on the sphere S(r) (monopole part) and A0 is amonopole–free function (“radiative” part of A0). Now, the boundary term A0δD

3 in (80)reads

∫

∂VλA0δD

3 =1

4π

∫

∂Vλr−2φδe +

∫

∂VλA0δD

3. (84)

Let us consider the first term in the r.h.s. of (84). In Appendix C we show that themonopole function φ(r) satisfies the following equation (cf. (C.19))

∂3φ(r) =√

1− v2

(e

4πr2+ ∂3φ(r)

), (85)

where φ(r) is regular in r = 0. The first term in the above formula corresponds to theCoulomb potential of the particle in its rest frame (factor

√1− v2 corresponds to the fact

that we use laboratory time instead of proper time). Therefore, we have

1

4π

∫

Vλr−2φδe = δ

(√1− v2

e2

32π2r0

)+ φ(∞)δe− φ(r0)δe . (86)

Finally, we perform the Legendre transformation between φ and the monopole part ofD3 at infinity. Hence, we control the total charge contained in S(r0) and the monopolefunction φ at infinity. Since the latter does not contain any physical information and isused only to fix the residual gauge, we may use the simplest possible choice: φ(∞) ≡ 0.We denote

Ψ1 = rB3 , (87)

Ψ2 = rD3, (88)

χ1 = −r4−1(DA||BεAB) , (89)

χ2 = r4−1(BA||BεAB) . (90)

Together with the value e of the electric charge contained in S(r0), they contain the entire(gauge invariant) information about the electromagnetic field. The first two quantitiesplay the role of field configurations. The corresponding Lagrangian has to be consideredas a function of these and their derivatives. The fields χ will appear as correspondingmomenta. They are equal to derivatives of the Lagrangian with respect to the timederivatives of the fields.

23

The response of the system to the control of the boundary values of the configurationsis given by the boundary momenta:

χr1 = −r4−1(

1

λF3A||BεAB) , (91)

χr2 = −r−1A0 , (92)

equal to the derivatives of the Lagrangian with respect to the radial derivatives of thefields.

Finally, formula (80) reads:

δ∫

VL =

∫

V∂0(λχAδΨA) +

∫

∂Vλχr

AδΨA + eδφ(∞) + φ(r0)δe (93)

where the new Lagrangian L equals

∫

VL =

∫

V

(L + ∂0(λD3A3)− ∂3(λD3A0)

)+√

1− v2e2

32π2r0

. (94)

We stress that that L remains finite, even if we put r0 = 0. Indeed, Maxwell La-grangian L may be decomposed into the regular part

Lreg =1

2λN

(D

2 −B2)

+ λNDD0 (95)

and the remaining part 12λN(D0)

2. This way we have

∫

VL =

∫

VLreg +

√1− v2

e2

32π2r0

. (96)

Finally, L is given by

L = Lreg + ∂0(λχAΨA) + ∂3(λχrAΨA)− ∂3

(λ

e

4πr2φ

). (97)

Formula (93) defines a generalized lagrangian system describing the evolution of theelectromagnetic field, when the dynamical variables ΨA and the electric charge e arecontrolled at the boundary ∂V .

¿From the lagrangian relation (93) we immediately obtain the hamiltonian one per-forming the Legendre transformation:

− δH =∫

Vλ(χAδΨA − ΨAδχA) +

∫

∂Vλχ3

AδΨA + eδφ(∞) + φ(r0)δe , (98)

where

H :=∫

V(λχAΨA − L) . (99)

It is easy to check (see [8]) that H is equal to the amount of electromagnetic energycontained in V .

24

7 The Lagrangian in the co-moving frame

We apply the above construction to the case of renormalized electrodynamics formulatedin the particle’s co-moving frame. Hence, the left-hand side of (99) has to be replaced byLH , expressed in terms of the renormalized generators H, Rk and Sm. The Lagrangianis given by the inverse Legendre transformation

∫

ΣL =

∫

ΣλχAΨA + LH . (100)

To simplify the notation we define a complex-valued configuration quantity

Ψ := Ψ1 + iΨ2 (101)

and a complex-valued momentum

χ := χ2 − iχ1 = −i(χ1 + iχ2) . (102)

