A unified approach to the Darwin approximation

A unified approach to the Darwin approximation

Todd B. Krause

Institute for Fusion Studies and Physics Department, The University of Texas at Austin, Austin, Texas 78712, USA

A. Apte

Centre for Applied Mathematics, Tata Institute of Fundamental Research, Bangalore, India P. J. Morrison

Physics Department and Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA

共Received 9 July 2007; accepted 25 September 2007; published online 30 October 2007兲

There are two basic approaches to the Darwin approximation. The first involves solving the Maxwell equations in Coulomb gauge and then approximating the vector potential to remove retardation effects. The second approach approximates the Coulomb gauge equations themselves, then solves these exactly for the vector potential. There is noa priorireason that these should result in the same approximation. Here, the equivalence of these two approaches is investigated and a unified framework is provided in which to view the Darwin approximation. Darwin’s original treatment is variational in nature, but subsequent applications of his ideas in the context of Vlasov’s theory are not. We present here action principles for the Darwin approximation in the Vlasov context, and this serves as a consistency check on the use of the approximation in this setting.

© 2007 American Institute of Physics.关DOI:10.1063/1.2799346兴 I. INTRODUCTION

The Darwin approximation of electrodynamics is the order-共v/c兲² approximation to the relativistic interaction of classical charged particles and the electromagnetic field they produce. There have been several different approaches to the study of the Darwin approximation. These fall into two basic types. In the first approach one starts with the full Maxwell equations in terms of the scalar and vector potentials, solves these in Coulomb gauge to obtain exact expressions for the potentials, and then finally approximates the vector potential to remove retardation effects. In the second approach one also begins with the full Maxwell equations in Coulomb gauge, but approximates the equations themselves by remov- ing the time derivatives of the vector potential, and then obtains expressions for the potentials by solving these ap- proximate equations. These two procedures do not necessar- ily commute, and it remains to see in what sense these ap- proaches may both be termed the Darwin approximation. A major objective of the present paper is to highlight the dif- ferences in the approaches and to demonstrate directly how they are related.

The specific setting considered here is that of charges located in an unbounded domain. This may cover a wide range of phenomena pertinent to an astrophysical plasma set- ting. In the laboratory setting, boundary conditions will in- evitably become important.^1–3 Qinet al.⁴in particular simu- late a system with a combination of cylindrical and periodic boundary conditions. Such situations often call for the intro- duction of harmonic or vacuum fields in addition to the lon- gitudinal and transverse fields considered here.

The idea of Darwin’s original approximation⁵has found application in the description of plasmas, where particle dy- namics is replaced by that of a phase space density governed

by a Vlasov-like dynamics. As is the case for all of the basic models of plasma physics, one expects this description to possess an action principle formulation.⁶A second objective of the present paper is to construct an action principle for the Vlasov–Darwin approach, thus returning to one of Darwin’s own stated purposes: “To reduce the problem. . .to a Lagrang- ian form, so that all the well-known theorems of general dynamics may be made applicable.” Along the way, we also obtain the field action for the approximated field equations and the combined particle-field action.

The paper is organized as follows. In Secs. II and III we undertake our first objective: To make clear the differences in the two basic approaches to the Darwin approximation and to show their equivalence in the specific case of an unbounded domain. Section II describes the approaches, and Sec. III demonstrates their equivalence. In Sec. IV we consider our second objective: To show that the various approaches all derive from an action principle. We conclude with Sec. V. In the remainder of the present section, we review the original Darwin approximation and its various applications.

In his original paper, Darwin derived an order-共v/c兲² approximation to the fully relativistic motion of charged par- ticles in an unbounded domain by constructing the approxi- mate Lagrangian

L=

兺

a=1

n

冉

^{ma2q˙a2+m8caq˙2a4}

冊

⁻¹²a

兺

⫽b

eaeb

rab

+ 1

2_a

兺

_⫽b2c^ea2e_r^b_ab^{关q˙a·q˙b+共q˙a·rˆab兲共q˙b·rˆab兲兴, 共1兲}

1070-664X/2007/14共10兲/102112/10/$23.00 14, 102112-1 © 2007 American Institute of Physics

whererabªqa−qb, rabª储rab储, and rˆabªrab/rab.共See also, e.g., Ref.7, Sec. 65, pp. 165–169, and Ref.8, Sec. 12.6, pp.

596–598.兲The expression for the Lagrangian of Eq.共1兲has a kinetic part deriving from an expansion in powers ofv/cof the term

冑

^{1 −共v/c兲2} in the fully relativistic Lagrangian; the interaction stems from the coupling with the fields of other particles through the potentialA^␮=共␾^,A兲, where

␾共x,t兲=

兺

_{b 储x−}^eb_qb储^,

共2兲 A共x,t兲=

兺

_b ^eb关q˙_2c储x^b+共q˙₋^b_q^·rˆ_b储^b兲rˆb兴,

with qb the position of the bth particle at time t and q˙bªdqb/dtits velocity. By working to this order, one elimi- nates the complexities of dealing with retarded quantities, and thus calculation becomes more practicable. Darwin pro- ceeded to apply the results to a problem of extreme interest at the time: The Bohr–Sommerfeld atom.

