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arXiv:2108.07382v2 [math-ph] 22 Jun 2022 A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus

William Barham

1, Philip J. Morrison2, and Eric Sonnendr¨ucker3,4

1 Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin

2Department of Physics and Institute for Fusion Studies, TheUniversity of Texas at Austin

3Max-Plank-Institut f¨ur Plasmaphysik

4Technische Universit¨at M¨unchen, Zentrum Mathematik

June 23, 2022

Abstract

A Hamiltonian field theory for the macroscopic Maxwell equations with fully general po- larization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulationinto differential forms is discussed in detail. We choose to distinguish between straight and twisted differential forms so that all integrals be taken over densities (i.e. twisted top forms). This ensures that the duality pairings, which are stated as integrals over densities, areorientation independent. The relationship between functional differentiation with respect to vector fields and with re- spect to differential forms is established using the chain rule. The theory is developed such that the Poisson bracket is metric and orientation independent with all metric dependence contained in the Hamiltonian. As is typically seen in the exterior calculus formulation of Maxwell"s equations, the Hodge star operator plays a key role in modeling the constitutive relations. As a demonstration of the kind of constitutive models this theory accommodates, the paper concludes with several examples.

1 Introduction

A host of electromagnetic phenomena occur in polarized and magnetized media. As the rational of introducing polarization and magnetization amounts to the hiding complicated microscopic behavior in constitutive laws, the equations describing electromagnetism in a medium are often called the macroscopic Maxwell equations. Typically, an empirical linear model is used for these constitutive models. However, in many plasma models, it 1 is useful to consider a self consistent model that can account for more complex couplings between the material, e.g. a charged particle model, and thefields. A systematic theory for lifting particle models to kinetic models and the Hamiltonian structure of these lifted models was given in [14]. It has been shown that many kinetic models of interest fit into this framework such as guiding center drift kinetics [14] orgyrokinetics [5]. Frequently, such models include nonlinear field dependent polarizations. Recently, structure preserving discretizations which replicate aspects of the continuous Hamiltonian structure in a discrete system have been a popular area of research [15]. In particular, such a discretization was found for the Maxwell-Vlasov equations [10]. These structure preserving methods generally rely on the characterization of Maxwell"s equations using differential forms (e.g. [8], [1], and [11]). The Maxwell component of the model given in [14] involves a somewhat complicated expression for the relationship between the polar- ization and magnetization with the electric and magnetic fields through functional deriva- tives of an energy functional. This adds difficulty when incorporating this model into the geometric language of differential forms necessary for structure preserving discretization. In particular, some care is needed when interpreting functional derivatives with respect to differential forms. This work seeks to provide a sound geometric interpretation of the macroscopic Maxwell equations and their Hamiltonian structure. A structure preserving discretization of the macroscopic Maxwell equations in Hamiltonian form is given in the accompanying paper [2]. The macroscopic Maxwell equations obtained from [14] are a Hamiltonian field theory. This paper seeks to express this Hamiltonian field theory in terms of differential forms. Hence, it follows that one must investigate the meaning of variational derivatives with respect to differential forms. In order to properly understand derivatives with respect to differential forms, it is necessary to give some attention to notions of duality in the double de Rham complex. The most obvious notion of duality is theL2inner product which pairs k-forms with themselves. Alternatively, one may use Poincar´e duality to pair differential k-forms with orientation dependent twisted (n-k)-forms (a pairing which does not utilize a metric). This leads to two alternative identifications of the functional derivative which are Poincar´e duals of each other through the Hodge star operator. Using the split exterior calculus formalism presented in [6], we obtain a Hamiltonian model for the macroscopic Maxwell equations which is orientation independent and whose Poisson bracket is explicitly metric independent. This manner of expressing a Hamiltonian theory is believed to be advantageous for discretization using the novel split Hamiltonian finite element framework [3] or the split mimetic spectral element framework developed in [2]. The resulting Hamiltonian theory is easily converted into a format based only on the primal complex, but certain features (e.g. explicit metric independence of the Poisson bracket) are lost in this conversion. Treatment of linear constitutive relations in electromagnetism through Poincar´e duality is also found in [8]. Finally, we conclude by considering several motivating examples of constitutive models that that this framework accommodates. 2

2 The double de Rham complex and Maxwell"s equationsIn this section, we discuss the basic objects of exterior calculus and use them to translate

Maxwell"s equations from the language of vector calculus toexterior calculus.