Observe, that∫

ΣλχAΨA = Im

∫

Σλχ∗Ψ , (103)

where “Im” denotes the imaginary part and “*” denotes complex conjugation.To express L given by (100) in terms of the field configurations Ψ and its derivatives

we rewrite Maxwell equations in the co-moving frame (see the Appendix B for the proof):

iΨ =√

1− v2

{4χ + ak

(rKkχ + i4−1Vk(rΨ),3

)+ ωmVmΨ

}, (104)

iχ = −√

1− v2

{1

r2Ψ +

1

r4−1(rΨ),33 + ak

(1

r4−1∂3

[rKk4−1(rΨ),3−

− ir2Vkχ]+

1

r2Ψxk + i

e

4πr3xk

)− ωmVmχ

}(105)

(operators Kk and Vk are defined by (B.22) and (B.23); together with 4, they are self-adjoint operators on the space C of monopole-free functions on the unit sphere S2). Now,we rewrite the quantities H, Rk and Sm in terms of the field configuration Ψ and thecorresponding momentum χ. Using the same techniques as in Appendix B we obtain:

H = m +1

2

∫

Σλ

{1

r2Ψ∗Ψ− 1

r2(rΨ∗),34−1(rΨ),3 − χ∗4χ

}, (106)

Rk =1

2Re

∫

Σλ

{xk

r2Ψ∗Ψ− 1

r(rΨ∗),34−1Kk4−1(rΨ),3 − rχ∗Kkχ−

− 2χ∗i4−1Vk(rΨ),3 − ie

2πr3xkΨ

}, (107)

Sm =∫

ΣRe (λΨ∗Vmχ) + ∂3(bm) , (108)

where the boundary term in the last equation reads

bm = λΨ2(Km4−1 − xm

r

)Ψ1 − λ

e

4πr2xmΨ1 . (109)

25

Inserting (104) into (100) and using (106)–(108) we derive the following formula for theLagrangian of the electrodynamics of moving particles:

L :=∫

ΣL = −m

√1− v2 +

√1− v2

∫

Σ

(L0 + akLk + eaklk + ∂3(ω

mbm))

, (110)

where

L0 =λ

2

{− 1

r2Ψ∗Ψ +

1

r2(rΨ∗),34−1(rΨ),3 − χ∗4χ

}, (111)

Lk =λ

2

{−xk

r2Ψ∗Ψ +

1

r(rΨ∗),34−1Kk4−1(rΨ),3 − rχ∗Kkχ

}, (112)

lk = − λ

4πr3xkΨ

2 . (113)

Let us notice that due to the asymptotic behaviour of the field Ψ, the boundary term in(110) vanishes. Nevertheless, we keep this term to secure the consistency of the theory.This is typical situation in the field theory. Functionals with and without this term arenumerically the same but functionally they are different.

To complete the inverse Legendre transformation (100) one has to express the mo-mentum χ in terms of the “velocity” Ψ. From (104) we obtain immediately

χ = (4+ rakKk)−1

[i√

1− v2Ψ− akiVk4−1(rΨ),3 − ωmVmΨ

]. (114)

Observe that L may be written in a similar form as Ltotal defined in (1):

L = Lparticle + LMaxwell + Lint + boundary term , (115)

where

LMaxwell :=√

1− v2

∫

Σ

(L0 + akLk

), (116)

and the interaction term

Lint :=√

1− v2

∫

Σeaklk = −

∫

ΣNDD0 d3x , (117)

but contrary to (1) the interaction term Lint is finite and, as we shall show in the nextpaper, the Hamiltonian for the electrodynamics of moving particles is always well defined.

This way we have constructed a consistent Lagrangian structure for our theory. Now,variation of L with respect to both fields and particles is well defined and does not lead toany contradiction. Varying L with respect to field configuration Ψ we reproduce Maxwellequations (105) and (114). Variation with respect to the particle trajectory gives thefundamental equation (13).

26

8 Particle in an external potential

Suppose now that the particle moves in an external (generalized) potential U = U(q, q, t).Then the Lagrange function is given by:

L = −m√

1− v2 +√

1− v2

∫

Σ

(L0 + akLk + eaklk + ∂3(ω

mbm))− U . (118)

Varying L with respect to the particle trajectory ζ we obtain the following “equations ofmotion”:

DP(mα + e2β) = − e√1− v2

(δk

i −√

1− v2ϕ(v2)vkvi

)Qk

xi

r, (119)

where

Qi = −∂U

∂qi+

d

dt

∂U

∂qi(120)

is a vector of the generalized force in the laboratory frame. Formula (119) is obviouslyequivalent to the laboratory-frame equation

d

dtpi(t) = Qi . (121)

The influence of the external potential is manifested in the non-homogeneous boundarycondition (119).