The Darwin approximation has found numerous applica- tions in the area of quantum mechanics. Fewer investigations have focused on the Darwin Lagrangian from a purely clas- sical mechanical or dynamical systems point of view. Some, such as Dettwiller,⁹have studied conserved quantities in the system.

By far the majority of literature using the Darwin system in a classical setting employs it as a simplifying approxima- tion, in essence as it was originally intended. Several of these investigations pertain specifically to plasma physics. For ex- ample, Appel and Alastuey^10,11have investigated the equilib- rium properties of low-density, one-component plasmas gov- erned by the Darwin interaction. Kaufman and Soda¹² applied the Darwin approximation to study the magnetic sus- ceptibility of an imperfect gas. Kaufman and Rostler¹³ in 1971 were among the first to propose the Darwin model as a self-standing basis of plasma simulation. Several have taken up the idea, and the Darwin model has found its greatest applications in particle-in-cell plasma simulation codes共cf.

Nielson and Lewis¹⁴兲. Hewett¹⁵ has given an overview of applications of the Darwin model to the simulation of low- frequency plasma phenomena: In Mirrortron simulations, simulating the fields of an ion-bunch accelerated by repul- sion from a trailing tail of ions; and in simulating instabilities arising in a plasma column with stationary ions and counter- streaming electron components. Pritchett¹⁶ points out that, although the Darwin approximation works well in two- dimensional simulations, it suffers from difficulties in three dimensions with nonperiodic boundary conditions. Weitzner and Lawson² take up the problem of boundary conditions, basing considerations on closeness of approximation to Max- well’s equations and on conservation of charge and energy.

In the mathematics literature, Degond and Raviart,³ Bena- chour et al.,¹⁷ and Bauer and Kunze¹⁸ have discussed the existence of solutions of the Darwin system.

II. APPROACHES TO THE DARWIN APPROXIMATION In Sec. II A we discuss the approach that leads directly to Darwin’s original form of the vector potentialA_C^qsof Eq.

共6兲. This method finds exact solutions to Maxwell’s equa- tions in Coulomb gauge over an unbounded region, then ap- proximates the vector potential. We then demonstrate with hindsight that the approximated vector potential obeys Eq.

共9兲. In Sec. II B we discuss the approach which first approxi- mates the equations of motion, resulting in Eq. 共11兲, with solutionATof Eq.共12兲. Finally, in Sec. II C we show that the two approaches yield the same equations共15兲for the poten- tials, which can be directly solved to giveADin terms of the full currentJas in Eq.共16兲; we show in Eq.共20兲that this is in fact equivalent to Darwin’s form of the interactionA_C^qs. A. Quasistatic approach

The quasistatic approach共cf. Ref.19兲begins with Max- well’s equations in terms of potentials,

ⵜ²␾⁺¹ c

⳵

⳵^{t共⵱·A兲= − 4}␲␳^,

共3兲 ⵜ²A− 1

c²

⳵^2A

⳵^{t2 −⵱}

冉

^{⵱·A+1c⳵␾}⳵^t

冊

^{= −4c␲J,}

and employs the Coulomb condition ⵱·A= 0, resulting in equations

ⵜ²␾= − 4␲␳^,

共4兲 ⵜ²A− 1

c²

⳵^2A

⳵^{t2 = −} 4␲

c J+1 c

⳵⵱␾

⳵^{t .}

The solutions to these Coulomb gauge equations in the case of an unbounded region can be written as

␾C共x,t兲=

冕

^{d3x⬘␳共xr⬘,t兲 共5兲}

and

AC共x,t兲=1

c

冕

^{d3x⬘1r兵J共x⬘,t⬘兲−rˆ关rˆ·J共x⬘,t⬘兲兴其ret}

+c

冕

^d3x⬘1r

冕

0 r/c

d␶␶兵3rˆ关rˆ·J共x⬘^,t−␶兲兴

−J共x⬘^,t−␶兲其,

where rª储x−x⬘^{储 and rˆ=共x−x}⬘^兲/r, “ret” denotes that the quantities in brackets are evaluated at the retarded time t⬘

=t−r/c, and the spatial integration extends over R³. The instantaneous or “quasistatic” form of the vector potential results from the substitutionJ共x⬘^{,t−␶兲→J共x}⬘^,t兲 in the sec- ond integral inACand removing “ret” from the first integral;

i.e., the current is evaluated at the timet. This gives A_C^qs共x,t兲= 1

2c

冕

^{d3x⬘1r兵J共x⬘,t兲+rˆ关rˆ·J共x⬘,t兲兴其. 共6兲}

This is clearly the continuum analog of the vector potential in Eq.共2兲. This approach has the virtue of yielding explicitly

the form of the potentials derived by Darwin in his original work, manifestly containing terms only up to order共v/c兲².

Given the quasistatic potentials ␾C and A_C^qs, one may turn the question around and seek the explicit equations which they satisfy. The scalar potential clearly satisfies Pois- son’s equation,

ⵜ²␾C= − 4␲␳^. 共7兲

A direct computation shows the quasistatic vector potential obeys

ⵜ²A_C^qs= −4␲ c J+1

c

冕

^{d3x⬘r13关J⬘− 3共J⬘·rˆ兲rˆ兴,}

withJ⬘ªJ共x⬘^,t兲.

