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1 Introduction 3

2 Mathematical context 4 2.1 Functionalderivatives ...... 5 2.1.1 The Fr´echet and Gˆateaux derivatives ...... 5 2.1.2 Chainrule...... 6 2.1.3 Functional chain rule examples ...... 7 2.2 Exteriorcalculus ...... 8 2.2.1 Tangent space and cotangent space ...... 8 2.2.2 Differentialforms...... 11 2.2.3 Orientation and orientability of manifolds ...... 14 2.2.4 Twisted forms: a dual space of differential forms ...... 15 2.2.5 A correspondence between vectors and differential forms...... 18 2.2.6 The adjoint of the flat and sharp operators ...... 19 2.2.7 Vector calculus in R3 anddifferentialforms ...... 20 2.3 Functional calculus with respect to differential forms ...... 23 2.3.1 First variation with respect to a differential form ...... 23 2.3.2 Maps between differential forms ...... 25 2.3.3 Second variation with respect to a differential form ...... 26 2.3.4 Partial functional derivatives ...... 29 2.3.5 Change of variables from (E, B) to (D, B) ...... 31

3 Geometrized macroscopic Maxwell equations 33 3.1 Macroscopic Maxwell equations ...... 34 3.1.1 Hamiltonianstructure ...... 35 3.2 3D geometrized Maxwell equations ...... 36 3.2.1 Hamiltonianstructure ...... 38 3.2.2 Derivatives of the Hamiltonian ...... 38 3.2.3 Equations of motion and Casimir invariants ...... 40 3.3 On the Poisson bracket: the correspondence between the two versions . . . 41 3.4 Theoryusingonlystraightforms ...... 43 3.4.1 Codifferentialoperator...... 43 3.4.2 Hamiltonian structure with straight forms ...... 43 3.5 1D geometrized Maxwell equations ...... 44 3.5.1 1DMaxwellequations ...... 44 3.5.2 Differentialformsin1D ...... 45 3.5.3 1D Maxwell’s equations as differential forms ...... 47

2 4 Example energy functional 49 4.1 Linearmodelforpolarization ...... 49 4.2 Linear polarization with dependence on derivatives ...... 50

5 Conclusion 53

6 Acknowledgements 55

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 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 the fields. A systematic theory for lifting particle models to kinetic models and the Hamiltonian structure of these lifted models was given in [13]. It has been shown that many kinetic models of interest fit into this framework such as guiding center drift kinetics [13] or gyrokinetics [4]. 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 [14]. In particular, such a discretization was found for the Maxwell-Vlasov equations [9]. These structure preserving methods generally rely on the characterization of Maxwell’s equations using differential forms (e.g. [7], [1], [10]). The Maxwell component of the model given in [13] 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 [13] 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. Given a Banach space X, the variational derivative of a functional on this space, e.g. K : X R, lives in the dual space X∗. Hence, in order → to properly understand derivatives with respect to differential forms, it is necessary to

3 give some attention to the pairing between differential forms and linear functionals on differential forms. One may consider spaces of differential forms on a Riemannian manifold to be self-dual through the metric dependent L2-inner product. Alternatively, one may use Poincar´eduality to pair differential k-forms with orientation dependent twisted (n k)- − forms (a pairing which does not utilize a metric). These two alternative identifications of the functional derivative are related to each other through the Hodge star operator and are thus Poincar´eduals of each other. Inspired by the formalism presented in [5], representation of functional derivatives via Poincar´eduality is taken in this paper because it provides a richer description of Maxwell’s equations. In particular, the use of twisted differential forms yields a Hamiltonian field theory which is orientation independent and whose Poisson bracket is explicitly metric independent. Each variable is associated with a space of differential forms on either the primal de Rham complex of straight differential forms or the dual de Rham complex of twisted differential forms. These two complexes are related to each other through the Hodge star operator. 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 the constitutive relations in electromagnetism through Poincar´eduality is also found in [7]. While we largely refrain from discussion of rigorous functional analytic concerns in order to focus discussion on the Hamiltonian structure of Maxwell’s equations in a geometric context, [7] discusses the appropriate rigorous mathematical setting in greater detail. We begin with a discussion of variational calculus (2.1) and differential geometry (2.2) to fix notation. The relationship between functional differentiation with respect to vector fields and with respect to differential forms is given (2.3). With the mathematical con- text thus established, the Hamiltonian structure of the macroscopic Maxwell equations is presented. First, in terms of vectors (3.1) and subsequently in terms of differential forms (3.2). The structure of the Poisson bracket is then discussed in greater detail (3.3). Next the Hamiltonian structure previously developed using both straight and twisted forms is rewritten in a format that only utilizes straight forms (3.4). An analogous discussion is presented for a simple one-dimensional version of Maxwell’s equations (3.5). Finally, we discuss several example energy functionals (4).

2 Mathematical context

In this section, we present a brief overview of the mathematical tools needed to develop a Hamiltonian theory for the macroscopic Maxwell equations using the language of differen- tial forms.

4 2.1 Functional derivatives We begin by considering functional derivatives on a general Banach space. We will later specialize to the case of derivatives on spaces of differential forms.

2.1.1 The Fr´echet and Gˆateaux derivatives Definition 1. Suppose that X and Y are normed linear spaces. Let B(X,Y ) be the space of bounded linear operators from X to Y . Let U X. Consider Φ : U Y . Let x U ⊂ → ∈ and h U. We say that Φ is Fr´echet differentiable at x if there exists a bounded linear ∈ operator D Φ[x] B(X,Y ) such that X ∈ Φ[x + h] Φ[x] D Φ[x]h k − − X kY 0 as h 0. (1) h → k kX → k kX

We call DX Φ[x] the Fr´echet derivative of Φ at x. If Φ is Fr´echet differentiable at every x U, then we say that Φ is differentiable on U. Since ∈ we assumed that the derivative is a bounded linear operator, this means that Φ C1(X,Y ). ∈ If we further assume that Φ is a bijection between U and V Y and D Φ−1(y) is exists, ⊂ Y then Φ is a C1-diffeomorphism between U and V .

Definition 2. We define the Gˆateaux derivative, D Φ[x] : U Y at x, to be the X → bounded linear operator D Φ[x] B(X,Y ) such that for t R 0 and h X X ∈ ∈ \ { } ∈ 1 Φ[x + th] Φ[x] tD Φ[x]h 0 as t 0. (2) t k − − X kY → → If Φ is Fr´echet differentiable, then it is Gˆateaux differentiable, but the reverse is not nec- essarily true. For our purposes throughout this article, we assume that Φ is Gˆateaux differentiable. Notice, if f : X R, then D f[x] X∗, the space of bounded linear functionals on → X ∈ X. If a duality pairing , ∗ has been specified, we use the following notation: h· ·iX ,X δf DX f[x]h = , h . (3) δx ∗  X ,X Throughout this article, the notation on the left side of the equality is used to emphasize that the quantity is independent of the duality pairing used, whereas the notation on the right is dependent on our choice of duality pairing. This will become useful in situations where there are multiple convenient identifications of the dual space.

5 2.1.2 Chain rule Theorem 1. (Chain Rule) Let X,Y,Z be normed linear spaces. Let U X and V Y ⊂ ⊂ be open. Suppose f : U Y and g : V Z. Let y = f[x] V . Suppose g is Fr´echet → → ∈ differentiable at y and f is Gˆateaux (Fr´echet) differentiable at x. Then g f is Gˆateaux ◦ (Fr´echet) differentiable at x and

D (g f)[x]= D g[y] D f[x]. (4) X ◦ Y ◦ X Corollary 1. Let X,Y be normed linear spaces. Let Φ : X Y be a C1-diffeomorphism. → Then −1 −1 (DX Φ[x]) = DY Φ [y]. (5) Proof: id = D (Φ−1 Φ)[x]= D Φ−1[y] D Φ[x]. (6) X X ◦ Y ◦ X

Corollary 2. Let X,Y be normed linear spaces. Let K : Y R and let Φ : X Y be a → → C1-diffeomorphism. Suppose y = Φ[x]. Then

D K[y] = ((D Φ[x])−1)∗D (K Φ)[x]. (7) Y X X ◦ That is, if h Y , ∈

D K[y], h ∗ = ((D Φ[x])−1)∗D (K Φ)[x], h ∗ h Y iY ,Y h X X ◦ iY ,Y = D (K Φ)[x], (D Φ[x])−1h ∗ . h X ◦ X iX ,X Proof: By the chain rule,

D (K Φ)[x]= D K[y] D Φ[x] X ◦ Y ◦ X Let k X. Let h Y such that h = D Φ[x]k. Thus, ∈ ∈ X D (K Φ)[x], k ∗ = D K[y], D Φ[x]k ∗ = D K[y], h ∗ . h X ◦ iX ,X h Y X iY ,Y h Y iY ,Y −1 But since k = (DX Φ[x]) h, it follows that

D K[y], h ∗ = D (K Φ)[x], (D Φ[x])−1h ∗ h Y iY ,Y h X ◦ X iX ,X = ((D Φ[x])−1)∗D (K Φ)[x], h ∗ . h X X ◦ iY ,Y

6 2.1.3 Functional chain rule examples Example 1. (functional derivative of object in the dual space) Let (X, ) be a normed k · kX linear space and let X∗ be its dual. Let f, g : X R. Moreover, assume that the space → is reflexive so that X∗∗ = X. Denote their Fr´echet derivative at v X in the direction ∈ u X by ∈ δf DX f[v]u = , u . δv ∗  X ,X If (X, ( , ) ) is an inner product space, then there exists a linear isomorphism R : X∗ X · · X → such that u, v ∗ = (Ru, v) . h iX ,X X Hence, if f ∗ : X∗ R, and we define → δf ∗ ∗ ∗ ∗ ∗ ∗ DX f [v ]u = u , ∗ , δv ∗  X ,X where we have used reflexivity of X, then if we let u = Ru∗, v = Rv∗ and f = f ∗ R−1, ◦ then by the chain rule, we have

∗ ∗ ∗ DX∗ f [v ]u = DX f[v]u.

Hence,

∗ ∗ δf δf u , ∗ = , u δv ∗ δv ∗  X ,X  X ,X δf ∗ δf δf δf ∗ = Ru∗, = R , u = R = . (8) ⇒ δv∗ δv ⇒ δv δv∗  X  X

The above makes it clear that if one is in Hilbert space, it makes little difference whether one defines the functional derivative with respect to the duality pairing , ∗ or the h· ·iX ,X inner product ( , ) as the two representations are related via the Riesz map. Likewise, · · X the functional derivative of a functional on the dual space may be recast on the primal space via the Riesz representation.

Example 2. (functional derivative with respect to gradient) Let K : C∞(Rn, Rn) R → such that g C∞(Rn, Rn), f C∞(Rn) such that K[g] = K[ f]. Note that g ∀ ∈ ∃ ∈ ∇ ∀ ∈ C∞(Rn, Rn) and f C∞(Rn), we have ∈

(g, f) 2 Rn Rn = ( g,f) 2 Rn . ∇ L ( , ) − ∇ · L ( )

7 So, we find that, the L2 adjoint of the gradient is the negative divergence. Let Kˆ = K ◦∇ so that Kˆ [f]= K[ f]. The chain rule implies ∇ δKˆ [f] δK[ f] DKˆ [f]ξ = ,ξ = ∇ , ξ δf δ f ∇ 2 Rn Rn !L2(Rn)  ∇ L ( , ) δKˆ [f] δK[ f] = = ∇ . (9) ⇒ δf −∇ · δ f ∇

Comment 1. This example assumed infinite regularity and an infinite domain. However, with some delicacy, this result may be extended to spaces on a finite domains (with appro- priate boundary conditions) and limited regularity. For the purposes of this paper, we do not need to generalize to finite domains or situations with limited regularity. However, we will later generalize this result to obtain an analogous expression for differential forms.

2.2 Exterior calculus In what follows, only enough exposition is given to fix our notation. Nearly all proofs are omitted since they may be found in most differential geometry books. For a more complete discussion of exterior calculus, see [6] [11].

2.2.1 Tangent space and cotangent space Definition 3. Given a set M, a chart on M is a an ordered pair (U, φ) where U M and ⊆ φ : U φ(U) Rn is a bijection. Given x M, we call the vector φ(x) = (x1, ,xn) → ⊂ ∈ · · · the coordinates of x.