As an example consider the particle interacting with an external electromagnetic fieldf ext

µν . The generalized potential is given by:

U(q, q, t) = eAext0 (q, t)− eqAext(q, t) , (122)

where Aext0 and Aext stand for the four-potential of the external field in the laboratory

frame. The generalized force (120) in terms of the laboratory-frame components Ei andBi of the external field now reads:

Qi = e(Ei(q, t) + εijkv

jBk(q, t))

. (123)

In the next paper we show that also in this case Maxwell equations for the radiation field,together with the non-homogeneous boundary condition (119), define an infinite-dimen-sional hamiltonian system. This means that initial data (Ψ, χ;q,v) for the radiation fieldand for the particle uniquely determine the entire history of the system if the externalpotential is given.

27

Appendixes

A Hamiltonian structure for a 2-nd order Lagrangian

theory

Consider a theory described by the 2-nd order lagrangian L = L(q, q, q). Introducing anauxiliary variable v = q we can treat our theory as a 1-st order one with a lagrangianconstraint φ := q − v = 0 on the space of lagrangian variables (q, q, v, v). Dynamics isgenerated by the following relation:

d (L(q, v, v) + µ(q − v)) =d

dt(p dq + π dv) , (A.1)

where µ is a Lagrange multiplier corresponding to the constraint φ = 0 and p, π aremomenta canonically conjugated to q and v respectively. From (A.1) we immediatelyobtain:

p = µ , π =∂L

∂v,

p =∂L

∂q, π =

∂L

∂v− p . (A.2)

From the last equation we get the formula for p

p =∂L

∂v− π =

∂L

∂v− d

dt

(∂L

∂v

), (A.3)

and, consequently,

p =d

dt

(∂L

∂v

)− d2

dt2

(∂L

∂v

). (A.4)

It is equivalent to the Euler-Lagrange equation:

δL

δq:=

d2

dt2

(∂L

∂v

)− d

dt

(∂L

∂v

)+

∂L

∂q= 0 . (A.5)

The hamiltonian description is obtained from the Legendre transformation applied to(A.1):

− dH = p dq − q dp + π dv − v dπ , (A.6)

where H(q, p, v, π) = p v+π v−L(q, v, v). In this formula we have to insert v = v(q, v, π),calculated from equation π = ∂L

∂v. Let us observe that H is linear with respect to the

momentum p. This is a characteristic feature of the 2-nd order theory.Euler-Lagrange equation (A.5) is of 4-th order. The corresponding 4 hamiltonian

equations have, therefore, to describe the evolution of q and its derivatives up to thirdorder. Due to Hamiltonian equations implied by relation (A.6), the information aboutsuccesive derivatives of q is carried by (v, π, p):

28

• v describes q

q =∂H

∂p≡ v (A.7)

hence, the constraint φ = 0 is reproduced due to linearity of H with respect to p,

• π contains information about q:

v =∂H

∂π, (A.8)

• p contains information about...q

π = −∂H

∂v=

∂L

∂v− p , (A.9)

• the true dynamical equation equals

p = −∂H

∂q=

∂L

∂q. (A.10)

B Maxwell equations in the co-moving frame

Formally, the Maxwell equations in the co-moving frame look identically as in Lorentziancoordinates:

∂µFνµ = 0 , (B.1)

but the relation between the tensor f and the tensor density F :

Fνµ =√− det g gναgµβfαβ , (B.2)

is given by the non-trivial metric tensor gµν on M. The components of the tensor gµν

are given by equations (19). Using them and formula (B.2), we obtain the followingexpressions for the components of Fµν :

F0j = Dj , (B.3)

Fmn =√

1− v2[(Dmεnk

l −Dnεmkl)ωkx

l + εmnk(1 + aix

i)Bk]

. (B.4)

Therefore, the Maxwell equation ∂0F0n + ∂mFmn = 0 implies:

Dn =√

1− v2∂

∂xm

[(εmk

lDn − εnk

lDm)ωkx

l − εmnk(1 + aix

i)Bk]

. (B.5)

29

Changing D to B and B to −D we obtain the remaining equation

Bn =√

1− v2∂

∂xm

[(εmk

lBn − εnk

lBm)ωkx

l + εmnk(1 + aix

i)Dk]

. (B.6)

The factor√

1− v2 corresponds to the fact, that the dot means the derivative with respectto the laboratory time t (which we have used to parameterize the trajectory) and not withrespect to the proper time τ on ζ.