2.1 The macroscopic Maxwell equations

Maxwell"s equations may be written in two equivalent formats: ? ·D= 4πρf, ? ·B= 0, -c? ×E=∂B ∂t, c? ×H=∂D ∂t+ 4πJf,or? ·E= 4πρ ? ·B= 0 -c? ×E=∂B ∂t c? ×B=∂E ∂t+ 4πJ.(1) The left column is frequently called the macroscopic Maxwell equations, and the right column are just the standard Maxwell equations. One relatesthe displacement field,D, and the magnetic field intensity,H, with the electric field,E, and magnetic field,B, by

D=E+ 4πPandH=B-4πM(2)

wherePis the polarization andMis the magnetization. Moreover,ρfandJf, the free charge and free current densities respectively, are related toρandJby

ρ=ρb+ρfandJ=Jb+Jf(3)

where b=-? ·PandJb=Jp+Jm=∂P ∂t+c? ×M.(4) It is straightforward to show that the two forms of Maxwell"sequations are equivalent using these definitions. The utility of the macroscopic Maxwell equations is that the terms which couple Maxwell"s equations to a matter model,ρfandJf, are prescribed by freely propagating charged particles rather than the, possibly complicated, bound charges and currents in the medium. These microscopic features bound tothe medium are instead encoded in constitutive relations forPandM. This paper is concerned with combining two distinct modeling frameworks to study this system. First, we translate this system into the language of split exterior calculus [6]. Second, we translate the Hamiltonian structure of this model, discovered in [14], into the language of split exterior calculus. We shall see that this approach helps to elucidate the rich structure of these equations. 3

2.2 The double de Rham complexThe modeling framework employed in this paper utilizing thedouble de Rham complex

is inspired by the split exterior calculus framework of [6].Discussion of the differential geometric framework is kept to a minimum here, and the readeris referred to the afore- mentioned split exterior calculus paper and to refereces such as [7] and [12]. All notation used herein will either be standard as found in the previous references, or shall use notation defined below. Let (Ω,g) be a Riemannian manifold of dimensionn. Throughout this paper, we assume either periodic, or homogeneous boundary conditions. Let{(Λk,dk)}nk=0be the

vector spaces of differential forms on Ω. We may define a second complex,{(˜Λk,˜dk)}nk=0,

called the complex of twisted differential forms. This dual complex differs from the first in that twisted forms change sign under orientation changing transformations. The two complexes are related to each other through the Hodge star operator. Diagrammatically, this is given by

···ΛkΛk+1···

˜Λn-k˜Λn-(k+1)···

dk

˜dn-(k+1)

(5) While the primal de Rham complex of straight differential forms,{Λk}nk=0, is a familiar

object in differential geometry, the dual complex of twisted differential forms,{˜Λk}nk=0would benefit from further discussion. Let{∂i}ni=1be a coordinate system on theTΩ,

the tangent space of Ω. We define an operatoro(∂), which encodes the orientation of the manifold, such that it transforms as

o(T∂) =o(T∂1,...,T∂n) = sign(det(T))o(∂1,...,∂n) = sign(det(T))o(∂).(6)