To proceed further, recall that a vector J, under condi- tions of sufficient differentiability, may be decomposed into transverse and longitudinal components asJ=JT+JL, where

JT共x,t兲= 1

4␲^{⵱ ⫻⵱⫻}

冕

^{d3x⬘J共xr⬘,t兲,}

共8兲 JL共x,t兲= − 1

4␲^⵱

冕

^{d3x⬘⵱⬘·J共xr ⬘,t兲.}

The longitudinal component may be directly re-expressed as J_L共x,t兲= 1

4␲

冕

^{d3x⬘r13关J⬘− 3共J⬘·rˆ兲rˆ兴,}

so thatA_C^qssatisfies the equation ⵜ²A_C^qs共x,t兲= −4␲

c JT共x,t兲. 共9兲

The vector potential therefore satisfies Poisson’s equation, where the source is not given by the full current, but only by its transverse componentJT.

It is worth noting, since we will have cause to return to the point in later sections, that charge conservation as given by the continuity equation

⳵␳

⳵^{t +⵱·J= 0}

follows automatically from the potential equations共3兲, a fact which remains true under the change of gauge giving Eq.共4兲. On the other hand, the equations satisfied by ␾C and A_C^qs 关Eqs.共7兲and共9兲兴do not enforce charge conservation.

B. Operator approximation approach

The second approach exploits the notion that the Darwin approximation neglects retardation, but, in contrast to Dar- win’s own derivation, it employs the approximation scheme at the level of the equations themselves. More specifically, one again begins with the full Maxwell equations for the scalar and vector field: Eq.共3兲. One then argues, by analogy with the case for the Lorenz gauge, that the term 共1 /c²兲⳵^2A/⳵^t2 is responsible for the retardation. This term may therefore be dropped from the equations, essentially leading to a change of the differential operator in the equa- tion. The fact thatA scales as 1 /c, so that the term overall

scales as 1 /c³, makes this more plausible in the context of the Darwin approximation. The resulting equations are therefore

ⵜ²␾⁺¹ c

⳵

⳵^{t共⵱·A兲= − 4}␲␳^,

共10兲 ⵜ²A−⵱

冉

^{⵱·A+1c⳵␾}⳵^t

冊

^{= −4c␲J.}

The removal of the time-derivative term from the full Maxwell equations implies that the continuity equation is no longer automatically valid. In fact,

⳵␳

⳵^{t +⵱·J= −} 1 4␲^c

⳵²

⳵^{t2⵱ ·A.}

If the sources␳ ^{and J} do indeed preserve charge conserva- tion, then the ansatz⵱·A= 0 becomes natural. With this an- satz, Eq.共10兲becomes

ⵜ²␾^{= − 4}␲␳^,

共11兲 ⵜ²A−1

c

⳵⵱␾

⳵^{t = −} 4␲

c J.

We now find a solution to these equations and show thatA does indeed satisfy the condition of the ansatz.

The solution of the first equation is the usual Coulomb potential given in Eq.共5兲. The solution of the second equa- tion is

AT共x,t兲=1

c

冕

^{d3x⬘J共xr⬘,t兲−4}␲^1c

⳵

⳵^t

冕

^{d3x⬘⵱⬘␾Cr共x⬘,t兲.}

共12兲 We then check to see that this is divergence free. Using the equation for␾C, one computes

⵱·AT=1

c

冕

^d3x⬘1r

冋

^{⵱⬘·J共x⬘,t兲+⳵␳共x⳵t⬘,t兲}

册

^.

Thus,AT is divergence free when the sources共␳,J兲 satisfy charge conservation.

Important to note in this context is that this theory, as expressed by Eq.共11兲, is not gauge invariant. In terms of the fields E and B, a gauge change by a function ␹ ^leaves

⵱·E⬘^{= 4}␲␳invariant, but not the second equation, 4␲

c J=⵱⫻B⬘⁺¹ c

⳵⵱␾⬘

⳵^{t =⵱⫻B+} 1 c

⳵⵱␾

⳵^{t +} 1 c²

⳵²␹

⳵^t2. Thus, Eqs. 共11兲 are invariant under gauge transformations when the function␹ is of the same order in powers of 1 /c asA.

Within this framework, we now see that, if we impose charge conservation, we may then rewrite the equation forA given in Eq.共11兲in the form of Eq.共9兲. To do so, we again decompose the current asJ=JT+JL. However, in the expres- sion forJLgiven in Eq.共8兲, we may use the continuity equa- tion to replace⵱⬘^·J共x⬘^{,t兲by −}⳵␳共x⬘^,t兲/⳵t, and then use the equation for␾C. This results in

1 4␲

⳵⵱␾C共x,t兲

⳵^{t =JL共x,t兲.}

Substituting this in the left-hand side of Eq.共11兲 finally re- sults in ⵜ²A= −共4␲^/c兲共J−JL兲= −共4␲^/c兲JT. This is exactly Eq.共9兲, showing thatA_C^qsandATsatisfy the same equation.