Definition 4. A differentiable n-manifold is a set M, along with a collection of coordi- nate charts (U , φ ) such that, given two charts (U, φ ), (V, φ ) such that U U = ∅, α Uα Uα Uβ α ∩ β 6 φ (U U ) and φ (U U ) are open subsets of Rn and Uα α ∩ β Uβ α ∩ β φ φ−1 : φ (U U ) φ (U U ) Uα Uβ Uβ α β Uα α β ◦ φU (Uα∩Uβ) ∩ → ∩ β and −1 φUβ φU : φUα (Uα Uβ) φUβ (Uα Uβ) ◦ α φUα (Uα∩Uβ) ∩ → ∩ ∞ are C . Moreover, the integer n is called the dimension of M. We may define the space C∞(M) to be the space of functions f : M R such that → f φ−1 C∞ for every coordinate chart φ. A curve on a manifold is simply a function ◦ ∈ c : R M. →

8 Definition 5. Given two curves c , c : R M, they are called equivalent at x M if 1 2 → ∈ d d c (0) = c (0) = x and (φ c ) = (φ c ) . 1 2 dt ◦ 1 dt ◦ 2 t=0 t=0

This defines an equivalence class of curves on the manifold. This equivalence class is called the tangent space to M at x, denoted T M. An element X T M is called a tangent x ∈ x vector. Given a coordinate chart such that φ(x) = (x1, ,xn) and a curve c which gives · · · rise to X T M as described above, the coordinates of X are ∈ x d Xi = (φ c)i . dt ◦ t=0

i n Given a local coordinate system on M, x , TxM has basis vectors that we shall denote { }i=1 ∂ n . This basis will also frequently be denoted { i}i=1 n ∂ i ∂ ∂ = = X TxM is written in coordinates as X = X . (10) i ∂xi ⇒ ∈ ∂xi Xi=1 Definition 6. The collection

T M = (x,y) : x M y T M { ∈ ∈ x } is called the tangent bundle of M.

Definition 7. If X : M T M is a smooth map, then we will call this a smooth vector → field. We will denote the space of smooth vector fields on M by X(M).

Smooth vector fields of this space will frequently be denoted with capital letters. We may equip a manifold with a pointwise inner product on the tangent space.

Definition 8. A Riemannian manifold is an ordered pair (M, g) where M is a manifold, and g, called the Riemannian metric, is a pointwise inner product on the tangent space. Since g depends on the base point in M, we shall frequently denote it gx. Hence, given x M, g : T M T M R linear, symmetric, and positive definite. ∈ x x × x → This inner product may be written for X,Y X(M), ∈ n g(X,Y )= X Y = Xig Y j (11) · ij Xi=1 where the second expression is in local coordinates. Occasionally, one uses the notation g = gx in order to make dependence of the metric on the base point in M explicit.

∗ Definition 9. The dual space to TxM is called the cotangent space and is denoted Tx M.

9 In local coordinates, it has a basis dxi n which is dual to ∂ n . Hence, dxi, ∂ = δi { }i=1 { i}i=1 h j i j and n ω T ∗M = ω = ω dxi. (12) ∈ x ⇒ i Xi=1 Definition 10. The collection

T ∗M = (x,y) : x M y T ∗M { ∈ ∈ x } is called the cotangent bundle. Analogous to the space of smooth vector fields, which were smooth maps M T M, we → may define a collection of smooth maps M T ∗M. → Definition 11. The space of differential 1-forms to be smooth maps ω : M T ∗M. → We will use Λ1(M) to denote this space. We typically denote elements of Λ1(M) with greek letters or lowercase letters. If X X(M) ∈ and ω Λ1(M), then ∈ ω(X)= ω X = ω, X = ω Xi (13) · h i i Xi are all notations for evaluating ω at X. Notice that the value of this expression depends on the base point x M on the manifold. Hence, Λ1(M) may be thought of either as a ∈ smooth map from M to T ∗M or a map sending X(M) to smooth functions on M. ij Definition 12. Because gij is positive definite, it has an inverse: g . That is,

kj j ik i gikg = δi and g gkj = δj. Hence, we may define an inner product on Λ1(M) as follows:

n ω, η Λ1(M) = g−1(ω, η)= ω η = ω gijη . (14) ∈ ⇒ · i j Xi=1 where the notation g−1 is used to emphasize that the matrix representation of the metric on 1-forms is the inverse of that on vector fields. One can see that the dot product notation can mean different things depending on its arguments. It can mean a pairing between a vector and a covector, the inner product of two vectors, or the inner product between two covectors. Context will usually make the intended meaning obvious. In coordinates, we may write

g( , )= g dxi dxj · · ij ⊗ and ∂ ∂ g−1( , )= gij . · · ∂xi ⊗ ∂xj

10 Definition 13. Given f : M R, we may define its differential by → n ∂f df(x) v = vi (15) · ∂xi Xi=1 where v X(M). Hence, df Λ1(M). ∈ ∈ Notice that if we consider the basis vectors of a vector field to be differential operators, then n ∂f X[f]= Xi = df(x) X. ∂xi · Xi=1 2.2.2 Differential forms Definition 14. For k 1, the space of differential k-forms on M, Λk(M), is defined ≥ to be the space of smooth, anti-symmetric, k-linear forms on T M. That is, ωk Λk(M) x ∈ if ωk :T M T M R x ×···× x → n times where for any permutation of σ : 1,...,k 1,...,k , and any X k T M, we {| }{z → { } } { i}i=1 ⊂ x have k k ω (X1,...,Xk)= sign(σ)ω (Xσ(1),...,Xσ(k)). We define Λ0(M) to be the space of scalar functions on M. For k < 0 and k>n, we define Λk(M)= ∅.

In the previous section, we defined Λ1(M). Notice that the definition here is consistent with our previous definition. When it is helpful we will frequently indicate a differential k-form with a superscript: ωk Λk(M). However, to avoid notational clutter this superscript will ∈ be used only when confusion is possible or multiple kinds of differential forms are at play.

Proposition 1. Given a manifold M with dim(M)= n, Λk(M) is a vector space and

n dim(Λk(M))) = . (16) k   Definition 15. The wedge product is a mapping : Λk(M) Λl(M) Λk+l(M) with ∧ × → the following properties:

1. Linearity: ω (aη + bν)= aω η + bω ν ∧ ∧ ∧ 2. Anticommutativity: ωk ηl = ( 1)klηl ωk ∧ − ∧ 3. Associativity: ω (η ν) = (ω η) ν. ∧ ∧ ∧ ∧

11 Note, anticommutativity plus linearity implies bilinearity. Linearity also implies distribu- tivity of scalar multiplication. Finally, anticommutativity implies that ωk ωk = 0 when ∧ k is odd or when k >n/2. Recall, the canonical basis for Λ1(M) was denoted dxi n . Hence, the canonical basis { }i=1 for Λk(M) is

dxi1 dxik : i k 1,...,n and i = i for l = m . (17) ∧···∧ { l}l=1 ⊂ { } l 6 m 6 n o Comment 2. One can see that the differential defined in the previous section is an opera- tion taking 0-forms to 1-forms: d : Λ0(M) Λ1(M). This idea may be generalized to all 0 → k-forms.

Definition 16. An exterior derivative: d : Λk(M) Λk+1(M) which satisfies the k → following:

1. For f Λ0(M), df Λ1(M) is the differential of f. ∈ ∈ 2. The spaces (Λk(M), d ) form a cochain complex: i.e. d d = 0. k k+1 ◦ k 3. The exterior derivative satisfies the Leibniz rule:

d(ωk ηl)= dωk ηl + ( 1)kωk dηl. (18) ∧ ∧ − ∧ Although the above definition does not directly give a formula to compute the exterior derivative of a general ωk Λk(M), one may recursively compute the exterior derivative ∈ using its three defining characteristics. We will typically omit the k-subscript from the exterior derivative notation, but it is important to remember that d : Λk(M) Λk+1(M) → refers to a hierarchy of operators acting on a sequence of interrelated vector spaces.

Definition 17. A differential form ωk such that ωk = dηk−1 is called exact. A differential form such that dωk = 0 is called closed.

Because the exterior derivative is a nilpotent operator, it is clear that every exact differential form is closed. The reverse is not true in general; however, the Poincar´elemma tells us that on a contractible manifold, all closed differential forms are exact.

Definition 18. Let φ : M N be a smooth mapping between two manifolds of dimension → n. The pullback, denoted φ∗ of a differential k-form is given by

φ∗ωk(v , . . . , v )= ωk (D φ v ,...,D φ v ) (19) x 1 k φ(x) x · 1 x · k where the x-subscript is included to emphasize dependence on the base point x M and ∈ φ(x) N. ∈

12 Proposition 2. Let ωk, ηk Λk(N) and let φ : M N be a smooth mapping between ∈ → differential n-manifolds. Then the pullback φ∗ has the following properties: 1. Linearity: φ∗(ωk + ηk)= φ∗ωk φ∗ηk ∧ 2. Algebra homomorphism: φ∗(ωk ηk) = (φ∗ωk) (φ∗ηk) ∧ ∧ 3. Composition rule: (φ φ )∗ = φ∗ φ∗. 1 ◦ 2 2 ◦ 1 Proposition 3. Let φ : M N be a smooth mapping between differential k-manifolds → and let ωk Λk(N). Then ∈ ωk = φ∗ωk. (20) Zφ(M) ZM Therefore, it is through the pullback that one changes coordinates. ∗ k ∗ k Proposition 4. The exterior derivative commutes with the pullback: dk(φ ω )= φ (dkω ). Hence, we find that the exterior derivative is independent of our choice of coordinate system. Theorem 2. (Stokes’ theorem) Let ω Λk(M) and let Σ be a manifold of dimension ∈ k + 1 with boundary ∂M. Then dω = ω. (21) ZΣ Z∂Σ Hence, with respect to the pairing between a differential k-form and a k-manifold defined by integration Stokes’ theorem implies that the exterior derivative and the boundary operator are in a certain sense adjoints of each other. Definition 19. The interior product of a vector field, X, with a differential k-form is a map i : Λk(M) Λk−1(M) such that X → k k (iX ω )x(v2, . . . , vk)= ωx(X(x), v2, . . . , vk). (22) Anti-commutativity of the wedge product immediately implies: i (ωk ηl) = (i ωk) ηl + ( 1)kωk (i ηl). X ∧ X ∧ − ∧ X Definition 20. Let X (M). Then ϕ is the flow of X if ∈X t d ϕ = X(ϕ ). dt t t Definition 21. The Lie derivative of a k-form ωk is d £ ωk = ϕ∗ωk . (23) X dt t t=0

Proposition 5. (Cartan’s magic formula) Let ω Λk(M) and X X(M). Then ∈ ∈ £X ω = diX ω + iX dω. (24) Physically, the Lie derivative represents the instantaneous rate of change of a quantity as it is dragged by the flow of a vector field.

13 2.2.3 Orientation and orientability of manifolds A shortcoming of differential forms is that their integration is an orientation dependent procedure. If this is not addressed, physical theories built using differential forms may contain an artificial dependence of measurements on the arbitrarily selected orientation of the ambient space. This is the motivation for defining an orientation independent volume form from which twisted differential forms may be defined. To begin with, it is helpful to provide a rigorous definition of orientation. This notion is intimately connected with the notion of coordinate systems on a vector space and trans- formations of that coordinate system. That is, if u n and v n are two coordinate { i}i=1 { i}i=1 systems of basis vectors for a vector space V , then there exists a bijection T : X X such → that n vi = Tijuj. (25) Xj=1 Definition 22. Let V be a vector space of dimension n. Define an equivalence class on the set of all ordered sets of basis vectors on V as follows: n u n = v n T : X X such that v = T u where det(T ) > 0. { i}i=1 ∼ { i}i=1 ⇐⇒ ∃ → i ij j Xj=1 If two sets of basis vectors are in the same equivalence class, they are said to have the same orientation. Because the entries of T area all real, det(T ) R. Hence, because det(T ) takes one of ∈ two signs, it is clear that the above definition partitions the set of all basis vectors on a vector space into two equivalence classes. One may pick arbitrarily which of these two orientations is the positive orientation and which is negative. This notion may be extended to certain manifolds as follows: Definition 23. Consider a differentiable n-manifold, M. If there exists an atlas of M A such that for any two charts (U , φ ), (U , φ ) in such that U U = , the Jacobian α α β β A α ∩ β 6 ∅ determinant of φ−1 φ is positive, then the manifold is said to be orientable. β ◦ α The choice of the atlas selects an orientation for each tangent space, TxM, to the manifold. The orientation of a manifold M which has boundary ∂M induces an orientation on ∂M. It is important to note that Stokes’ theorem requires that one use the compatible induced orientation of ∂M. It is not always possible to define a global orientation on a manifold. Perhaps the most famous example of a non-orientable manifold is the M¨obius strip. Comment 3. There is a notion of inner and outer orientation of manifolds important in the field of mimetic discretization [10]. While this will not be discussed here, this idea is complementary to the discussion of straight versus twisted differential forms in the following section.

