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A Hamiltonian Model for the Macroscopic Maxwell Equations

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A Hamiltonian Model for the Macroscopic Maxwell Equations A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus William Barham1, Philip J. Morrison2, and Eric Sonnendr¨ucker3,4 1Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin 2Department of Physics and Institute for Fusion Studies, The University of Texas at Austin 3Max-Plank-Institut f¨ur Plasmaphysik 4Technische Universit¨at M¨unchen, Zentrum Mathematik August 18, 2021 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 variables into 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, are orientation 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 consti- tutive relations. In addition to the three-dimensional macroscopic Maxwell equations, a one-dimensional variant, and several example polarizations are considered. arXiv:2108.07382v1 [math-ph] 17 Aug 2021 1 Contents 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.
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