We may view the above system directly in terms of the fieldsEandB.^3,17,18,20We may decompose the electric field as E=ET+EL, with the conditions ⵱·ET= 0 and⵱⫻EL=0.

Given this decomposition, the system 1

c

⳵^EL

⳵^{t −⵱⫻B= −} 4␲

c J, ⵱ ·EL= 4␲␳^, 共13兲 1

c

⳵^B

⳵^{t +⵱⫻ET=0, ⵱ ·B= 0, 共14兲} is equivalent to the Darwin system 关Eq. 共11兲兴 upon writing EL= −⵱␾,ET= −共1 /c兲⳵^A/⳵^{t, and B=}⵱⫻A.

C. General approach

Both the above approaches lead to field equations ⵜ²␾D共x,t兲= − 4␲␳共x,t兲,

共15兲 ⵜ²A_D共x,t兲= −4␲

c J_T共x,t兲,

for the Darwin system, where the transverse current JT is given by the first expression in Eq.共8兲. This is supplemented by the Coulomb gauge condition, which in this setting is equivalent to imposing charge conservation.

In this form, the equations for ␾ ^andA are decoupled, and thus can be solved easily. The equation for the scalar potential is Poisson’s equation, so that inversion of the ⵜ² operator yields␾D=␾C, with the latter given by Eq.共5兲. The equation for the vector potential may be solved in the same way, component by component. Using the form ofJT given in Eq.共8兲, we find

AD共x,t兲= 1

4␲^c

冕

^d3rx⬘

冋

^{⵱⬘⫻⵱⬘⫻}

冕

^{d3x⬙J共xrx}⬘^⬙x,t兲⬙

册

^,

共16兲 wherer_x⬘x⬙ª储x⬘^−x⬙储and⵱⬘ª⳵^/⳵^x⬘. One shows by direct computation thatA_Ddoes satisfy⵱·A_D= 0.

III. EQUIVALENCE OF DARWIN FORMULATIONS The approaches of both Secs. II A and II B may be char- acterized by Eq.共15兲, whereJTis the transverse component of the full current, and is thus a shorthand for the expression in Eq.共8兲. However, we now have two expressionsA_C^qsand ADfor the vector potential, Eq.共6兲and Eq.共16兲respectively, each in some sense a valid representation of the salient fea- tures of the Darwin approximation. SinceA_C^qshas the explicit form found in Darwin’s original calculations, we may inquire whetherADcan be converted to this form. We know at the outset that it must be possible, sinceA_C^qsandADsatisfy the

same equations, and are in the same gauge. They can at most, then, differ by a constant. In the present section we show they are indeed equivalent.

Starting withADas given in Eq.共16兲, use of the identity

⑀ijk⑀klm=␦il␦jm−␦im␦jl and ⵜ²共1 /储x−x⬘^{储兲= −4}␲␦^共3兲共x−x⬘^兲 gives

AD=1

c

冕

^d3x⬘储x^J共x−^⬘x^,t兲⬘储 + 1

4␲^c

冕

^d储x^3x⬙^⬙−dx3x⬘^⬘储共J⬘^·⵱⬙兲⵱⬙_储x ¹

−x⬙^储.

Application of the identity⳵i⬙共1 /储x−x⬙储兲= −⳵i共1 /储x−x⬙储兲al- lows the derivatives in the second term to be extracted from one integral,

A_D=1

c

冕

^d3x⬘储x^J共x−^⬘x^,t兲⬘^储 + 1

4␲^c

冕

^{d3x⬘共J⬘·⵱兲⵱}

冕

储x−x⬙^d储储x^3x⬙⬘^−x⬙储. 共17兲 It remains at this point to perform the integral over x⬙^. Before doing so, however, we note that the gradient with respect tox acts on this integral. Given this, adding a term independent ofx to the integrand will not change the result.

We thus add the term −1 /储x⬙^{储2 to the integrand} 1 /储x−x⬙^储储x⬘^−x⬙储. This gives

AD=1

c

冕

^d3x⬘储x^J共x−^⬘x^,t兲⬘储 + 1

4␲^c

冕

^{d3x⬘共J⬘·⵱兲⵱}

冋 冕

^储x−xd⬙^储储x3x⬙⬘^−x⬙^储

−

冕

储^dx³⬙^x储^⬙2

册

^{. 共18兲}

The addition of the term −1 /储x⬙^储2has the effect of canceling the divergence as储x⬙^储→⬁. Postponing the calculation for a moment and simply using the result that the term in square brackets is equal to −2␲储x−x⬘^{储 关see Eq. 共21兲} below兴, this becomes

A_D=1

c

冕

^d3x⬘储x^J共x−^⬘x^,t⬘^兲储− 1

2c

冕

^{d3x⬘共J⬘·⵱兲⵱储x−x⬘储,}

共19兲

which finally gives

A_D= 1

2c

冕

^{d3x⬘J共x⬘,t兲+}储x^rˆ⬘−^关rˆx^⬘⬘^{·储J共x⬘,t兲兴=AC}

qs. 共20兲

Thus, we find that the two formsA_C^qsandAD of the vector potential are equivalent.