14 2.2.4 Twisted forms: a dual space of differential forms It is also possible to show that orientability of a differentiable n-manifold is equivalent to the existence of a nowhere vanishing n-form: i.e. the existence of a volume form. However, there should be a way to define a volume measuring form that does not depend on orientation nor even requires orientability of a manifold. Indeed, one can imagine measuring areas of on M¨obius strip everywhere except on a set of measure zero: e.g. by cutting the strip and flattening it. This motivates the following discussion of densities.

Definition 24. If V is a vector space, a density on V is a map

ω :V V R ×···× → n times such that for any linear map A : V V , if Y = AX for i = 1,...,n, then →| {z i } i ω(Y ,...,Y )= det(A) ω(X ,...,X ). (26) 1 n | | 1 n Proposition 6. The space of densities associated with V , denoted Ω(V ), is a one-dimensional vector space.

Definition 25. A density such that ω(X ,...,X ) > 0 for all linearly independent X n 1 n { i}i=1 is called positive.

If ωn Λn(V ), then we may associate a unique positive density with ωn. Namely, ∈ ωn (X ,...,X )= ωn(X ,...,X ) . | | 1 n | 1 n | Definition 26. If M is a smooth manifold, then we may define a space of densities, known as the density bundle, associated with each tangent space as follows: x M, then ∈ Ω(TxM) is the space of densities on TxM. We define

Ω(M)= Ω(TxM). x [ Just as a vector field is a section of the tangent bundle, a density on M is defined to be a section of Ω(M).

Definition 27. We define the space of densities on M to be smooth maps ω : M → Ω(M). We will use Λ˜ n(M) to denote this space. Hence, if ω˜n Λ˜ n(M), then ω˜n ∈ x ∈ Ω(TxM). Proposition 7. For any smooth manifold Reimannian manifold (M, g), there exists a n unique positive density, call it vol , such that for any local orthonormal frame ∂ n , { i}i=1 n g vol (∂1,...,∂n) = 1. (27)

g 15 n Definition 28. This density, vol , will be called the volume form. The space Λ˜ n(M) is also known as the space of twisted n-forms. g Definition 29. Let ∂ n be a coordinate system on the tangent space (which we also { i}i=1 denote ∂). We define an operator o(∂) which encodes the orientation of the manifold. This operator transforms as

o(T ∂)= o(T ∂1,...,T∂n)= sign(det(T ))o(∂1,...,∂n)= sign(det(T ))o(∂). (28) This operator is useful in writing coordinate representations of twisted differential forms. Example 3. On a Riemannian manifold, (M, g), in local coordinates xi n , the volume { }i=1 form is written n vol = o(∂) det gdx1 . . . dxn. (29) ∧ ∧ Using the volume form, we may introducep a new hierarchy of vector spaces called the g twisted differential forms. To avoid confusion, we call the previously defined orientation independent differential forms straight forms. Twisted forms go by other names in the literature. Frankel calls them pseudoforms, and twisted top forms are called densities as noted above. The name, pseudoform, is perhaps preferable because it emphasizes the analogous character of pseudovectors and pseudoforms. However, because of prevalent usage, this paper will favor the term twisted forms. Moreover, we will usually represent twisted forms with a tilde and the space of twisted k-forms by Λ˜ k(M): e.g.ω ˜k Λ˜ k(M). ∈ The space of straight forms will continue to be denoted by Λk(M), and a k-form without a tilde will generally refer to a straight form. The space of twisted (n k)-forms, Λ˜ n−k(M), may be defined as a dual space to Λk(M) − as follows. Definition 30. The metric tensor induces a pointwise scalar product on k-forms. If

ω = ω dxik . . . dxik and η = η dxik . . . dxik i1,...,ik ∧ ∧ i1,...,ik ∧ ∧ i1<...bilinear form is non-degenerate, it induces an injective map M˜ n−k,k : Λ (M) ∗ → Λ˜ n−k(M) defined by   ηk η , =: M˜ ηk. (34) 7→ h k ·i n−k,k ∗ We define M : Λ˜ n−k(M) Λk(M) similarly. k,n−k → Definition 34. One may define the twisted
hodge star operator ⋆˜ : Λk(M) Λ˜ n−k(M) k → such that k n−k k k k k ω , η˜ = ω ⋆˜kη = (ω , η ) 2 k . (35) k ˜ n−k L Λ (M) Λ ,Λ M ∧ D E Z The Hodge star defines an isomorphism between k-forms and twisted (n k)-forms. − ˜ −1 ˜ Proposition 8. One can see that ⋆˜k = Mn−k,n−kMn−k,k. Using the twisted Hodge star, one finds an alternative way to write the L2 inner product on k-forms: Proposition 9. k k k k (ω , η ) 2 k = ω ⋆˜ η . (36) L Λ (M) ∧ k ZM As with other notation used in this paper, when it is clear from context, we will omit the k label on the Hodge star operator. In coordinates, the Hodge star may be written:

ωk = ω dxi1 . . . dxik i1,...,ik ∧ ∧ i1<...

17 where

k i1,...,ik (⋆˜kω )ik+1,...,in = ω ǫi1,...in o(∂) det(g) i <...

⋆˜−1 = ( 1)k(n−k)⋆˜ . (38) k − n−k Proposition 11. n n 1= ⋆˜nvol and ⋆˜01= vol .

Because straight and twisted differentialg forms areg essentially the same mathematical object modulo orientation dependence, most constructs defined with straight forms carry over to twisted differential forms with little to no modification. Abusing notation slightly, we may also define ⋆˜ : Λ˜ k(M) Λn−k(M) such that k → n ω˜k ⋆˜ η˜k = ω˜k η˜kvol . ∧ k · ZM ZM In general, we will see that one can interchange the rolesg of twisted and straight forms in most formulas presented in this paper. For example:

k n−k k n−k k k k k ω˜ , η k n−k = ω˜ η and (˜ω , η˜ ) k = ω˜ ⋆˜ η˜ . (39) h iΛ˜ ,Λ ∧ L2Λ˜ (M) ∧ k ZM ZM As with other notation established in this paper, when it is clear from context, the subscript on the inner product and duality pairing indicating the function spaces in question will be omitted.

2.2.5 A correspondence between vectors and differential forms Using the metric tensor, one may define an isomorphism between X(M) and Λ1(M). This allows one to convert expressions from the language of vector calculus to that of differential forms and vice versa.

18 Definition 35. We define the index lowering operator or flat operator ( )♭ : X(M) · → Λ1(M) to be v1 = g( ,V )= g V jdxi := V ♭. (40) · ij The inverse operation is called the index raising operator or sharp operator ( )♯ : · Λ1(M) X(M): → ∂ V = g−1( , v1)= gijv1 := (v1)♯. (41) · j ∂xi This definition establishes a notational choice that will be used throughout the remainder of this article. Any quantity which is needed both as a vector and as a differential form will be written as a capital letter when meant as a vector (e.g. V ), and with a lowercase letter when meant as a differential form (e.g. v1). From proposition 1, we know that dim(Λ1(M)) = dim(Λn−1(M)) = n. Therefore, there is also a natural identification of vectors with (n 1)-forms. More particularly, there is a − natural identification of vectors with twisted (n 1)-forms: − Definition 36. We may define an isomorphism between vector fields with twisted n (n 1)-forms by i vol : X(M) Λ˜ n−1(M) by − (·) → n n−1 ♭ i d 1 d i d n v˜ = iV volg= ⋆˜1V = V o(∂) det(g) x . . . x . . . x (42) ∧ ∧ ∧ ∧ i X p c where the hat symbolg means omission of “dxi” from the wedge product. The inverse oper- ation is given by n−1 ♯ V = ⋆˜n−1v˜ . (43) Recall that many physically significant quantities are
orientation dependent. The magnetic field and vorticity of a fluid are notable examples of so called pseudovectors. If V is a 1 ♭ n−1 n pseudovector, then one can see thatv ˜ = V is twisted while v = iV vol is straight. Hence, pseudovectors are naturally identified with straight (n 1)-forms. − g 2.2.6 The adjoint of the flat and sharp operators Definition 37. Let W, V X(M). We define the L2 inner product of vector fields to ∈ be n (W, V ) 2 = W V vol (44) L (M) · ZM where the reader is reminded that the dot product dependsg on the metric tensor: W V = W ig V j. · ij Proposition 12. With respect to the duality pairing between Λ1(M) and Λ˜ n−1(M) and the L2-inner product on X(M),

∗ n −1 ∗ n ( )♭ = i vol and ( )♯ = i vol . (45) · (·) · (·)       g 19 g 1 ♭ n−1 n Proof: Consider v = V andw ˜ = iW vol . Then

n−1 1 n−1 1 w˜ , v − = gw˜ v Λ˜ n 1,Λ1 ∧ ZM n = i vol V ♭ W ∧ ZM n = W gV vol = (W, V ) 2 . · L (M) ZM n g Therefore, sincew ˜n−1 = ⋆˜ W ♭ = i vol W = (⋆˜ w˜n−1)♯, 1 W ⇐⇒ n−1 −1 n−1 ♭ gn−1 ♯ n n−1 w˜ ,V = (⋆˜n−1w˜ ) ,V = i(·)vol w˜ ,V ˜ n−1 1 2 Λ ,Λ L (M) L2(M) D E      and similarly g 1 n 1 ♯ v , iW vol = (v ) ,W . Λ1,Λ˜ n−1 L2(M) D E   g Comment 5. Recalling (7), one can see that this relationship will be useful when investi- gating the calculus of variations for functionals of differential forms.

2.2.7 Vector calculus in R3 and differential forms We now turn our attention to vector calculus in R3. Notice that exterior differentiation is inherently metric independent whereas it is well known that vector calculus objects such as the gradient, divergence, and curl depend on one’s coordinate system – often in a very complicated manner. By defining a differential form related to a vectorial quantity, one may obtain simpler, metric-independent expressions for their derivatives. The following definitions allow us to reconstruct the standard vector calculus operations from the exterior derivative.

Definition 38. Let f : M R be a scalar field on a Riemannian manifold. We define its → associated 0-form to be f 0 = f. The gradient of f is defined by

df 0 = ( f)♭. (46) ∇ Definition 39. Let V X(M). If we define its associated 1-form by v1 = V ♭, then the ∈ curl of V is defined by 3 dv1 = i vol = ⋆˜ ( V )♭. (47) ∇×V 1 ∇× Hence, inverting this expression, we find g V = (⋆˜ (d(V ♭)))♯. ∇× 2

20 Definition 40. Let V X(M). If we define its associated twisted (n 1)-form by v˜n−1 = 3 ∈ − iV vol , then the divergence is defined to be

3 3 g dv˜2 = di vol = £ vol = ⋆˜ ( V ). (48) V V 0 ∇ · 3 ♭ Hence, recalling that iV vol = ⋆˜1(V g), we find g

V = ⋆˜ d(⋆˜ (V ♭)). g ∇ · 3 1 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). Notice, the curl of a straight vector yields a pseudovector according to our definition. Moreover, we have that 3 3 £ vol = ( V )vol . (49) V ∇ · That is, we defined the divergence of a vector field such that it physically corresponds with g g the the change in the volume element as it is dragged by that vector field. This corresponds with our physical intuition for these quantities.

Proposition 13. Let U, V, W X(M) and let u1, v1, w1 be their corresponding 1-forms. ∈ Then 3 ⋆˜ (u1 v1 w1)= vol (U, V, W ). (50) 3 ∧ ∧ Proof: g u1 v1 w1 = (u dxi) (v dxj) (w dxk) ∧ ∧ i ∧ j ∧ k Xijk = u v w dxi dxj dxk i j k ∧ ∧ Xijk = ⋆˜ (u1 v1 w1)= o(∂) det(g) U iV jW kǫ ⇒ 3 ∧ ∧ ijk i

g Hence, the triple wedge product of three 1-forms measures the volume of the parallelepiped spanned by their vector proxies.

Proposition 14. Let U, V X(M), and let u1 and w˜n−1 be their associated 1-form and ∈ twisted (n 1)-form respectively. Then − 3 u1 w˜2 = U W vol . (51) ∧ ·

21 g Proof:

3 u1 v˜2 = u1 i vol ∧ ∧ V 3 = i vol u1 V g∧ 3 3 = i (vol u1)+ vol i u1 V g ∧ ∧ V 3 = U V vol . ·g g g Hence, one may write the inner product in terms of a wedge product. Proposition 15. Let U, V X(M), and let u1, v1 be their associated 1-forms. Then ∈ 3 u1 v1 = i vol . (52) ∧ U×V Proof: A geometrically intuitive definition of the cross product is that U V is the unique g × vector such that W X(M), ∀ ∈ 3 U V W = vol (U, V, W ). × · But we already know that g 3 vol (U, V, W )= ⋆˜ u1 v1 w1. 3 ∧ ∧ Hence, we find g 3 W U V vol = u1 v1 w1. · × ∧ ∧ But according to proposition 14, g 3 3 U V W vol = i vol w1. × · U×V ∧ Hence, g3 g u1 v1 i vol w1 = 0 w1 Λ1(M). ∧ − U×V ∧ ∀ ∈  3  Hence, it follows that u1 v1 = i volg . ∧ U×V Notice that this implies that if U and Vgare standard vectors, then U V is a pseudovector. × Corollary 3. Let U, V X(M) and let u1, v1 be their associated 1-forms. Then ∈ 3 u1 dv1 = U V vol . (53) ∧ ·∇× This identity will later be useful when writing the Born-Infg eld bracket in terms of differential forms.