We may consider twisted differential forms to be straight forms which have been multiplied by this operator. Because the orientation rarely plays a role in the algebraic manipulation of differential forms, for conciseness, we suppress any further use of this operator to the background. We can see that the sign of a twisted form depends on the parityof the coordinate system of the ambient space. Therefore, integration over a setU?Ω of a twistedn- form will return a value which is independent of the orientation of space. Hence, densities are modeled as twisted forms. As an example, the volume form,thatn-form such that

vol(∂1,...,∂n) = 1 for any orthonormal frame{∂i}ni=1, is a twisted form. We shall typically

denote a straightk-form asωkand a twistedk-form as ˜ωk. An important feature of the split exterior calculus is two distinct notions of duality on the double de Rham complex. First, we have the standardL2inner product, (·,·) : 4

Vk×Vk→R, which is defined

(ωk,ηk) =? g x(ωk,ηk)voln(7) wheregxis the pointwise inner product onk-forms induced by the Riemannian metric.

The second notion of duality is Poincar´e duality,?·,·?:VkטVn-k→R, which is defined

ωk,˜ηn-k?

ωk?˜ηn-k.(8)

TheL2inner product, because of its dependence on the Riemannian metric and volume form, is a metric dependent quantity. On the other hand, the Poincar´e duality pairing, built from the wedge product structure alone, is purely topological. Moreover, as both duality pairings are expressed as an integral of a twistedn-form, they are independent of the orientation of the coordinate system. Finally, the Hodge star operator?:Vk→˜Vn-kis defined such that

ωk,ηk?

ωk,?ηk?

.(9) Note that the Hodge star is not a single operator, but rather afamily of operators. A more precise notation might be?n-k,k:Vk→˜Vn-k, however we opt for the more concise notation where confusion is unlikely. These notions of duality are essential for discussing the Hamiltonian structure of the macroscopic Maxwell equations.

2.3 The correspondence between vector calculus and exterior calculus

We denote the space of vector fields on Ω byX. Vector fields and 1-forms are isomorphic to each other. The index lowering operator or flat operator (·)?:X→Λ1is defined by v

1=g(·,V) =gijVjdxi:=V?.(10)

The inverse operation is called the index raising operator or sharp operator (·)?: Λ1→X:

V=g?(·,v1) =gijv1j∂

∂xi:= (v1)?(11) whereg?is the dual metric on the covectors. We may likewise define an isomorphism between vector fields and twisted (n-1)-forms. We definei(·)voln:X→˜Λn-1by

˜vn-1=iVvoln=?

iV i? det(g)dx1?...??dxi?...?dxn(12) where the hat symbol means omission of "dxi" from the wedge product andiVαis the interior product. It is possible to show thatiVvoln=?V?. Hence, the inverse operation is given by

V=??˜vn-1??.(13)

5 Many physically significant quantities are orientation dependent. The magnetic field and vorticity of a fluid are notable examples of so called pseudovectors. IfVis a pseudovec- tor, then one can see that ˜v1=V?is twisted whilevn-1=iVvolnis straight. Hence, pseudovectors are naturally identified with straight (n-1)-forms. It is helpful to establish the following adjoint relationships between the isomorphisms we just defined. We define theL2inner product of vector fields to be (W,V) =? M

W·Vvoln

whereW·V=WigijVj. Proposition 1.With respect to the duality pairing betweenΛ1and˜Λn-1and theL2-inner product onX,? (·)???=?i(·)voln?-1and? (·)???=i(·)voln.(14)

Proof:Considerv1=V?and ˜wn-1=iWvoln. Then

˜wn-1,v1?=?

M

˜wn-1?v1=?

M i

Wvoln?V?=?

M

W·Vvoln= (W,V).

Therefore, since ˜wn-1=?W?=iWvoln??W= (?˜wn-1)?,

˜wn-1,V??

(?˜wn-1)?,V? =??i(·)voln?-1˜wn-1,V? and similarly ?v1,iWvoln?

1,˜Λn-1=?

(v1)?,W? L 2(M).