We now return to the calculation of the integral IDª

冕

^d3x⬙

冉

储x⬙⁻x储储x¹ ⬙^−x⬘储− 1

储x⬙储²

冊

^共21兲

in Eq. 共18兲. We break the integral into two parts as I_D=I₁

−I₂, with

I₁ª

冕

储x⬙^{−x储储xd3x⬙}⬙^−x⬘^{储 and I2ª}

冕

储x^d3⬙^x储⬙2.

Note that bothI₁andI₂are divergent because of insufficient decay at large 储x⬙储, but the divergence is removed upon subtraction.

A simple approach to the calculation of I₁ involves the use of Legendre polynomialsPl. We first make the substitu- tionsyªx⬙^−x⬘^andzªx−x⬘. We also employ the orthogo- nality condition

冕

−1 1

P_k共␰兲P_l共␰兲d␰^{= 2}␦kl

2l+ 1 and the expansion

1 储y−z储=

兺

l=0

⬁ r_⬍^l

r_⬎^l+1Pl共cos␥兲,

withr_⬍ªmin兵储y储,储z储其,r_⬎ªmax兵储y储,储z储其, and␥^{the angle} betweeny andz. Recalling that P₀共␰兲= 1, a straightforward calculation shows

I₁=

冕

储y储储y^d3y−z储

= 2␲

兺

l=0

⬁

冕

0

⬁ r_⬍^l r_⬎^l+1ydy

冕

−1

1

P₀共␰兲Pl共␰兲d␰

= 4␲

冕

0

⬁ y r_⬎dy,

whereyª储y储.

For the integralI₂, we change the variablex⬙^{toyto find} I₂=兰d³x⬙^/储x⬙^{储2=兰d3y/y2= 4}␲兰₀^⬁dy. Putting the integrals to- gether and writingzª储z储, we find

ID=I₁−I₂= 4␲

冕

0

⬁

冋

^ry⬎− 1

册

^dy

= 4␲

冕

0

z

冋

^{yz− 1}

册

^{dy+ 4␲}

冕

z

⬁关1 − 1兴dy

= − 2␲^z

= − 2␲储x−x⬘储, which is the result used in Eq.共19兲.

IV. ACTION PRINCIPLES

Given the various approaches to the Darwin approxima- tion, it is natural to seek a unified viewpoint via action prin- ciples. This is facilitated by the analysis of Secs. II and III, culminating in the equivalence demonstrated in Eq.共20兲. Es- sential motivation comes from the fact that an action prin-

ciple provides a natural method by which to ensure consis- tency. In approximation schemes such as the Darwin approximation, lack of a top-down viewpoint can open the door to inconsistency in the order of approximation, and therefore to the rise of artifacts of calculation not properly due to physical processes.

Moreover, Darwin’s original formulation focused on the interaction between particles, while the discussion of Secs. II and III focused on the field equations. In Sec. IV A we dem- onstrate an action for the Darwin field equations共10兲. In Sec.

IV B we describe a combined action principle for particles interacting with the fields of other particles all subject to the Darwin approximation. Both the field and particle equations follow from a single action principle. The particle equation of motion produced from this action differs somewhat from that found elsewhere in the literature. This is then general- ized in Sec. IV C, where we discuss a noncanonical action principle and include a particle distribution function.

A. Darwin field action

By analogy with the Maxwell field action S_Mf关␾,A兴=

冕

^dtd3x

冋

^{−␳␾+1cJ·A}

册

+ 1

8␲

冕

^{dtd3x关E2−B2兴}

giving the Maxwell equations 共3兲 for the electromagnetic potentials coupled to sources, we may look for an action that yields the Darwin field equations as given in Eq.共10兲. If we decompose E into an irrotational part ELª−⵱␾ ^{and a} divergence-free partETª−共1 /c兲⳵^A/⳵t, we can then see that the termE_T²must be dropped since, by an argument similar to the one at the beginning of Sec. II B, it scales as 1 /c³and is responsible for the retardation. Thus, we obtain the following action:

S_Df关␾,A兴=

冕

^dtd3x

冋

^{−␳␾+1cJ·A}

册

+ 1

8␲

冕

^{dtd3x关E2−ET2−B2兴,}

whose variation with respect to its arguments yields Eq.共10兲.

B. Darwin particle-field and particle actions

The initial impetus of Darwin’s original work was to describe the motion of charged particles in the field of other charged particles. This remains the main thrust of most ap- plications of the Darwin approximation, where one seeks to evolve many-particle systems without the computationally expensive complication of retardation. With this in mind, we formulate a combined particle-field action for the Darwin system, by analogy with the full Maxwell particle-field ac- tion. Expanding the kinetic term

冑

^{1 −}共q˙a/c兲²to second order inq˙a/cand dropping the constant, we find

S_D关q;␾^,A兴=

冕

^dt

再 兺

^a

冋

^{ma2q˙a2+m8caq˙2a4}

册

+

兺

_a ^ea

冕

^{d3x␦共3兲共x−qa兲}

冋

^{−␾共x,t兲}

+q˙_a

c ·A共x,t兲

册

^+81␲

冕

^{d3x关E2共x,t兲}

−E_T²共x,t兲−B²共x,t兲兴

冎

^.