Defining the complex vector field

F := B + iD , (B.7)

we may rewrite equations (B.5) and (B.6) in a more compact way

F n =√

1− v2∂

∂xm

[(εmk

lFn − εnk

lFm)ωkx

l − iεmnk(1 + aix

i)F k]

. (B.8)

Now, we rewrite Maxwell equations (B.8) in terms of unconstrained degrees of freedom(by F we denote, as usual, the monopole-free part of F):

Ψ = rF3

,

χ = r4−1(FA||BεAB

). (B.9)

The time derivative ψ of any scalar quantity ψ may be decomposed as the sum

ψ =√

1− v2⊥ψ + Nk∂kψ , (B.10)

where by⊥ψ we denote the time derivative of ψ with respect to the proper time τ along ζ in

the Fermi-propagated frame. This frame is characterized by vanishing of the shift vectorNk = 0 (actual shift Nk is given by (19)). The lapse function in the Fermi-propagatedframe equals N = 1 + akxk. Hence, we will first rewrite the Maxwell equations in theFermi-propagated frame and then, using (B.10), we will calculate Ψ and χ.

Equation (B.8) applied to the Fermi-propagated frame gives us

⊥F k = i∂l

(Nε lm

k Fm

). (B.11)

We will use the spherical coordinates (ξa) = (ξA, ξ3 = r) introduced in the Section 6.On each sphere S(r) the 2-dimensional complex covector field FA may be decomposedinto its “longitudinal” and “transversal” part:

FA = Z,A + ε BA W,B . (B.12)

The functions Z and W are defined up to additive constants and fulfil the followingidentities:

r2FA||A = 4Z , (B.13)

r2FA||BεAB = 4W (B.14)

30

(we remind that 4 denotes the 2-dimensional Laplacian normalized to the unit sphere -it contains only the derivatives over angles and, therefore, commutes with ∂3). Due tothe Gauss law

0 = divF = r−2((r2F

3),3 +4Z

)(B.15)

the longitudinal part of FA is fully determined by the radial part F3. Therefore, using

(B.9) and (B.15) we obtain

FA = −[4−1(rΨ),3],A + rε CA χ,C . (B.16)

Moreover, according to this “2 + 1” decomposition we have

(curl F)3 = εBAFA||B , (B.17)

(curl F)A = εBA(FB,3−F 3,B) . (B.18)

Now, from (B.11) we have

−i⊥Ψ= −ir

⊥F

3= −ir

⊥F 3 = xk∂l

(Nε lm

k Fm

)= xkalε

lmk Fm + rN(curl F)3 . (B.19)

The last term may be calculated from (B.17). In the first term we may replace Fm by thefollowing covector

Gm := −[4−1(rΨ),3],m + xnε snm χ,s . (B.20)

Indeed, due to (B.16), both covectors F and G differ only by the radial component, whichis anihilated by the term xkε lm

k . Finally, we obtain

⊥Ψ = −i4χ− ak

(irKkχ−4−1Vk(rΨ),3

), (B.21)

where we have defined following r-independent operators

Kk :=xk

r4+

(δmk −

xmxk

r2

)r ∂m =

r2

λ∂A

(xk

rλgAB∂B

), (B.22)

Vk := iεklmxl∂m . (B.23)

Let C denote the space of complex functions on the unit sphere S2. Observe that for anyψ ∈ C both Kkψ and Vkψ belong to C (the space of monopole-free complex functions onS2). Therefore, the dynamics given by (B.21) lives on C. We will see in the sequel, thatthe same is true for the remaining Maxwell equations.