22 2.3 Functional calculus with respect to differential forms What is meant by functional derivatives with respect to differential forms is presented in this section. The key tool in this discussion is the functional chain rule. The ideas in this section are developed from [5].

2.3.1 First variation with respect to a differential form Definition 41. Using the inner product given by (36), one may define an inner prod- 2 k uct space of k-forms (L Λ (M), ( , ) 2 k ). This endows the space with a norm, · · L Λ (M) 2 k . k · kL Λ (M) Consider a functional K : L2Λk(M) R. The Fr´echet derivative of this quantity, → DK[ωk] (L2Λk(M))∗ is defined such that: ∈ K[ω + η] K[ω] DK[ω]η = O( η ). (54) | − − | k k Definition 42. Let K : L2Λk(M) R. Using the two duality pairings on L2Λk(M), → (36) and (33), one may identify DK[ω] (L2Λk(M))∗ with either an element of L2Λk(M) ∈ or L2Λ˜ n−k(M): δK DK[ω]η = η, (55) δω 2 k  L Λ (M) and δK˜ DK[ω]η = η, . (56) δω * +Λk,Λ˜ n−k The tilde notationally distinguishes the two identifications.

One can see that δK δK˜ δK˜ δK L2Λk(M), L2Λ˜ n−k(M), and = ⋆˜ . (57) δω ∈ δω ∈ δω δω Notice, δK/δω˜ is a twisted form while δK/δω is straight.

Definition 43. We will call δ/δω˜ the twisted functional derivative and δ/δω the straight functional derivative.

The same procedure may be done for a functional which takes a twisted form as an argument: K : L2Λ˜ k(M) R. In this case, the twisted functional derivative yields a → straight form and the straight functional derivative yields a twisted form. That is, if ω˜ = ⋆ω˜ L2Λ˜ n−k(M), then ∈ δK δK˜ δK˜ δK L2Λ˜ k(M), L2Λn−k(M), and = ⋆˜ . (58) δω˜ ∈ δω˜ ∈ δω˜ δω˜

23 Proposition 16. Let K be as above and ω˜ = ⋆ω˜ . Then the chain rule implies δK δK = ⋆˜ . (59) δω˜ δω Corollary 4. Under the same hypotheses as before, one finds

δK˜ δK = . (60) δω˜ δω

Proposition 17. Let k n 1 and let ω be a k-form. Suppose K[ω]= Kˆ [dω]. Then ≤ − δK˜ δ˜Kˆ = ( 1)k+1d . (61) δω − δdω Proof: The Leibniz rule and chain rule imply that η L2Λk(M), ∀ ∈ ˜ ˜ ˆ δK d δK η, = η, d * δω + * δ ω + δ˜Kˆ δ˜Kˆ = η, ( 1)k+1d + tr η − δdω ∂M ∧ δdω * + Z∂M ! where tr : Λn−1(M) Λn−1(∂M) is the trace operator [1]. With enough regularity, this ∂M → map is just restriction to the boundary. If we assume homogeneous boundary conditions or that tr∂M (δ˜K/δˆ dω) = 0, we find that

δK˜ δ˜Kˆ = ( 1)k+1d . (62) δω − δdω

Corollary 5. Assume the same hypotheses as in the previous proposition. Then δK/δω˜ is an exact differential form.

Proposition 18. Suppose E X(M) where M R3. Let K : X(M) R. Define ∈ ⊂ → the variational derivative, δK/δE, with respect to the previously defined inner product on vector fields: 3 (W, V ) 2 = W V vol . (63) L (M) · ZM Now, let e1 = E♭ and let Kˆ [e1]= K[E]. Then the variationalg derivative of Kˆ with respect to e1 is a twisted 2-form given by

δ˜Kˆ 3 = i vol . (64) δe1 δK/δE

24 g Proof: Using (7) and (45), one finds

δ˜Kˆ ∗ −1 δK 3 = ( )♭ = i vol . δe1 · δE δK/δE    g Proposition 19. Again, let M R3. Suppose B X(M) is a pseudovector and K : ⊂ ∈ X(M) R. Again, define the variational derivative, δK/δB with respect to the L2 inner → 2 3 2 product on vector fields. Suppose b = iBvol Λ (M). Then the variational derivative ∈ of Kˆ with respect to b2 is the twisted 1-form g δ˜Kˆ δK ♭ = . (65) δb2 δB   Proof: Using (7) and (45), one finds ♭ δ˜Kˆ 3 ∗ −1 δK δ˜Kˆ 2 = i(·)vol = 2 . (66) δb δB δb !    g

Notice, b2 is a straight form because the orientation dependence of the volume form cancels with the orientation dependence of the pseudovector B.

2.3.2 Maps between differential forms Proposition 20. Consider a map φ : L2Λk(M) L2Λl(M) (or L2Λ˜ l(M)). If φ is → sufficiently regular (e.g. convex in ω), then Fr´echet derivative of φ is Dφ[ω] B(L2Λk(M),L2Λl(M)), ∈ and is defined by

φ[ω + η] φ[ω] Dφ[ω]η 2 l = O η 2 k . k − − kL Λ (M) k kL Λ (M) Comment 6. The distinction of twisted or untwisted functional derivati ves is less mean- ingful here and will not be made. Hence, we will use the notation: δφ Dφ[ω]η = η L2Λl(M). (67) δω ∈ Maps of this form appear when changing coordinates in a Hamiltonian field theory. For example, suppose e1 L2Λ1(M) and d˜2 L2Λ˜ 2(M). Suppose further that d˜2 = d˜2[e1]. ∈ ∈ Then δd˜2 B(L2Λ1(M),L2Λ˜ 2(M)) (68) δe1 ∈ appears in the chain rule formula for computing derivatives with respect to d˜2.

25 2.3.3 Second variation with respect to a differential form Definition 44. By taking the Fr´echet derivative of DK[ω]η (L2Λk(M))∗, one obtains ∈ the second variation:

2 DK[ω + ξ] DK[ω] D K[ω]( ,ξ) ∗ = O ξ 2 k − − · (L2Λk(M)) k kL Λ (M)   where D2K [ω]( , ) (L2Λk(M) L2Λk(M))∗ and the norm on the dual space is · · ∈ × Aη A (L2Λk(M))∗ = sup | | . k k kηk =1 η L2Λk(M) L2Λk(M) k k In general, it is easier to check if

2 DK[ω + ξ]η DK[ω]η D K[ω](η,ξ) = O η 2 k ξ 2 k . (69) − − k kL Λ (M)k kL Λ (M)   Definition 45. Using the two duality pairings on k-forms, there are two natural ways of writing D2K(ω):

δ2K δ˜2K D2K[ω](η,ξ)= η, ξ = η, ξ . (70) δωδω 2 k δωδω  L Λ (M) * +Λk,Λ˜ n−k From this definition, it is clear that

δ2K δ˜2K B(L2Λk(M),L2Λk(M)) and B(L2Λk(M),L2Λ˜ n−k(M)). (71) δωδω ∈ δωδω ∈ Moreover, the first identification maps between like forms (straight to straight or twisted to twisted) while the latter changes the twistedness (straight to twisted or twisted to straight). Proposition 21. Using these two representations of the second variation, (69) becomes

δK δK δ2K η, [ω + ξ] [ω] [ω]ξ = O η L2Λk(M) ξ L2Λk(M) (72) δω − δω − δωδω L2Λk(M) k k k k     if we use the L2-pairing and

δK˜ δK˜ δ˜2K η, [ω + ξ] [ω] [ω]ξ = O η 2 k ξ 2 k (73) δω δω δωδω L Λ (M) L Λ (M) * − − + k n−k k k k k Λ ,Λ˜   if we use the pairing between k-forms and twisted (n k)-forms. − Proposition 22. The following relationship holds:

δ˜2K δ2K = ⋆˜ . (74) δωδω k δωδω

26 Proposition 23. The second variation is a symmetric operator:

D2K[ω](η,ξ)= D2K[ω](ξ, η). (75)

Corollary 6. Symmetry of the second variation implies

δ2K δ2K η, ξ = η,ξ δωδω 2 k δωδω 2 k  L Λ (M)  L Λ (M) δ˜2K δ˜2K and η, ξ = ξ, η . δωδω δωδω * +Λk,Λ˜ n−k * +Λk,Λ˜ n−k

Change of variables from E to D The following results allow us to perform a crucial change of coordinates for the Hamiltonian formulation of the macroscopic Maxwell equations.

Proposition 24. Suppose that E, D X(M) where M R3. Moreover, let K : X(M) ∈ ⊂ → R such that δK D = E 4π − δE and assume K has two continuous derivatives. Then δD δ2K V = I 4π V. δE − δEδE   Proof: The linearity of differentiation immediately implies the result.

Corollary 7. Under the same hypotheses as in the previous proposition, if Kˆ [D]= K[E], then ∗ δKˆ δD −1 δK δ2K −1 δK = = I 4π . δD δE δE − δEδE δE   !   Proof: The self-adjointness of the second variation implies

δ2K ∗ δ2K I 4π = I 4π − δEδE − δEδE   from which the result immediately follows.

Proposition 25. Consider the same hypotheses as before. Now let e1 = E♭ and d˜2 = 3 iDvol . Then δK˜ d˜2 = ⋆˜ e1 4π . g 1 − δe1

27 Moreover, ♯ ˜2 δD δd ♭ ( )= ⋆˜2 1 ( ) . (76) δE · δe · ! Hence, if v1 = V ♭, then

δD δ2K δd˜2 δ˜2K V = I 4π V v1 = ⋆˜ 4π v1. (77) δE − δEδE ⇐⇒ δe1 1 − δe1δe1   ! Proof: First, notice that ♭ ˜ 2 3 δK 1 3 1 δK d˜ = iDvol = ⋆˜ E 4π = ⋆˜ e 4πi vol = ⋆˜ e 4π . 1 − δE 1 − δK/δE 1 − δe1   Let V,W X(Mg) be arbitrary and let ω1 = W ♭ and v1 g= V ♭. Since δ˜2K/δe1δe1 ∈ ∈ B(L2Λ1(M),L2Λ˜ 2(M)), it follows that

2 ˜2 δ K 3 1 δ K 1 W V vol = ω , 1 1 v M · δEδE δe δe Z * +Λ1,Λ˜ 2 g δ˜2K = ω1 v1 ∧ δe1δe1 ZM ♯ δ˜2K 3 = W ⋆˜ v1 vol · 2 δe1δe1 ZM ! ♯ g δ˜2K 3 = W ⋆˜ V ♭ vol . · 2 δe1δe1 ZM ! Hence, g ♯ 2 ˜2 ˜2 δ K δ K ♭ δ K 1 3 ( )= ⋆˜ ( ) or v = i 2 vol . 2 1 1 1 1 δ K V δEδE · δe δe · ! δe δe δEδE The same argument as above shows that g

♯ ˜2 δD δd ♭ ( )= ⋆˜2 1 ( ) . δE · δe · ! Hence, it follows that

δD δ2K δd˜2 δ˜2K V = I 4π V v1 = ⋆˜ 4π v1. δE − δEδE ⇐⇒ δe1 1 − δe1δe1   !

28 Corollary 8. Under the same hypotheses as before, if Kˆ (d˜2)= K(e1), then

∗ −1 −1 δ˜Kˆ δd˜2 δK˜ δ˜2K δK˜ = 1 1 = ⋆˜1 4π 1 1 1 . (78) δd˜2 δe ! ! δe − δe δe ! δe Proof: Corollary 6 implies ∗ δ˜2K δ˜2K ⋆˜1 4π 1 1 = ⋆˜1 4π 1 1 (79) − δe δe ! − δe δe from which the result follows.