Variation with respect to the field variables␾ ^andArepro- duces Eq.共10兲as before, with the imposition of the Coulomb condition giving共15兲. The density␳ and currentJare given in terms of particles by

␳共x,t兲=

兺

b

eb␦^共3兲关x−qb共t兲兴,

共22兲 J共x,t兲=

兺

_b ^{ebq˙b共t兲␦共3兲关x−qb共t兲兴.}

We may solve the resulting field equations for the fields

␾ ^andA due to the particles. Solution in terms of␳ ^andJ yields the usual Coulomb scalar potential␾C, and use of the continuity equation gives the equation for the vector poten- tial the form in Eq.共15兲. The solutionADis therefore given by Eq.共16兲, which may be rewritten asA_C^qs. Insertion of the expressions共22兲then produces the expressions given in Eq.

共2兲, so that substitution back intoSDyields an action in terms of particles alone. This action has the form SD关q兴=兰dt L, withL given by Eq.共1兲, bringing the procedure full circle.

We may use this action formulation to derive the equa- tion of motion for the particles. It is convenient to write the solutions of the field equations as

␾共x,t兲=

冕

^{d3x⬘K共x兩x⬘兲␳共x⬘,t兲,}

共23兲 Ai共x,t兲=1

c

冕

^{d3x⬘Kij共x兩x⬘兲Jj共x⬘,t兲,}

with kernels

K共x兩x⬘兲ª 1 储x−x⬘储,

共24兲 Kij共x兩x⬘^兲ª 1

2储x−x⬘^储

冋

^{␦ij+共xi−储xxi⬘−兲共xx}⬘^{j储−2xj⬘兲}

册

^.

This makes explicit the fact that the kernels of the two po- tentials are symmetric in their arguments. Insertion of the relations共22兲yields the potentials centered on the particles:

␾共x,t兲=兺bebK共x兩qb兲andAi共x,t兲=共1 /c兲兺bebKij共x兩qb兲q˙bj. In this form the action becomes

SD关q兴=

冕

^dt

再 兺

^{a T共q˙a兲+12a}

兺

^⫽beaeb

⫻

冋

^{K共qa兩qb兲+q˙aicq˙2bjKij共qa兩qb兲}

册 冎

^,

withT共q˙a兲ªmaq˙_a²/ 2 +maq˙_a⁴/ 8c². Using the symmetry of the kernels, so that ␦^qbl⳵K共qa兩qb兲/⳵^qbl=␦^qbl⳵K共qb兩qa兲/⳵^qbl

=␦^qal⳵K共qa兩qb兲/⳵^qalandKij=Kji, then the condition of sta- tionary variationdS_D关q_a+⑀␦^qa兴/d⑀兩_⑀=0= 0 gives

d dt

⳵T共q˙a兲

⳵^q˙a

= −ea

⳵␾共qa,t兲

⳵^qa

+ ea

c

⳵

⳵^qa

关q˙a·A共qa,t兲兴

−ea

c d

dtA共qa,t兲.

Using standard vector identities and the definitions ofE_L,E_T, andB in terms of potentials, this gives

d dt

⳵T共q˙兲

⳵^{q˙ =e共EL+ET兲+} e cq˙⫻B,

where we have dropped the particle label a. It remains to compute the derivatives of the kinetic termT. We find

d dt

⳵^T共q˙兲

⳵^q˙j

= d

dt

冋 冉

^{1 +2cq˙22}

冊

^mq˙j

册

^=mq¨iAij,

whereAijª␦ij+共q˙²/c²兲aijandaij=␦ij/ 2 +q˙iq˙j/q˙². Searching for the inverse ofAijto second order in q˙/cgivesA_ij⁻¹=␦ij

−共q˙²/c²兲aij. The equation of motion for theith component of any particle then becomes

mq¨_i=eE_i+e

c共q˙⫻B兲i−eq˙²

c²a_ijE_Lj 共25兲 to orderq˙²/c². This expression differs from some²¹so-called nonrelativistic implementations of Darwin by the presence of a term proportional to共q˙/c兲²ELj.

C. Vlasov–Darwin action

Following Low²² and Ye and Morrison,²³ we formulate the action principle for both Vlasov–Maxwell and Vlasov–

Darwin systems by modifying the usual action for Maxwell’s equations coupled to particles to account for a continuum of particles; i.e., the characteristics of Vlasov’s theory. In these actions, the particles and fields are conventionally varied in- dependently. However, in the case of the Vlasov–Darwin system, because the fields are instantaneously determined by particle states, there is a fairly simple action principle in terms of particle variations alone, and we present this also.

This latter action is a continuum action based on the idea of Darwin’s original Lagrangian. Lastly, we present the nonca- nonical Hamiltonian field description of the Vlasov–Darwin system.