Moreover, one can easily prove that 4, Kk and Vk are self-adjoint operators on C withrespect to the following scalar product:

< ψ1|ψ2 >:=∫

S(1)λψ∗1ψ2 , (B.24)

31

for any ψ1, ψ2 ∈ C. It is obvious that the generator of rotations Vk commutes with theLaplace-Beltrami operator 4. All the three operators commute with ∂3, because theycontain only differentiation over angles.

Finally, using (B.21), (B.10) and observing that Nk∂k = −i√

1− v2 ωmVm we obtain

Ψ =√

1− v2

{−i4χ− ak

(irKkχ−4−1Vk(rΨ),3

)− iωmVmΨ

}. (B.25)

Using again (B.11) we have

−i⊥χ = −ir4−1

(⊥FA||B εAB

)= r4−1

{[curl(NF)

]A||BεAB

}=

= r4−1[εABεCA

((NFC),3 − (NF 3),C

)]||B=

= −r4−1[(NFB)

||B,3 − r−24(NF 3)

]=

= −r−14−1∂3

(r2(NFA)||A

)+ r−1(NF 3) . (B.26)

The factor r2 appears when we change the order of lowering and rising of the indices underthe differentiation ∂3. This is due to the fact that gAB is proportional to r2 and gAB isproportional to r−2. The last term in the above equation denotes the monopole-free partof the function NF 3, which we obtain as the result of the operator 4−14 acting on it.We have

NF 3 =1

r

(Ψ + akΨxk + i

e

4πrakxk

). (B.27)

Moreover,

(NFA)||A = ak(∂Axk)FA + NFA||A (B.28)

The first term may be calculated as follows

(∂Axk)FA = gAB(∂Axk)FB =(gmn − xmxn

r2

)(∂mxk)Fn =

(gkn − xkxn

r2

)Fn . (B.29)

In the last expression, we may again replace Fn by Gn. This way we finally obtain

iχ = −√

1− v2

{1

r2Ψ +

1

r4−1(rΨ),33 + ak

(1

r4−1∂3

[rKk4−1(rΨ),3−

− ir2Vkχ]+

1

r2Ψxk + i

e

4πr3xk

)− ωmVmχ

}. (B.30)

C Boundary momenta

In this Appendix we compute boundary momenta χrA, which describe the response of

the system to the control of the boundary values of the configurations ΨA. ¿From the

32

definition (91) we have

χr1 = −r4−1

(1

λF3A||BεAB

)= −r4−1

[(D3NA −DAN3 + NεACBC)||BεAB

]=

= −r4−1[(D3NA)||BεAB + (NBA)||A

], (C.1)

since the radial component of the shift vector vanishes. The second term in the aboveformula we have already computed (cf. (B.28)). To compute the first term let us observethat

NA = Nk(∂Axk) =√

1− v2 εklmωlxm(∂Axk) =√

1− v2 rε BA ∂B(xmωm) (C.2)

and

NA||BεAB =√

1− v21

r4(xmωm) . (C.3)

Hence, we obtain

(D3NA)||BεAB = (D3||BNA + D3NA||B)εAB =

=√

1− v2

{rD3||A(xmωm)||A +

1

rD34(xmωm)

}=

=√

1− v2 ωm{KmD3 − xm

r4D3 +

1

rD34xm

}=

=√

1− v21

rωm

{KmΨ2 − xm

r4Ψ2 − 2

1

rΨ2xm − e

2πr2xm

}, (C.4)

due to 4xm = −2xm. It is easy to prove using the definition (B.22) that the operator Kk

satisfies the following identities:

4Kk + Kk4− 21

r4xk4 = 0 , (C.5)

xk4+4xk − 2rKk + 2xk = 0 . (C.6)

¿From (C.6) we obtain immediately

(D3NA)||BεAB =√

1− v21

rωm

{4

(Ψ2xm

r

)−KmΨ2 − e

2πr2xm

}. (C.7)

Thus, using (B.28) and (C.7) we get finally

χr1 =

√1− v2

1

r

{4−1(rΨ1),3 + ak4−1

(Kk4−1(rΨ1),3 − r2iVkχ2

)−

− ωm(Ψ2xm −4−1rKmΨ2 +

e

4πr2xm

) }. (C.8)

To compute χr2 = −r−1A0 observe that

∂3A0 = A3 − f03 , (C.9)

33

and

f0k = −NDk + Nmfmk . (C.10)