2.3.4 Partial functional derivatives Definition 46. Suppose that ω is a k-form and η is an l-form. Moreover, let

K : L2Λk(M) L2Λl(M) R × → so that that K = K[ω, η]. We may define partial functional derivatives using the standard definition of a Fr´echet derivative, but we vary only one argument at a time. Let ν be a k-form and ξ be an l-form. Then

K[ω + ν, η] K[ω, η] D K[ω, η]ν = O ν 2 k (80) | − − 1 | k kL Λ (M)   and K[ω, η + ξ] K[ω, η] D K[ω, η]ξ = O ξ 2 l . (81) | − − 2 | k kL Λ (M) Notice, we indicate the partial functional derivatives with subscripts to indicate with respect to which argument we are varying. Definition 47. The total functional derivative defined to be

DK[ω, η](ν,ξ)= D1K[ω, η]ν + D2K[ω, η]ξ. (82)

As usual, there are two possible identifications using the two duality pairings: δK δK D1K[ω, η]ν + D2K[ω, η]ξ = ν, + ξ, δω 2 k δη 2 l  L Λ (M)  L Λ (M) and δK˜ δK˜ D K[ω, η]ν + D K[ω, η]ξ = ν, + ξ, . 1 2 δω δη * +Λk,Λ˜ n−k * +Λl,Λ˜ n−l The total functional derivative is just the standard Fr´echet derivative on a tensor product space.

29 Definition 48. Suppose that ω is a k-form and η is an l-form. Moreover, let

φ : L2Λk(M) L2Λl(M) Λm(M). × → As was done previously, we define the partial functional derivatives by varying each argu- ment individually. One obtains

Dφ[ω, η](ν,ξ)= D1φ[ω, η]ν + D2φ[ω, η]ξ (83) δφ δφ = ν + ξ. (84) δω δη Recall that in the case of maps from differential forms to differential forms, we chose not to distinguish between twisted and straight functional derivatives.

Notice, δφ δφ B(L2Λk(M),L2Λm(M)) and B(L2Λl(M),L2Λm(M)). δω ∈ δη ∈ Definition 49. A mixed second partial derivative is defined, for example, by taking the variation of D1K[ω, η] with respect to the second argument:

2 D K[ω, η + ξ]ν D K[ω, η]ν D K[ω, η](ν,ξ) = O ν 2 k ξ 2 l . (85) 1 − 1 − 1,2 k kL Λ (M)k kL Λ (M)   As before, there are two natural identifications of the second functional derivative:

2 ˜2 2 δ K δ K D1,2K(ω, η)(ν,ξ)= ν, ξ = ν, ξ . δηδω 2 k δηδω  L Λ (M) * +Λk,Λ˜ n−k Proposition 26. The mixed second partial functional derivative is symmetric with respect to interchange of order of differentiation:

2 2 D1,2K = D2,1K.

Comment 7. When dealing with partial functional derivatives, the formal statement of the chain rule, δE δE δE = δD + δB, (86) δD δB is a helpful device to remember the more rigorous expression (83).

30 2.3.5 Change of variables from (E, B) to (D, B) Using the results from the previous section, we can find another crucial formula involved changing coordinates in the Hamiltonian formulation of the macroscopic Maxwell equa- tions.

Proposition 27. Suppose that E, B, D X(M), and let K : X(M) X(M) R so that ∈ × → K = K[E, B]. Moreover, suppose that

δK D = E 4π . − δE If we think of E as an implicit function of (D, B), then

δE δ2K −1 = I 4π (87) δD − δEδE   and δE δ2K −1 δ2K = 4π I 4π . (88) δB − δEδE δBδE   Proof: Let Φ : X(M) X(M) X(M) X(M) be the map from (E, B) to (D, B). × → × That is, Φ[E, B] = (D, B). We assume that Φ is a diffeomorphism. Hence, we also have Φ−1[D, B] = (E, B). We have that

DΦ[E, B](δE, δB) = (D1Φ[E, B]δE + D2Φ[E, B]δB, δB) D Φ[E, B] D Φ[E, B] δE = 1 2 . 0 1 δB    Hence, it follows that

δE δE DΦ−1[D, B](δD, δB) = δD + δB, δB δD δB   D Φ[E, B]−1 D Φ[E, B]−1D Φ[E, B] δD = 1 − 1 2 0 1 δB    If we compute the entries of this matrix, one finds

δE δ2K −1 = I 4π δD − δEδE   in agreement with the result from the previous section, and we have further found that

δE δ2K −1 δ2K = 4π I 4π . δB − δEδE δBδE   31 1 ♭ 2 3 Let E, B X(M), and assume that B is a pseudovector. Define e = E and b = iBvol . ∈ Notice, both e1 and b2 are straight forms. Then it follows that g δe1 B(L2Λ2(M),L2Λ1(M)). (89) δb2 ∈ The following proposition tells us how this quantity is related to δE/δB.

Proposition 28. Under above assumptions, we find

δE δ2K −1 δ2K V = 4π I 4π V (90) δB − δEδE δBδE   −1 ♯ δ˜2K δ˜2K δe1 ♯ = 4π ⋆˜ 4π v˜2 = v˜2 . (91)  1 − δe1δe1 δb2δe1  δb2 !     3 Proof: Let V,W X(M) where V be a pseudovector be arbitrary. Define v2 = i vol = ∈ V ♭ 2 3 ⋆˜1V andw ˜ = iW vol . Then g 1 1 2 δe 2g 2 δe 2 w˜ , 2 v = w˜ 2 v δb 2 1 ∧ δb  Λ˜ ,Λ ZM ♯ δe1 3 = W ⋆˜ V ♭ vol · δb2 1 ZM   ♯ ♯ δe1 g δE δe1 = W, ⋆˜ V ♭ = ( )= ⋆˜ ( )♭ . δb2 1 ⇒ δB · δb2 1 ·   !L2(M)   Note that δ˜2K B(L2Λ2(M),L2Λ˜ 2(M)). δb2δe1 ∈ Let U, V X(M) where V is a pseudovector be arbitrary. Define u1 = U ♭ and v2 = ∈

32 3 ♭ iV vol = ⋆˜1V . Then we find

˜2 ˜2 g 1 δ K 2 1 δ K 2 u , 2 1 v = u 2 1 v δb δe M ∧ δb δe * +Λ1,Λ˜ 2 Z ♯ δ˜2K 3 = U ⋆˜ ⋆˜ V ♭ vol · 2 δb2δe1 1 ZM ! ♯ ˜2 g δ K ♭ = U, ⋆˜2 2 1 ⋆˜1V  δb δe !  L2(M)   ♯ 2 ˜2 δ K δ K ♭ = ( )= ⋆˜2 2 1 ⋆˜1( ) . ⇒ δBδE · δb δe · !

From proposition 25, we recall that

−1 ♯ δ2K −1 δ˜2K I 4π V = ⋆˜ 4π v˜2 . − δEδE  1 − δe1δe1    !   Together, these results imply

δE δ2K −1 δ2K V = 4π I 4π V δB − δEδE δBδE   −1 ♯ δ˜2K δ˜2K δe1 ♯ = 4π ⋆˜ 4π v˜2 = v˜2 .  1 − δe1δe1 δb2δe1  δb2 !    

3 Geometrized macroscopic Maxwell equations

The model for material laws used in this formulation of the macroscopic Maxwell equations was taken from [13]. There, Maxwell’s equations are presented as a component of a kinetic plasma model. Here, we are only considering the Maxwell component in isolation. We first present the theory in terms of vector calculus before recasting the theory in terms of split exterior calculus [5].

33 3.1 Macroscopic Maxwell equations Definition 50. Maxwell’s equations may be written in two equivalent formats: D = 4πρ , E = 4πρ ∇ · f ∇ · B = 0, B = 0 ∇ · ∇ · ∂B ∂B c E = , c E = (92) ∇× ∂t ∇× ∂t ∂D ∂E c H = + 4πJ , c B = + 4πJ. ∇× ∂t f ∇× ∂t The left column is frequently called the macroscopic Maxwell equations, and the right column are just the standard Maxwell equations. One relates the displacement field, D, and the magnetic field intensity, H, with the electric field, E, and magnetic field, B, by D = E + 4πP and H = B 4πM (93) − where P is the polarization and M is the magnetization. Moreover, ρf and Jf , the free charge and free current densities respectively, are related to ρ and J by

ρ = ρb + ρf and J = Jb + Jf (94) where ρ = P (95) b −∇ · and ∂P J = J + J = + c M. (96) b p m ∂t ∇× It is straightforward to show that the two forms of Maxwell’s equations 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, ρf and Jf , are prescribed by freely propagating charged particles rather than the, possibly complicated, bound charges and currents in the medium. These microscopic features bound to the medium are instead encoded in constitutive relations for P and M. Definition 51. In this paper, we consider a general matter model prescribed by

3 K[E, B]= (x, E, B, E, B, ...)vol . (97) K ∇ ∇ ZM We define the polarization and magnetization by g δK δK P (x,t)= and M(x,t)= (98) − δE −δB where the functional derivatives are defined via the usual L2(M) pairing:

δK 3 δK 3 DK[E, B](δE, δB)= δE vol + δB vol . (99) · δE · δB ZM ZM g g 34 3.1.1 Hamiltonian structure Definition 52. We may define a non-canonical Hamiltonian structure for the macro- scopic Maxwell system defined above. The Hamiltonian of this system is defined to be

δK 3 3 H[E, B]= K E vol + H = K + E P vol + H (100) − · δE v · v ZM ZM where g g 1 3 H = (E E + B B) vol . (101) v 8π · · ZM The Poisson bracket is defined to be g

δF δG δG δF 3 F, G = 4πc vol . (102) { } δD ·∇× δB − δD ·∇× δB ZM   This is a variant of the Born-Infeld bracket. g Proposition 29. Let H[D, B]= H[E, B]. Then one can show that

δH E δH H = and = . (103) δD 4π δB 4π The proof is omitted here, but the corresponding result in terms of differential forms is given later. With this, we find the evolution of an arbitrary observable: Corollary 9. Let F = F [D, B]. Then

δF δF 3 F˙ = F, H = c H E vol . (104) { } δD ·∇× − ·∇× δB ZM   Noting that, under the assumption of homogeneous boundary conditions,g

3 3 U V vol = U V vol , ·∇× − ∇× · ZM ZM we immediately have the following corollary.g g Corollary 10. The equations of motion are given by ∂D = D,H = c H ∂t { } ∇× (105) ∂B = B,H = c E. ∂t { } − ∇× Since D and B are not functionals, we cannot actually plug them into the Poisson bracket which only takes functionals as arguments. To get around this limitation, we consider a delta sequence of fuctionals in order to obtain the strong form of the equations of motion.

35 Proposition 30. This bracket has Casimir invariants:

D = [ D] and B = [ B] (106) C F ∇ · C G ∇ · where and are arbitrary smooth functionals. F G Proof: This follows from the result in example 2.

Hence, Gauss’s law and the absence of magnetic monopoles hold as long as they are enforced in the initial conditions. In this paper, Maxwell’s equations have not been coupled to a dynamical model governing the source terms. Hence, because we are only considering self-consistent dynamics generated by a non-canonical Poisson bracket, we set ρf = 0 and Jf = 0. One may augment the theory by adding, for example, a fluid or kinetic model for the source terms.

3.2 3D geometrized Maxwell equations The above Hamiltonian model for the macroscopic Maxwell equations may be rewritten in terms of split exterior calculus as in [5]. This yields a Hamiltonian theory which is coordinate and orientation independent, has a metric independent Poisson bracket, and the governing equations are defined either on the complex of straight forms, or on the complex of twisted forms. Metric dependence in the theory ultimately comes from the Hodge star operator which relates the straight and twisted forms. This way of expressing a Hamiltonian theory is believed to be advantageous for discretization using the novel split Hamiltonian finite element framework [3].

Definition 53. The split exterior calculus formulation of Maxwell’s equations may be written in two equivalent formats:

d ˜2 3 d 1 3 d = 4πr˜f , ⋆˜1e = 4πr˜ db2 = 0, db2 = 0 ∂b2 ∂b2 (107) cde1 = , cde1 = − ∂t − ∂t ˜2 1 1 ∂d 2 2 ∂⋆˜1e 2 cdh˜ = + 4πj˜ , cd⋆˜2b = + 4πj˜ . ∂t f ∂t

The displacement field twisted 2-form and magnetic intensity field twisted 1-form, d˜2 and h˜ 1 respectively, are related to the electric field 1-form, e1, and magnetic field 2-form, b2, by d˜2 = ⋆˜ e1 + 4πp˜2 and h˜ 1 = ⋆˜ b2 4πm˜ 1 (108) 1 2 −

36 where p˜2 is the polarization twisted 2-form, and m˜ 1 is the magnetization twisted 1-form. 3 ˜2 Moreover, the free charge twisted 3-form and free current twisted 2-form, r˜f and jf respec- tively, are related to r˜3 and j˜2 by 3 3 3 ˜2 ˜2 ˜2 r˜ =r ˜f +r ˜b and j = jf + jb (109) where ∂p˜2 r˜3 = dp˜2 and j˜2 = j˜2 + j˜2 = + cdm˜ 1. (110) b − b p m ∂t Straightforward algebraic manipulation shows that the two columns of Maxwell’s equations are equivalent. The left column is the split exterior calculus formulation of the macroscopic Maxwell equations.