Because the Darwin equations of motion 关Eq. 共25兲兴, as well as the corresponding fully relativistic equations,

mq¨i=

冑

^{1 −q˙}c²²

冉

^␦ij−q˙c^iq˙2j

冊 冋

^{eEj+ec共q˙⫻B兲j}

册

^{, 共26兲}

require an initial position and velocity, these may be used to label particles. However, the volume in these velocity phase space coordinatesd³qd³q˙is not conserved. For convenience, we seek volume preserving coordinates for these dynamics.

This is most naturally achieved by starting, in the relativistic case, from the standard fully relativistic Lagrangian for par- ticles in specified fields␾ andA,

LR= −mc²

冑

^{1 −q˙}c²²−e␾+e cA·q˙,

and from a corresponding Lagrangian in the case of the Dar- win approximation with an expansion that parallels that of Eq.共1兲. Using a standard Legendre transformation, we obtain the respective Hamiltonians

HR共q,p,t兲=mc²

冑

^{1 +}m¹²c²

冋

^{p−ecA共q,t兲}

册

^{2+e␾共q,t兲}

共27兲 and

HD共q,p,t兲= 1

2m

冉

^p−ecA

冊

²⁻8m¹³c²

冉

^p−ecA

冊

⁴

+e␾共q,t兲. 共28兲

The equations of motion for 共q,p兲 derived from these Hamiltonians obscure the elementary form of either the Lor- entz law or its approximate Darwin expression. However, we can perform the following noncanonical yet volume preserv- ing transformation:

P=p−e

cA共q,t兲, 共29兲

and obtain the equations of motion in the following familiar form:

q˙= P

m

冉

^{1 +}m^P22c²

冊

^−1/2,

共30兲 P˙=e

冋

^E+

冉

^{1 +mP22c2}

冊

^{−1/2mcP ⫻B}

册

^,

in the Lorentz case, and in the Darwin case q˙= P

m

冉

^{1 −}2m^P22c²

冊

^,

共31兲 P˙=e

冋

^E+

冉

^{1 −2mP22c2}

冊

^{mcP ⫻B}

册

^.

It can readily be verified that the volume in these phase space coordinates Zª共q,P兲 关as well as in the zª共q,p兲 coordi- nates兴 is conserved by the above dynamics. However, Eqs.

共30兲and共31兲are not in canonical Hamiltonian form because they arise from an action that is not of the standard phase space action form, but rather of the type

S关Z兴=

冕

t₀ t₁

关

^␪␮共Z,t兲Z˙^␮−H共Z,t兲

兴

^dt,

whereSis defined on pathsZ共t兲with appropriately fixed end points. Variation of this general form yields

␻␮␯Z˙^␯=H_,␮+⳵t␪␮,

where ␻_␮␯ª␪_␯,␮⁻␪_␮,␯ ^satisfies ␻_␯␮,␴⁺␻_␮␴,␯⁺␻_␴␯,␮^{= 0. If}

␻_␮␯ is invertible, then the equations of motion become Z˙^␯=J^␯␮共H_,␮+⳵t␪_␮兲, 共32兲 whereJ^␯␴␻␴␮=␦_␮^␯.

In the present setting, the noncanonical relativistic action takes the form

SR关q,P兴=

冕

^dt

冋 冉

^P+ecA

冊

^·q˙−mc2

冑

^{1 +mP22c2−e␾}

册

^,

while the noncanonical Darwin action becomes

SD关q,P兴=

冕

^dt

冋 冉

^P+ecA

冊

^{·q˙−2mP2 +8mP34c2−e␾}

册

^.

In both cases, we identify Zª共q,P兲 and ␪共Z,t兲ª共P +eA/c,0兲, and we note that␾⁼␾共q,t兲andA=A共q,t兲. The form␻, and henceJ, is the same in both cases,

␻⁼

冢

^{1ec3B⫻ij3 −0133⫻⫻33}

冣

^{, Jª}

冢

^{−013⫻33⫻3 1ec3⫻3Bij}

冣

withBijª⑀ijkBk. Such symplectic two-forms and the cosym- plectic dualJwere used by Littlejohn in the context of guid- ing center perturbation theory.²⁴ With the above expression forJ, Eq.共32兲is

冉

^q˙P˙

冊

⁼

冢

^{−013⫻33⫻3 1ec3⫻3Bij}

冣 冢 冉

^{1 +e⳵␾⳵qmP2+c22ec}

冊

^{⳵−1/2⳵At mP}

冣

in the relativistic case, reproducing Eq.共30兲; and

冉

^q˙P˙

冊

⁼

冢

^{−013⫻33⫻3 1ec3⫻3Bij}

冣 冢 冉

^{1 −e⳵␾⳵q2m+P2ec2c⳵2⳵A}

冊

^tPm

冣

in the Darwin case, reproducing Eq.共31兲.