Since in our gauge A3 = A3 = rχ2, thus

∂3A0 = rχ2 + ND3 −NA(A3,A − AA,3) =

= rχ2 + ND3 − rNA∂Aχ2 + ∂3(NAAA) (C.11)

due to the fact that NA does not depend on r. To calculate the last term in the aboveformula let us remind that in our gauge

AA = −r2ε BA ∂B(4−1B3) . (C.12)

Therefore, using (C.2) we have

NAAA = −√

1− v2 r2εAB(rxmωm)||Bε CA (4−1B3)||C =

= −√

1− v2 ωm[rKm4−1Ψ1 −Ψ1xm] . (C.13)

Now, since Nk∂k = NA∂A and using (B.30) we obtain

∂3A0 = −√

1− v2 ∂3

{4−1(rΨ2),3 + ak4−1

(rKk4−1(rΨ2),3 + ir2Vkχ1

)+

+ ωm(rKm4−1Ψ1 −Ψ1xm

) }+ ND3 −ND3 . (C.14)

Taking the monopole–free part of ∂3A0 we obtain finally

χr2 =

√1− v2

1

r

{4−1(rΨ2),3 + ak4−1

(Kk4−1(rΨ2),3 + r2iVkχ1

)+

+ ωm(Km4−1Ψ1 −Ψ1xm

) }. (C.15)

If we define the complex boundary momentum

χr := χr1 + iχr

2 , (C.16)

then one can easily prove that

χr =δL

δΨ∗,3

. (C.17)

We stress that the above formula is not true without keeping the boundary term in (110).This term is responsible for terms linear in ωm in (C.8) and (C.15).

Using complex variables Ψ, χ and χr and the identity (C.5) we may rewrite theformulae (C.8) and (C.15) in a compact form

χr =√

1− v21

r

{4−1(rΨ),3 + ak4−1

(Kk4−1(rΨ),3 − r2iVkχ

)+

+ iωm(rKm4−1Ψ−Ψxm + i

e

4πrxm

) }. (C.18)

34

Observe that from (C.14) we may compute the monopole part of the scalar potentialφ = mon(A0). Namely

∂3 φ = mon(ND3) +√

1− v2 ∂3

[ωmmon(Ψ1xm)

]=

=√

1− v2

[e

4πr2+

1

rakmon(Ψ2xk) + ωm∂3 mon(Ψ1xm)

]≡

≡√

1− v2

[e

4πr2+ ∂3φ

]. (C.19)

D Proof of the conservation laws

To calculate the time derivative of H, Pk, Rk and Sm given by (39)–(42) we first do it forthe integrals extended over the region {r > r0} and then finally go to the limit r0 → 0.1) Conservation of the energy H:

H(r0) =∫

{r>r0}(D

nDn + BnBn) d3x =

∫

{r>r0}(DnDn + BnBn) d3x . (D.1)

Here, we used the fact that the time derivative of D0 vanishes and that the scalar product

of D0 with D vanishes when integrated over any sphere S(r) (the field D0 is angle-

independent, whereas D contains the dipole and higher harmonics only). Using equations(B.5) and (B.6) we obtain

H(r0) =√

1− v2

∫

{r>r0}

{∂m

[1

2(DnDn + BnBn)εmk

l ωkxl−

− εmnk(1 + aix

i)DnBk]− εnkmDnBkam

}d3x . (D.2)

Using the Gauss theorem and calculating the limit r0 → 0 we obtain

H =√

1− v2

∫

Σ

{amPm + lim

r0→0

∫

S(r0)

xm

r

[εmnk(1 + aix

i)DnBk]dσ

}, (D.3)

where dσ denotes the surface measure on the sphere S(r0). Observe that the contributionfrom D0 vanishes since it is parallel to xn. The remaining field D behaves like r−1 andthe surface element dσ like r2. Therefore, the surface integral vanishes in the limit andwe have:

H =√

1− v2amPm , (D.4)

which proves (28).2) Conservation of the momentum Pj:

Pj(r0) =√

1− v2

∫

{r>r0}

{(DiBj −BiDj)ω

i − aj1

2(DiD

i + BiBi)−

− ∂m

[(DmDj + BmBj − 1

2δm

j(DiDi + BiB

i))

(1 + akxk)

]}d3x (D.5)