1 ♭ 2 3 3 ˜2 3 Proposition 31. Suppose we let e = E , b = iBvol , r˜f = ⋆˜0ρf , and jf = iJ vol (and similarly for r˜3 and j˜2). Moreover, let Kˆ [e1, b2] = K[E, B] where K is as in definition 51. Then if we let g g δ˜Kˆ δ˜Kˆ p˜2 = and m˜ 1 = , (111) −δe1 −δb2 then the split exterior calculus formulation of Maxwell’s equations given in definition 53 is equivalent to the standard vector calculus definition given in definition 50. Proof: We will demonstrate the equivalence for the macroscopic Maxwell equations. This is sufficient since the standard Maxwell equations are equivalent to the macroscopic for- mulation. First, note that

♭ δ˜Kˆ 3 δ˜Kˆ δK p˜2 = = i vol and m˜ 1 = = . δe1 δK/δE δb2 δB   Hence, g

˜ 2 1 2 3 1 δK d˜ = ⋆˜ e + 4πp˜ = iDvol = ⋆˜ e 4π 1 1 − δe1 δK˜ g and h˜ 1 = ⋆˜ b2 4πm˜ 1 = H♭ = ⋆˜ b2 + 4π . 2 − 2 δb2 Using the identities from section 2.2.7, the first column of equation (107) becomes ⋆˜ ( D 4πρ ) = 0 0 ∇ · − f ⋆˜ ( B) = 0 0 ∇ · ∂B ♭ ⋆˜ c E = 0 (112) 1 ∇× − ∂t   ∂D ♭ ⋆˜ c H 4πJ = 0. 1 ∇× − ∂t − f   37 Hence, the split exterior calculus formulation is equivalent to the vector calculus formula- tion.

3.2.1 Hamiltonian structure As with the the vector formalism, the geometrized macroscopic Maxwell equations are endowed with a non-canonical Hamiltonian structure.

Definition 54. The Hamiltonian may be written

δ˜Kˆ 1 Hˆ [e1, b2]= H[E, B]= Kˆ e1 + e1 ⋆˜ e1 + b2 ⋆˜ b2 . (113) − ∧ δe1 8π ∧ 1 ∧ 2 ZM ZM
The Poisson bracket may be written

δF˜ δG˜ δG˜ δF˜ F, G = 4πc , d , d . (114) { }  δd˜2 δb2 − δd˜2 δb2  * +Λ1,Λ˜ 2 * +Λ1,Λ˜ 2   In the following sections, we will show that this Hamiltonian and Poisson bracket do in fact recover the macroscopic Maxwell equations.

3.2.2 Derivatives of the Hamiltonian Proposition 32.

∗ δ˜Hˆ δ˜Kˆ ⋆˜ e1 δ˜Hˆ δ˜Kˆ δ˜2Kˆ ⋆˜ b2 ˜ 1 e1 2 1 = ⋆1 4π 1 1 and 2 = 2 2 1 + . (115) δe − δe δe ! 4π δb δb − δb δe ! 4π

Proof: Taking the variation of Hˆ , we find

δ˜Hˆ δ˜Kˆ δ˜Kˆ D Hˆ [e1, b2]δe1 = δe1, = δe1, δe1, 1 δe1 δe1 − δe1 * +Λ1,Λ˜ 2 * +Λ1,Λ˜ 2 * +Λ1,Λ˜ 2 δ˜2Kˆ 1 e1, δe1 + δe1, ⋆˜ e1 . − δe1δe1 4π h 1 iΛ1,Λ˜ 2 * +Λ1,Λ˜ 2

Hence, we find 1 δ˜Hˆ δ˜Kˆ ⋆˜1e 1 = ⋆˜1 4π 1 1 . δe − δe δe ! 4π

38 Similarly,

δ˜Hˆ δ˜Kˆ D Hˆ [e1, b2]δb2 = δb2, = δb2, 2 δb2 δb2 * +Λ2,Λ˜ 1 * +Λ2,Λ˜ 1 δ˜2Kˆ 1 δe1, δb2 + δb2, ⋆˜ b2 . − δb2δe1 4π h 2 iΛ2,Λ˜ 1 * +Λ1,Λ˜ 2 Hence, ∗ δ˜Hˆ δ˜Kˆ δ˜2Kˆ ⋆˜ b2 e1 2 2 = 2 2 1 + . δb δb − δb δe ! 4π

Proposition 33. Define Hˆ [d˜2, b2]= Hˆ [e1, b2]. Then one finds

δ˜Hˆ e1 δ˜Hˆ ⋆˜ b2 δ˜Kˆ h˜ 1 = and = 2 + = . (116) δd˜2 4π δb2 4π δb2 4π Proof: The chain rule implies

δ˜Hˆ δ˜e1 δ˜Hˆ δ˜e1 δ˜Kˆ e1 = 1 = ⋆˜1 4π 1 1 δd˜2 δd˜2 δe δd˜2 − δe δe ! 4π and δ˜Hˆ δ˜Hˆ δ˜e2 δ˜Hˆ = + δb2 δb2 δb2 δe1 ∗ δ˜Kˆ δ˜2Kˆ ⋆˜ b2 δ˜e2 δ˜Kˆ e1 e1 2 = 2 2 1 + + 2 ⋆˜1 4π 1 1 . δb − δb δe ! 4π δb − δe δe ! 4π Applying (77), we find δ˜Hˆ e1 = . δd˜2 4π Applying (90), we find δ˜Hˆ ⋆˜ b2 δ˜Kˆ h˜ 1 = 2 + = . δb2 4π δb2 4π

The derivation of (103), which was omitted, is entirely analogous. The above calculuation allows us to write e1 h˜ 1 DHˆ [d˜2, b2](δd˜2, δb2)= , δd˜2 + , δb2 . (117) 4π 1 ˜ 2 4π  Λ ,Λ * +Λ˜ 1,Λ2

39 Comment 8. Therefore, (4π)−1(e1, h˜ 1) may be thought of as the gradient of Hˆ [d˜2, b2]. That is, (e1, h˜ 1) is related to (d˜2, b2) as the response to variation of the electromagnetic energy, Hˆ .

3.2.3 Equations of motion and Casimir invariants Theorem 3. Let F = F [d˜2, b2]. Then

δF˜ δF˜ F˙ = F, Hˆ = c , dh˜1 e1, d . (118) { }  δd˜2 − δb2  * +Λ1,Λ˜ 2 * +Λ1,Λ˜ 2   Proof: The result follows immediately from using the above calculated derivatives of Hˆ in equation (114).

Corollary 11. If F = F [d˜2], we obtain Amp`ere’s law in weak form:

δF˜ F˙ = F [d˜2], Hˆ = c , dh˜1 . (119) { } δd˜2 * +Λ1,Λ˜ 2

If F = F [b2], then we obtain Faraday’s law in weak form:

δF˜ F˙ = F [b2], Hˆ = c e1, d . (120) { } − δb2 * +Λ1,Λ˜ 2

Hence, using a, delta sequence for F , one obtains the strong form of Amp`ere’s and Faraday’s laws. This recovers the evolving equations of (107). The constraint equations are given by the Casimir invariants of the Poisson bracket.

Proposition 34. The Poisson bracket (114) has Casimir invariants

2 2 2 = F [db ] and = F [dd˜ ] (121) Cb Cd˜2 for any smooth functional F .

Proof: This follows from (61).

These Casimir invariants imply the non-evolving constraints in (107): that is, Gauss’s law and the nonexistence of magnetic monopoles.

40 3.3 On the Poisson bracket: the correspondence between the two ver- sions We will now examine the structure of the Poisson bracket (114) in greater detail. To begin, recall that δF˜ δF ♭ δF˜ δF ♭ = and = . (122) δd˜2 δD δb2 δB Definition 55. If we write δF i ∂ δF ∂ = gij , (123) δD ∂xi δDj ∂xi then one finds a coordinate representation

δF˜ δF = dxi. (124) δd˜2 δDi

Hence, unsurprisingly, the coordinates of δF/δ˜ d˜2 are just the components of the covector representation of δF/δD.

The write the coordinate expression for δF/δ˜ b2 in the same manner.

Proposition 35. Using the above coordinate expressions, we find that

δF ∂ δG δ ∂ δ F, G = 4πc G F dxi dxj dxk. (125) { } δDi ∂xj δBk − δDi ∂xj δBk ∧ ∧ ZM      Proof:

δF d i d δG d k δG d i d δF d k F, G = 4πc i x , k x i x , k x { } δD δB 1 2 − δD δB 1 2 "  Λ ,Λ˜   Λ ,Λ˜ # δF d i ∂ δG d j d k = 4πc i x , j k x x δD ∂x δB ∧ 1 2 − " Λ ,Λ˜ δG d i ∂ δF d j d k i x , j k x x − δD ∂x δB ∧ 1 2  Λ ,Λ˜ # δF ∂ δG δ ∂ δ = 4πc G F dxi dxj dxk. δDi ∂xj δBk − δDi ∂xj δBk ∧ ∧ ZM     

Comment 9. This coordinate representation makes it clear that the Poisson bracket has no metric dependence. As the Poisson bracket is a purely topological entity, this is one of the core advantages of the split exterior calculus framework.

41 Proposition 36. The Poisson bracket for the exterior calculus formulation is equivalent to that of the vector calculus formulation:

δF˜ δG˜ δG˜ δF˜ F, G = 4πc , d , d (126) { }  δd˜2 δb2 − δd˜2 δb2  * +Λ1,Λ˜ 2 * +Λ1,Λ˜ 2  δF δG δG δF 3 = 4πc vol . (127) δD ·∇× δB − δD ·∇× δB ZM   Proof: Because g δF˜ δF ♭ = , (128) δd˜2 δD it follows from (47) that ˜ d δF 3 = i∇× δF vol . (129) δd˜2 δD 2 Similar results hold for the functional derivativesg with respect b . Hence, from (53),

δF˜ δG˜ δF δG 3 d = vol . (130) δd˜2 ∧ δb2 δD ·∇× δB From this follows the result. g Proposition 37. The Poisson bracket satisfies the Jacobi identity. Proof: If we consider a functional F = F (d˜2, b2), then one can see that

2 2 2 2 2 2 2 2 2 2 DF [d˜ , b ](˜u , v )= D1F [d˜ , b ]˜u + D2F [d˜ , b ]v (131) δF˜ δF˜ = , u˜2 + , v2 (132) δd˜2 δb2 * +Λ1,Λ˜ 2 * +Λ˜ 1,Λ2 δF˜ =: , (˜u2, v2) (133) δ(d˜2, b2) * +X∗,X where X = L2Λ˜ 2(M) L2Λ2(M) and X∗ = L2Λ1(M) L2Λ˜ 1(M) and the duality pairing × × on the tensor product space (133) is given by the previous line. Hence, the Poisson bracket may also be written

δF˜ 0 d δG˜ F, G = , . (134) { } δ(d˜2, b2) d 0 δ(d˜2, b2) * −  +X∗,X The application of the Poisson tensor written in this notationally suggestive 2 2 matrix × format should be contextually clear. This way of writing the Poisson bracket makes it entirely clear that the Poisson tensor is independent of the fields. Hence, it follows that antisymmetry is sufficient for the Jacobi identity to hold [15].

42 3.4 Theory using only straight forms For the purposes of computation, it may be more convenient to write the Hamiltonian structure of Maxwell’s equations in terms of straight forms alone. This is because FEEC is built entirely in terms of the primal Hilbert complex of straight forms [1]. This section shows that all formulas found in this paper may be written in a manner that is compati- ble with this framework. However, the two formulations are not entirely interchangeable. Converting to a theory entirely stated in terms of straight forms obscures the metric inde- pendence of the Poisson bracket and requires use of the codifferential operator.

3.4.1 Codifferential operator Definition 56. The codifferential operator d∗ : Λk(M) Λk−1(M) is defined to be the k → formal L2-adjoint of the exterior derivative: d∗ d (η, ω)L2Λk(M) = ( η,ω)L2Λk+1(M) . (135)

Hence, while d : Λk(M) Λk+1(M), one instead has that d∗ : Λk(M) Λk−1(M). As k → k → usual, one omits the subscript “k” when there is no danger of confusion.

Proposition 38. On a manifold without boundary, or under homogeneous boundary con- ditions, d∗ = ( 1)k⋆˜−1 d ⋆˜ = ( 1)n(k−1)+1⋆˜ d ⋆˜ . (136) k − k−1 n−k k − n−k+1 n−k k For more information on the codifferential operator, see [1].