Now, as alluded to above, instead of a discrete particle label a, we envision a particle trajectory passing through

every point of a six-dimensional phase space; i.e., a con- tinuum of particles. Natural continuum particle labels are the initial position and momentum,Z₀ª共q0,P₀兲; i.e., the trajec- toriesq共q0,P₀,t兲andP共q0,P₀,t兲, which we write compactly asZ共Z0,t兲, define an invertible map of the phase space onto itself. We assume that associated with each trajectory is a phase space number density 共distribution function兲 f₀共Z₀兲, which, like the initial conditions, is constant on the trajec- tory. To find the phase space density at a phase space obser- vation point ⌶ª共x,⌸兲 at time t, one needs to know the phase space density associated with the trajectory that will be there at that instant in time. Setting x=q共q₀,P₀,t兲 and ⌸

=P共q₀,P₀,t兲, or more succinctly,⌶=Z共Z₀,t兲, inverting, and substituting into f₀, gives

f共⌶,t兲d⁶⌶=f₀„Z₀共⌶,t兲…d⁶Z₀, 共33兲 which implies

f共⌶,t兲=

冨 冏

^{f0⳵⳵共ZZZ00兲}

冏 冨

^{Z0共⌶,t兲, 共34兲}

where 兩⳵^Z/⳵^Z0兩 is the Jacobian determinant, which will be seena posteriorito be unity.

The quantity f共⌶,t兲=f共x,⌸,t兲 is the usual phase space density of Vlasov’s theory, and Eq.共33兲 or 共34兲establishes the connection between the Lagrangian phase space trajecto- ries Z共Z0,t兲 and the Eulerian distribution function f共⌶,t兲.

Later we will find it convenient to use the canonical La- grangian phase space trajectories and their Eulerian counter- parts, so we record these here. The canonical trajectories are z共z₀,t兲ª关q共q₀,p₀,t兲,p共q₀,p₀,t兲兴, 共35兲 where we recall thatp=P+eA/c; corresponding to these tra- jectories is the Eulerian phase space observation point ␨ ª共x,␲兲, whence the Eulerian phase space density is given by

f共␨^,t兲d⁶␨^=f0†z₀共␨^,t兲‡d⁶z₀. 共36兲 The phase space action for the relativistic Vlasov–

Maxwell system has three parts:

SVR关q,P;␾,A兴=SVRp关q,P兴+SVRc关q,P;␾,A兴

+SVRf关␾,A兴; 共37兲

the first contains only the particle or the trajectory degrees of freedom, the second provides the coupling, and the third de- pends only on the fields. Specifically, these are

S_VRp=

冕

^dt

冕

^{d6Z0fR0共Z0兲}

冋

^P·q˙−mc2

冑

^{1 +mP22c2}

册

^,

SVRc=

冕

^dt

冕

^{d6Z0fR0共Z0兲}

冋

^{−e␾共q,t兲+ecA共q,t兲·q˙}

册

⁼

冕

^dt

冕

^{d6Z0fR0共Z0兲}

冕

^{d6⌶␦共⌶−Z兲}

冋

^{−e␾共q,t兲+ecA共q,t兲·q˙}

册

=

冕

^dt

冕

^{d6⌶fR共⌶,t兲}

冋

^{−e␾共x,t兲+ec兩A共x,t兲·q˙兩Z0共⌶,t兲}

册

^{, 共38兲}

SVRf= 1

8␲

冕

^dt

冕

^{d3x关E2−B2兴.}

Observe that the coupling term is written first in a form suit- able for variation with respect to q共Z0,t兲 and P共Z0,t兲, and last in a form suitable for variation with respect to ␾共x,t兲 andA共x,t兲. The middle form uses the Dirac delta function on the trajectories to show the equivalence of the two. After deriving the particle equations of motion from this action, we can replaceq˙兩Z₀共⌶,t兲, which occurs in the current, by an ex- pression in terms of the Eulerian variable⌶by using the first of Eqs.共30兲, and we can also show that the Jacobian兩⳵^Z/⳵^Z0兩 of Eq.共34兲is unity, as mentioned earlier.

Similarly, for the Vlasov–Darwin action SVD关q,P;␾,A兴=SVDp关q,P兴+SVDc关q,P;␾,A兴

+SVDf关␾,A兴, 共39兲

we write

SVDp=

冕

^dt

冕

^{d6Z0fR0共Z0兲}

冋

^{P·q˙−2mP2 +8mP34c2}

册

^,

SVDc=SVRc, 共40兲

SVDf= 1

8␲

冕

^dt

冕

^{d3x关E2−ET2−B2兴.}

In SVDf, the E²−E_T²−B² is shorthand for an expression in terms of 共␾,A兲, using the usual definitions given in Sec. IV A.

Variation ofSVR andSVDwith respect toq andP, with the usual condition ofqbeing fixed at initial and final times,

yields the equations of motion 共30兲 and 共31兲, respectively.

These equations together with Eq.共33兲imply the distribution functions satisfy the Vlasov equations

⳵^fR

⳵^{t +}

冉

^{1 +}m^⌸2c²²

冊

^−1/2⌸m·⳵^fR

⳵^x

+e

冋

^E+

冉

^{1 +m⌸2c22}

冊

^{−1/2mc⌸ ⫻B}

册

^·⳵⳵⌸^fR= 0 共41兲 in the relativistic case, and

⳵^fD

⳵^{t +}

冉

^{1 −}2m^⌸22c²

冊

^⌸m·⳵^fD

⳵^{x +e}

冋

^E