2.4 Vector calculus inR3and differential forms

We now turn our attention to vector calculus inR3. Exterior differentiation is inherently metric independent whereas it is well known that vector calculus operators such as the gradient, divergence, and curl depend on one"s coordinate system - often in a very com- plicated manner. By defining a differential form related to a vectorial quantity, one may obtain simpler expressions for their derivatives which disentangle the metric dependent and independent portions of the operation. The following definitions allow us to reconstruct the standard vector calculus operations from the exterior derivative. Iff: Ω→Rbe a scalar field on a Riemannian manifold, its gradient and exterior derivative are related to each other via df= (?f)??? ?f= (df)?.(15) 6 LetVbe a vector field andv1=V?. Then the curl ofVis defined by dv1=i?×Vvol3=?(? ×V)??? ? ×V= (?(dv1))?.(16) Letting ˜v2=iVvol3, then the divergence is defined to be d˜v2=diVvol3=£Vvol3= (? ·V)vol3=?(? ·V)?? ? ·V=?d˜v2(17) where£Vis the Lie derivative. The divergence of a vector field physically corresponds with the the change in the volume element as it is dragged by that vector field. It is relatively straightforward to show that these formulas obtain the correct expressions for the gradient, divergence, and curl in Euclidean space and standard curvilinear coordinate systems (e.g. cylindrical and spherical coordinates). We next establish a relationship between the wedge product of differential forms and the typical vector and scalar products inR3. Proposition 2.LetU,V,W?Xand letu1,v1,w1be their corresponding1-forms. Then ?(u1?v1?w1) =vol3(U,V,W).(18)

Proof:

u

1?v1?w1=?

ijk(uidxi)?(vjdxj)?(wkdxk) =? ijku ivjwkdxi?dxj?dxk =??(u1?v1?w1) =? det(g)? iProposition 3.u1?˜v2=U·Vvol3.

Proposition 4.u1?v1=iU×Vvol3.

Proof:A geometrically intuitive definition of the cross product isthatU×Vis the unique vector such that?W?X,

U×V·W=vol3(U,V,W).

But we already know that

vol

3(U,V,W) =?u1?v1?w1??u1?v1?w1=U×V·Wvol3=iU×Vvol3?w1.

according to proposition 3. Hence, ?u1?v1-iU×Vvol3??w1= 0?w1?Λ1??u1?v1=iU×Vvol3. This implies that ifUandVare standard vectors, thenU×Vis a pseudovector. 7

2.5 The macroscopic Maxwell equations in the language of split exterior

calculus We now have the machinery to directly translate the standardvector calculus formulation of Maxwell"s equations into the language of split exterior calculus. Let e If we transform the equations by applying the appropriate isomorphisms as follows: c? ×E-∂B ∂t? = 0 ?(? ·B) = 0?? c? ×H-∂D∂t-4πJf? = 0 ?(? ·D-4πρf) = 0. Then, we may write the equations in either of two different formats: d

˜d2= 4π˜ρ3f,

db2= 0, -cde1=∂b2 ∂t, cd˜h1=∂˜d2 ∂t+ 4π˜j2f,ord?e1= 4π˜ρ3 db2= 0 -cde1=∂b2 ∂t cd?b2=∂ ?e1 ∂t+ 4π˜j2.(20)

The displacement field twisted 2-form and magnetic intensity field twisted 1-form,˜d2and˜h1respectively, are related to the electric field 1-form,e1, and magnetic field 2-form,b2,

by˜d2=˜?1e1+ 4π˜p2and˜h1=˜?2b2-4π˜m1(21) where ˜p2=iPvol3is the polarization twisted 2-form, and˜m1=M?is the magnetization twisted 1-form. Moreover, the free charge twisted 3-form and free current twisted 2-form, ˜ρ3fand˜j2frespectively, are related to ˜ρ3and˜j2by ˜ρ3= ˜ρ3f+ ˜ρ3band˜j2=˜j2f+˜j2b wherequotesdbs_dbs47.pdfusesText_47

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