3.4.2 Hamiltonian structure with straight forms Proposition 39. The Hamiltonian may be written

δKˆ 1 Hˆ [e1, b2]= Kˆ e1, + e1, e1 + b2, b2 , (137) − δe1 8π L2Λ1(M) L2Λ2(M) !L2Λ1(M) h

i and the Poisson bracket may be written

δF d∗ δG δG d∗ δF F, G = 4πc 1 , 2 1 , 2 . (138) { } δd δb 2 1 − δd δb 2 1 " L Λ (M)  L Λ (M)# Proof: These formulas are found simply by using the identities (57) and (58).

Proposition 40. Let F = F [d1, b2]. Then

δF δF ˙ ˆ d∗h2 e1 d∗ F = F, H = c 1 , , 2 . (139) { } δd 2 1 − δb 2 1 " L Λ (M)  L Λ (M)#

43 Proof: Using (57) and (58), we find

δHˆ e1 = , (140) δd1 4π and δHˆ b2 δKˆ h2 = + = . (141) δb2 4π δb2 4π The result immediately follows from substituting Hˆ into the Poisson bracket. Proposition 41. Let F = F [ω] such that F [ω] = Fˆ[dω] (alternatively, F [ω] = Fˆ[d∗ω]). Under the assumption of homogeneous boundary conditions or if the manifold has no bound- ary, δF δFˆ δF δFˆ = d∗ alternatively = d . (142) d d∗ δω δ ω δω δ ω ! Proof: This is obtained from (61) using (57) and (58). Corollary 12. The Poisson bracket has Casimir invariants F [d∗d1] and F [db2] for arbi- trary smooth functional F .

3.5 1D geometrized Maxwell equations A one dimensional version of Maxwell’s equations is presented in this section. These equations offer a simpler system that may be useful to investigate computational schemes that respect the geometric structures explored in this paper. While the derivation of the geometrized equations is similar to that presented in the 3D system, there are enough differences that a detailed treatment is warranted.

3.5.1 1D Maxwell equations

Proposition 42. Suppose x = (x, 0, 0), E(x,t) = (E1, E2, 0), and B(x,t) = (0, 0,B3). Then one has that δK δK δK D = E 4π = E 4π , E 4π , 0 = (D , D , 0) (143) − δE 1 − δE 2 − δE 1 2  1 2  and δK δK H = B + 4π = 0, 0,B + 4π = (0, 0,H ). (144) δB 3 δB 3  3  Moreover, Maxwell’s equations become

∂tD1 = 0

∂tD2 = c∂xH3 − (145) ∂ B = ∂ E t 3 − x 2 ∂xD1 = ρ.

44 Notice that the constraint B = 0 is automatically satisfied in this setting. Moreover, D ∇· 1 becomes a non-evolving constraint. Hence, the following Hamiltonian theory only concerns the dynamical variables D2 and B3. Proposition 43. Under the assumptions above, on a manifold M R, the Poisson bracket ⊂ for the one-dimensional theory is

δF ∂ δG δG ∂ δF F, G = 4πc dx (146) { } δD ∂x δB − δD ∂x δB ZM  2 2  and the Hamiltonian is δK δK 1 H = K E + E dx + (E2 + E2 + B2)dx. (147) − 1 δE 2 δE 8π 1 2 3 ZM  1 2  ZM

As alluded to before, E1 and D1 do not appear in the bracket. In fact, the appearance of E1 in the Hamiltonian could be eliminated since this is a non-evolving quantity. It is retained here because it is needed if one couple’s the system to a matter model.

3.5.2 Differential forms in 1D In one dimension, the Riemannian metric reduces to a positive scalar quantity: g. Hence, the notion of coordinates reduces to measuring distances on the real line. For concreteness, we provide a brief recapitulation of the pertinent geometric quantities in this setting.

Proposition 44. Suppose we have a one dimensional Riemannian manifold M with metric g, and U, V X(M). We write this in coordinates as ∈ ∂ ∂ U = u and V = v . (148) ∂x ∂x Their inner product is given by ∂ ∂ ∂ ∂ U V = v u = uv = uvg. (149) · ∂x · ∂x ∂x · ∂x Proposition 45. Let u1, v1 Λ1(M). In coordinates, we write u1 = udx and v1 = vdx. ∈ Their inner product is given by uv u1 v1 = udx vdx = uvdx dx = uvg−1 = . (150) · · · g Proposition 46. The volume form is defined in coordinates by

1 vol = o(∂)√gdx. (151)

g 45 Proposition 47. Let v1 Λ1(M) and u0 Λ0(M). The Hodge star operator is given by ∈ ∈ 1 0 o(∂)v 0 1 ⋆˜1v :=v ˜ = and ⋆˜0v :=v ˜ = o(∂)√gvdx. (152) √g A zero form u0 Λ0(M) is simply u0 = u and the dot product is pointwise multiplication. ∈ Proof: The first equality follows from

1 1 1 1 1 1 1 −1 o(∂)v u ⋆˜1v = u v vol = u v o(∂)√gdx = uvg o(∂)√gdx = udx . (153) ∧ · · ∧ √g The second follows fromg 1 u0 ⋆˜ v0 = u0 v0vol = o(∂)uv√gdx = u o(∂)√gvdx. (154) ∧ 0 · ∧ g While it may seem strange to distinguish between vectors and scalars in 1D, it is ∂ necessary because of transformation rules. A vector quantity U = u ∂x has a component, u, which is scaled when coordinates change. A scalar quantity u (which does not have a different coordinate expression) does not change when coordinates change. The reason for this seemingly unnecessary distinction is that a 1D vector is easily associated with a twisted 0-form or a straight 1-form whereas a scalar quantity is easily associated with a straight 0-form or a twisted 1-form. ∂ Definition 57. A vector U = u ∂x is most naturally associated with either a twisted 0-form or a straight 1-form: 1 u˜0 = i vol = o(∂)√gu U (155) u1 = U ♭ = gudx. g Now, suppose we have a scalar quantity u. This is most naturally associated with a straight 0-form or a twisted 1-form: u0 = u 1 (156) u˜1 = uvol = o(∂)√gudx. Proposition 48. With u0, u˜0, u1, and u˜1 defined as above, one has that g 0 1 1 0 ⋆˜0u =u ˜ and ⋆˜1u =u ˜ . (157) For the purpose of applying the functional chain rule, we will need the adjoints of our operators sending vectors do differential forms. Recall from (45) that

∗ 1 −1 ( )♭ = i vol . (158) · (·)     1 A similar (almost trivial) result holds for u0 = ugandu ˜1 = uvol :

0 1 0 1 1 u , v˜ = u v˜ = uvvol = (u, v) 2 . (159) h iΛ0,Λ˜ 1 ∧ g L (M) ZM ZM g 46 3.5.3 1D Maxwell’s equations as differential forms

It is most convenient to let E1 be associated with a straight 1-form, E2 with a straight 0-form, and B with a straight 1-form.

Definition 58. Suppose the vectorial and scalar variables are given by ∂ E = e , E = e , and B = b , (160) 1 1 ∂x 2 2 3 3 Define 1 ♭ 0 1 1 e1 = E1, e2 = e2, and b3 = b3vol . (161) Then the coordinate expressions are given by g 1 d 0 1 d e1 = ge1 x, e2 = e2, and b3 = o(x)√gb3 x. (162)

Notice, in this 1D setting, only E1 transforms like a vector whereas E2 and B3 are scalars. Using the functional chain rule, which in turn utilizes the adjoint expressions derived in the previous section, one finds the following.

Proposition 49. δK˜ 1 δK˜ δK 1 = iδK/δE1 vol = 1 = o(∂)√g , (163) δe1 ⇒ δe1 δe1 ˜ g ˜ δK δK 1 δK δK d 0 = vol = 0 = o(∂)√g x, (164) δe2 δE2 ⇒ δe2 δe2 and g δK˜ δK 1 = . (165) δb3 δb3

Defining D1, D2, and H3 as in the standard 1D Maxwell formulation, we obtain the following results.

Proposition 50. Suppose the vectorial and scalar variables are given by ∂ D = d , D = d , and H = h . (166) 1 1 ∂x 2 2 3 3 Define ˜0 1 ˜1 1 ˜0 d1 = iD1 vol , d2 = d2vol , and h3 = h3. (167) Then the coordinate expressions are given by g g ˜0 ˜1 d ˜0 d1 = o(∂)√gd1, d2 = o(∂)√gd2 x, and h3 = h3. (168)

47 Notice that these relations connecting the vectorial and differential forms expressions for D1, D2 and H3 repeat the patterns seen previously in the expressions for E1, E2, and B3. Proposition 51. δK˜ δK ♭ δK˜ δK = = = g dx, (169) ˜0 δD ⇒ ˜0 δd δd1  1  δd1 1 δK˜ δK = , (170) ˜1 δd2 δd2 and δK˜ δK 1 δK˜ δK = vol = = o(∂)√g dx. (171) ˜0 ˜0 δh3 δH3 ⇒ δh3 δh3 With the definitions above, oneg may directly translate Maxwell’s equations into the language of differential forms.

Proposition 52. The 1D Maxwell equations may be written

˜0 ∂td1 = 0 1 0 ∂td˜ = cdh˜ 2 − 3 (172) ∂ b1 = de0 t 3 − 2 d ˜0 1 d1 =ρ ˜ .

d 1 1 Note that b3 = 0 is immediately satisfied since b3 is a top form. The Hamiltonian structure is also straightforward to translate.

Proposition 53. The Poisson bracket becomes,

δF˜ δG˜ δG˜ δF˜ F, G = 4πc , d , d (173) ˜1 1 ˜1 1 { } "*δd2 δb3 + − *δd2 δb3 +#

and the Hamiltonian is ˜ ˜ 1 δK 0 δK 1 1 1 0 0 1 1 H = K e1, 1 + e2, 0 + e1, ⋆˜1e1 + e2, ⋆˜0e2 + b3, ⋆˜1b3 . (174) − * δe1 + * δe2 + 8π h i h i h i   We refrain from showing that this Hamiltonian and Poisson bracket yield the above 1D Maxwell equations because the process is entirely analogous to that given in the 3D setting.

48 4 Example energy functional

The utility of the formulation of the macroscopic Maxwell equations given in [13] is its flexibility in specifying the relationship between the pure fields, (E, B), and their associ- ated materially dependent counterparts, (D, B). So far, we have left unspecified, but K for concreteness, it is helpful to consider several examples. Throughout this section, we consider functionals of E alone. It is straightforward to extend to models that include dependence on B. Moreover, we consider only which are quadratic in E. This yields K polarizations that are linear in E. While the theory may accommodate much more general expressions for that give rise to nonlinear effects, they are not considered here. K 4.1 Linear model for polarization Proposition 54. Let

(x, E)= f(x)+ U(x) E(x)+ E(x) χ(x)E(x) (175) K · · where f R, U R3, and χ R3×3. If ∈ ∈ ∈ 3 K[x, E]= (x, E)vol , (176) K ZM then if χ is symmetric g

δK δ2K = U(x)+ χ(x)E and = χ. (177) δE δEδE If χ is not symmetric, functional differentiation extracts its symmetric part: (χ + χT )/2.

Hence, one obtains a simple linear relationship between E and the polarization. In this model, U represents a permanent polarization of the medium, and χE is polarization induced by the presence of the electric field itself.

Proposition 55. If f L1(M), U, E L2(M) and χ L∞(M), then L1(M) and ∈ ∈ ∈ K ∈ δK/δE L2(M). ∈ We now wish to convert this expression to the language of differential forms.

Definition 59. Let 3 ˜3 = (x, E)vol (178) K K so that g K = ˜3. (179) K ZM

49 Proposition 56. Define

3 3 2 3 1 ♭ f˜ = f(x)vol , u˜ = iU vol , and e˜ = E , (180)

and let χ1,1 is a linear map fromg1-forms to 1-formsg defined by

1,1 1 1,1 j i kj j i χ e = χ gijE dx = gikχ E dx (181)
so that χ1,1e1 = (χE)♭. Then

3 ˜3 = [f(x)+ U(x) E(x)+ E(x) χ(x)E(x)] vol K · · (182) 3 2 1 1 1,1 1 = f˜ + u˜ e + e ⋆˜1χ e . ∧ ∧ g ♭ 3 ♭ 3 Proof: This follows directly from observing that ⋆˜1V = iV vol and W iV vol = 3 ∧ W V vol . · g g g Proposition 57. If χ is a symmetric matrix, then

δK˜ 3 3 = u˜2 + ⋆˜ χ1,1e1 = i vol = i vol (183) δe1 1 U+χE δK/δE and g g δ˜2K = ⋆˜ χ1,1. (184) δe1δe1 1 If χ is not symmetric, then functional differentiation extracts the symmetric part.