35

and

Pj =√

1− v2(εmk

jPmωk − aj(H−m))−√

1− v2 limr0→0

{1

2

∫{r > r0}|D0|2 d3x

+∫

S(r0)

xm

r

(DmDj + BmBj − 1

2δm

j(DiDi + BiB

i))

(1 + akxk) dσ

}. (D.6)

The contribution of the non-singular part of the fields to the surface integral in (D.6)vanishes in the limit r0 → 0. Hence

Pj =√

1− v2(εmk

jPmωk − aj(H−m))−√

1− v2 limr0→0

{e2

8πr0

aj+

+∫

S(r0)

1

r

(xmDmDj +

1

2DmDmxj

)(1 + akx

k) dσ

}. (D.7)

Inserting (7) into (D.7) one can easily calculate the limit on the right hand side:

Pj =√

1− v2

(εmk

jPmωk − aj(H−m)− e

4πlimr0→0

∫

S(r0)

1

r2Dj dσ

), (D.8)

where D stands for the regular part of D. To calculate the surface integral let us decom-pose the field D into the radial component D3 and the 2-dimensional field DA tangent tothe sphere S(r):

Dj =xj

rD3 + ∂AxjDA . (D.9)

The contribution from the radial part equals

limr0→0

∫

S(r0)

xj

r3D3 dσ = lim

r0→0

∫

S(r0)

xj

r3(β + O(r))dσ =

4π

3βj . (D.10)

The tangent components give:∫

S(r0)

1

r2∂AxjDA dσ = −

∫

S(r0)

xj

r2DA

||A dσ =∫

S(r0)

1

r4xj(r

2D3),3 dσ =

=∫

S(r0)

xj

r3(2β + O(r)) dσ =

8π

3βk + O(r0) . (D.11)

Finally in the limit r0 → 0 we obtain

Pj =√

1− v2(εmk

jPmωk − ajH + (maj − eβj))

. (D.12)

This way we proved that the conservation of momentum (29) is equivalent to the funda-mental equation maj − eβj = 0.3) Conservation of the static moment Rk:

Rk(r0) =√

1− v2

∫

{r>r0}

{εkijx

iωj(DnDn + BnBn)− (1 + aixi)εmn

kDnBm−

− alxkεmnlDmBn + ∂l

[1

2xkε

lijxiωj(DnDn + BnBn)+

+ xk(1 + aixi)εnlmDnBm

]}d3x (D.13)

36

and

Rk =√

1− v2

(Pk − εkimaiSm − εkilω

iRl)

+

+√

1− v2 limr0→0

∫

S(r0)

xl

r

[(1 + aix

i)xkεnlmDnBm

]dσ . (D.14)

The contribution from D0 to the surface integral vanishes since it is parallel to xn. Dueto the asymptotic behaviour of the fields this integral vanishes in the limit r0 → 0 whichproves (30).4) Conservation of the moment of momentum Sm:

Sm(r0) = −√

1− v2

∫

{r>r0}εm

lk

{εlrs

[xiBsDrε

kit + xk(BsDjεjt

r + DrBjεjt

s )]ωt+

+1

2εmkla

lxk(DnDn + BnBn) + ∂l[1

2xk(1 + aix

i)(DnDn + BnBn)]−

− ∂j

[xk(1 + aix

i)(BjBl + DjDl) + xkεlrsεjitBsDrxiωt

]}d3x (D.15)

and

Sm =√

1− v2

(εmklakRl − εmklωkSl

)+

+√

1− v2 limr0→0

∫

S(r0)

xj

r

[(1 + aix

i)xkεmkl(B

jBl + DjDl)]dσ . (D.16)

To calculate the surface integral let us observe that the contribution from the non-singularpart of the fields vanishes in the limit r0 → 0. The only possibility to obtain nonzerovalue of this integral is to integrate the Coulomb component of Dj and r−1 component ofDl. Then, due to (7) we have

− e

8πlimr0→0

∫

S(r0)(1 + aix

i)xk

r3εm

klal dσ = − e

8πεm

klal∫

S(1)

xk

r3dσ ≡ 0 , (D.17)

which ends the proof of (31).

Acknowledgments

The authors are very much indebted for the financial support, which they got from theEuropean Community (HCM Contract No. CIPA–3510–CT92–3006) and from the PolishNational Committee for Scientific Research (Grant No. 2 P302 189 07).

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37

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