4.2 Linear polarization with dependence on derivatives Definition 60. We extend the model from the previous section with a quadratic term in E: ∇ 1 (x, E)= f(x)+ U(x) E(x)+ E(x) χ (x)E(x)+ E : χ E (185) K · · 1 2∇ 2∇ where χ2 is rank-4 tensor with the symmetry that if A and B are arbitrary rank-2 tensors,

A : χ2B = B : χ2A. (186)

kl In coordinates, this symmetry means (χ2)ij is symmetric with respect to the interchanges i k and j l. We shall see later why we write the indices of χ in this manner. ↔ ↔ 2 The term containing E merits further discussion in order to understand its general- ∇ ization to an arbitrary Riemannian manifold.

50 Definition 61. On a general Riemannian manifold, the coordinate expression for the gradient of a vector field is given by ∂ ∂ ∂ ∂ ∂ E := ( E)ij := gikgjm g El . (187) ∇ ∇ ∂xi ⊗ ∂xj ∂xm kl ∂xi ⊗ ∂xj   Proposition 58. With respect to the above definition for the gradient of a vector field,

3 3 E : E vol = ( E) ( E) vol = de1 ⋆˜ de1 (188) ∇ ∇ ∇× · ∇× ∧ 2 where, for the basis vectors,g tensor contraction means g ∂ ∂ ∂ ∂ : = g g . (189) ∂xi ⊗ ∂xj ∂xk ⊗ ∂xl ik jl     1 ♭ 1 i j Proof: Let e = E so that e = gijE dx . Then

3 1 ∂ i j k i Evol = de = g E dx dx . ∇× ∂xj ik ∧
Hence, g 1 ∂ ( E)♭ = ⋆˜ de1 = o(∂) det(g) ǫjk g Ei dxl. ∇× 2 2 | | l ∂xj ik Therefore, p
♯ 1 ∂ ∂ E = ⋆˜ de1 = o(∂) det(g) ǫjkl g Ei . ∇× 2 2 | | ∂xj ik ∂xl This is an unwieldy expression to
use for computations.p The following
shortcut to com- puting the norm of the curl on a general Riemannian manifold demonstrates of the value of differential forms in complicated geometries:

3 ( E) ( E) vol = de1 ⋆˜ de1 ∇× · ∇× ∧ 2 3 = de1 de1 vol g · ij 3 = de1 de1 vol ij g ∂Ei
ik jl ∂E
k 3 3 = g g gvol = E : Evol . ∂xj ∂xl ∇ ∇ g g It is clear from the foregoing discussion that we should consider the coordinates of the 1 i Electric field in covariant form, e = Eidx , so we will continue to do so. Notice, this corresponds with the derivative term in our matter model when χ2 = id.

51 Definition 62. Define χ2 such that ∂E ∂ ∂ χ E = χijkl l . (190) 2∇ 2 ∂xk ∂xi ⊗ ∂xj 2,2 Moreover, define χ2 such that ∂E χ2,2de1 = (χ ) kl l dxi dxj. (191) 2 2 ij ∂xk ∧ Hence, the components of the differential 2-form representation are obtained by lowering the i and j indices. Recall, (χ ) kl is symmetric with respect to interchanges i k and 2 ij ↔ j l. ↔ Proposition 59. ∂E ∂E 3 E : χ E = de1 ⋆˜ χ2,2de1 = χijkl j l vol . (192) ∇ 2∇ ∧ 2 2 2 ∂xi ∂xk To find the functional derivative of this quantity, we need onlyg consider the functional derivative of the term with derivatives since the other terms are identical to the previous case. As the presence of derivatives gives rise to complicated interactions with the metric, it is easier to first consider the derivative in the context of differential forms. Proposition 60. The matter model may be equivalently rewritten 1 (x, e1)= f˜3 + u˜2 e1 + e1 ⋆˜ χ1,1e1 + de1 ⋆˜ χ2,2de1 (193) K ∧ ∧ 1 1 2 ∧ 2 2 where all terms are defined as they were in the previous section. Proposition 61. Suppose that e1 H1Λ1(M) such that ∈ 1 tr∂M de = 0 (194) where tr : Λ2(M) Λ2(∂M) is the trace operator. Then ∂M → δ˜ 1 de1 ⋆˜ de1 = d⋆˜ de1. (195) δe1 2 ∧ 2 2  ZM  Proof: from Leibniz rule for exterior derivatives and Stokes’ theorem, one obtains

d(e1 + η1) ⋆˜ d(e1 + η1) de1 ⋆˜ de1 ∧ 2 − ∧ 2 ZM   = de1 ⋆˜ dη1 + dη1 ⋆˜ de1 + O dη1 2 ∧ 2 ∧ 2 k k ZM  
= 2 dη1 ⋆˜ de1 + O dη1 2 ∧ 2 k k ZM
= 2 η1 ⋆˜ de1 + η1 d⋆˜ de1 + O dη1 2 . ∧ 2 ∧ 2 k k Z∂M ZM 
52 The boundary term disappears because of the assumed boundary conditions. Because e1 ∈ H1Λ1(M), it follows that the exterior derivative is a bounded operator: dη1 = O( η1 ). k k k k Hence, the result follows.

1 Comment 10. For more about the trace operator, see [7]. The vector proxy of tr∂M de is E nˆ where nˆ is the unit normal vector to ∂M. ∇× · The exact same procedure yields the following result.

Corollary 13. Suppose that e1 H1Λ1(M) and χ2,2 B(H1Λ1(M),H1Λ1(M)) such that ∈ 2 ∈ 2,2d 1 tr∂M χ2 e = 0. (196)

Then δ˜ 1 de1 ⋆˜ χ2,2de1 = d⋆˜ χ2,2de1. (197) δe1 2 ∧ 2 2 2 2  ZM  Corollary 14. With the same assumptions as before,

δK˜ = u˜2 + ⋆˜ χ1,1e1 + d⋆˜ χ2,2de1, (198) δe1 1 1 2 2 and δ˜2K = ⋆˜ χ1,1 + d⋆˜ χ2,2d. (199) δe1δe1 1 1 2 2 1,1 2,2 If χ1 and χ2 are positive definite, is a generalization of the Laplacian operator. Proposition 62. In coordinates, one has that 1 ∂ ⋆˜ χ2,2de1 = o(∂) det(g) ǫ χijlm (g En) dxk (200) 2 2 2 | | ijk 2 ∂xl mn p so that 1 ∂ ∂ d⋆˜ χ2,2de1 = o(∂)ǫ det(g) χijlm (g En) dxp dxk. (201) 2 2 2 ijk ∂xp | | 2 ∂xl mn ∧   p 5 Conclusion

This work brings together two distinct modeling formats for Maxwell’s equations. Namely, the Hamiltonian structure of Maxwell’s equations which has been known at least since the work of Born and Infeld, and the structure of Maxwell’s equations in terms of differential forms whose formal elegance thus stated makes plain the relevance of exterior calculus to modeling in the physical sciences. This work provides a theory for the macroscopic Maxwell equations which is flexible in specifying a broad class of polarizations and magnetizations;

53 has Hamiltonian structure; is stated in the language of exterior calculus in a coordinate free, geometric language; is orientation independent (indeed the expressions are valid even on non-orientable manifolds); and separates the portions of the theory which are metric independent (the Poisson bracket) and metric dependent (the Hamiltonian). The desire to provide orientation independent expressions and a metric free Poisson bracket motivated the use of twisted differential forms. The integral of a twisted form over a manifold is independent of the orientation of that manifold. To be more explicit, our models are orientation independent because the Hamiltonians and Poisson brackets presented in this paper are written as the integral of a density (a twisted top form) over a manifold. Regarding metric independence in the Poisson bracket, the benefit of using Poincar´eduality as opposed to L2 duality seen in comparing (114) and (138). The former makes no use of the metric while the later is metric dependent (thus obfuscating the metric independent nature of the Poisson bracket). The Poincar´eduality pairing between k-forms and twisted (n k)-forms is metric free. Hence, so is the twisted functional derivative. − Thus, being only the composition of twisted functional derivatives, exterior derivatives, and the metric-free duality pairing, the bracket given by (114) does not depend on the metric. It is however the case that the functionals the bracket generally acts on, e.g. the Hamiltonian, may contain metric dependence. Hence, as previously stated, we have a partition of our theory into metric-free and metric-dependent components. As the Poisson bracket is a purely topological feature which is coordinate system independent, the notationally explicit metric independence of the bracket is advantageous in our theory. Since K generally depends on the metric, it follows that metric dependence is generally present in the mapping from (D, B) to (E, H). Further, as seen in [13], the simplest expression for the Poisson bracket is given when (D, B) are taken to be the evolving quantities, and (E, H) are considered to be derived quantities which are only computed when necessary. Because the Lorenz force law depends on (E, B), it follows that coupling to a particle model would at least require evaluation of E(D, B). In order to model a greater variety of phenomena, future work considering a greater variety of functionals K (in particular choices of K which give rise to nonlinear constitutive relations) and coupling to Hamiltonian models for charged particles is needed. Beyond the many theoretical benefits of a model possessing Hamiltonian structure (e.g. see [12], [8]), there is great value in leveraging Hamiltonian structure to design structure preserving discretizations. The geometric electromagnetic particle-in-cell method (GEMPIC) [9] is a Poisson integrator developed for the Maxwell-Vlasov equations. Such a numerical scheme has built in stability and does not introduce unphysical dissipation. The goal of developing GEMPIC-like discretizations for more general kinetic models models of the kind described in [13] and, in particular, a Hamiltonian gyrokinetic model [4] is a principle motivation behind the geometric treatment of the macroscopic Maxwell equations presented in this paper. Structure preserving spatial discretization relies on the structure of the de Rham complex of differential forms (e.g. see [7], [1], [10]). This in turn requires expressing the equations of motion in terms of differential forms. Hence, this this work

54 studies the geometric structure of the macroscopic Maxwell equations as a preliminary step towards incorporating the lifted kinetic models from [13] into a computational framework like that found in [9]. The paper [2] proposes a structure preserving discretization of the Hamiltonian formulation of the macroscopic Maxwell equations presented in this paper.

6 Acknowledgements

We gratefully acknowledge the support of U.S. Dept. of Energy Contract # DE-FG05- 80ET-53088 and NSF Graduate Research Fellowship # DGE-1610403.

References

[1] Douglas N. Arnold. Finite element exterior calculus / Douglas N. Arnold (University of Minnesota, Minneapolis, Minnesota). CBMS-NSF regional conference series in applied mathematics ; 93. Society for Industrial and Applied Mathematics SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104, Philadelphia, Pennsylvania, 2018.

[2] William Barham, Philip J. Morrison, and Eric Sonnendr¨ucker. A mimetic discretiza- tion of the macroscopic maxwell equations in hamiltonian form, In preperation.

[3] Werner Bauer, J¨orn Behrens, and Colin J Cotter. A structure-preserving split finite element discretization of the rotating shallow water equations in split Hamiltonian form. working paper or preprint, February 2019.

[4] J.W Burby, A.J Brizard, P.J Morrison, and H Qin. Hamiltonian gyrokinetic vlasov–maxwell system. Physics letters. A, 379(36):2073–2077, 2015.

[5] Christopher Eldred and Werner Bauer. Variational and Hamiltonian Formulations of Geophysical Fluids using Split Exterior Calculus. working paper or preprint, December 2018.

[6] Theodore Frankel. The Geometry of Physics: An Introduction. Cambridge University Press, 3 edition, 2011.

[7] Ralf Hiptmair. Maxwell’s equations: Continuous and discrete. In Computational Electromagnetism, Lecture Notes in Mathematics, pages 1–58. Springer International Publishing, Cham, 2015.

[8] Darryl Holm, Jerrold Marsden, Tudor Ratiu, and Alan Weinstein. Nonlinear stability of fluid and plasma equilibria. Phys. Rep., 123, 07 1985.

55 [9] Michael Kraus, Katharina Kormann, Philip J Morrison, and Eric Sonnendr¨ucker. Gempic: geometric electromagnetic particle-in-cell methods. Journal of plasma physics, 83(4), 2017.

[10] Jasper Kreeft, Artur Palha, and Marc Gerritsma. Mimetic framework on curvilinear quadrilaterals of arbitrary order. arXiv preprint arXiv:1111.4304, 2011.

[11] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to mechanics and symmetry : a basic exposition of classical mechanical systems / Jerrold E. Marsden, Tudor S. Ratiu. Texts in applied mathematics ; 17. Springer, New York, 2nd ed. edition, 1999.

[12] P. J Morrison. Hamiltonian description of the ideal fluid. Reviews of modern physics, 70(2):467–521, 1998.

[13] P. J Morrison. A general theory for gauge-free lifting. Physics of plasmas, 20(1):12104– , 2013.

[14] P. J Morrison. Structure and structure-preserving algorithms for plasma physics. Physics of plasmas, 24(5):55502–, 2017.

[15] Philip J. Morrison. Poisson brackets for fluids and plasmas. AIP Conference Proceed- ings, 88(1):13–46, 1982.

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