Download Maxwell equations in Hamiltonian form

Maxwell equations in Hamiltonian form
©
A. Bossavit, Jan. 2007
The Hamiltonian formulation of Maxwell's theory is part of many curricula
in Electromagnetism, but the physicists' approach, well documented for
instance in [Be], tends to ignore the possibility of non-homogeneous
constitutive coefficients ε and µ, which leaves engineers frustrated. The
present note addresses this problem.
Let's review Hamiltonian formalism in finite dimension. There is a
manifold Q, configuration space, and a fiber P q = T*qQ over each point
q. Points p of Pq are covectors. Phase space is T*Q, made of points
{q, p}. Let's denote by {δq, δp} the tangent vectors, elements of
TT*Q. For such a vector anchored at {q, p}, there is a natural linear
form θ, namely 1 {δq, δp} → 〈δq ; p〉. This is the one denoted "p dq"
in Treatises, because this expression is a shorthand for ∑i pi dq i, and
〈δq ; p〉 = ∑ i pi dqi indeed. The d of this 1-form, dθ, is the symplectic
form, a 2-form living on T*Q, often denoted dp ∧ dq, here Ω. By
definition, Ω({δq, δp}, {δq', δp'}) = 〈δq' ; δp〉 – 〈δq ; δp'〉, which makes
perfect sense (integrate θ along the boundary of the parallelogram).
On the other hand, consider the Maxwell equations. Working in
1 + 3 with differential forms, we have
(1)
– ∂td + dh = j,
(3)
d = εe,
(2)
∂tb + de = 0,
(4)
b = µh,
field is j, the current density. Electric charge ρ = dd, a twisted 3-form,
derives from j by integration in time: ρ(t) = – ∫0t dj(s) ds, assuming all
fields null for t ≤ 0, and using the "conservation of electricity" relation,
∂tρ + dj = 0, implied by (1).
Comparing (1)(2) with classical Hamilton equations, ∂tq = ∂ pH and
∂tp = – ∂qH, where the Hamiltonian H is a given function of p and q,
one cannot escape a feeling of déjà vu. Yet, the connexion is not so
straightforward, as we shall see.
The framework of standard Hamiltonian dynamics is the cotangent
bundle T*Q of a configuration manifold Q. In trying to formulate the
Maxwell equations in Hamiltonian form, the first question to settle is
therefore, what plays the role of Q.
To this effect, the strongest cue comes from studying the structure of
Yang–Mills equations, where a connection a, a Lie-algebra valued 1-form,
holds the center of the stage. In electromagnetism, this a is the 1-form
whose vector proxy is the classical vector potential. This suggests taking
the space of (straight) 1-forms living on 3D affine space A3, perhaps with
some qualifications, as "manifold" Q, of course infinite-dimensional, whose
points we denote by q, instead of a, to pay lip-service to the Hamiltonian
tradition. To deal with the cotangent space, we need some duality features.
Let's start, as first guess, from the vector space F1(A3) of smooth,
compactly supported 1-forms on A3. A natural dual is F 2(A3), the space
of smooth twisted 2-forms with compact support. Denoting by q and p
the 1-form and 2 -form3, the duality bracket is 〈q ; p〉 = ∫ q ∧ p, with
summation4 over A3, as will be assumed all along when the integration
domain is not specified. Putting on Fp the L2-norm ||u|| 2 = (∫ u ∧ ∗u)1/2,
where ∗ is the Hodge operator, and completing to get Hilbert spaces
Fp(A3) and F p(A3), we get a satisfactory functional framework (with two
spaces in duality, in the sense of [Br]). Introducing a Hodge operator at
Ÿ
Ÿ
where ε and µ are understood as (local, invertible, symmetric, positive
definite2) Hodge operators. Forms e and h have degree 1, forms j,
b, d have degree 2. Forms j, d, and h are twisted, whereas e and b
are straight. Recall that ε and µ map 1-forms, respectively straight and
twisted, to 2-forms, respectively twisted and straight. The source of the
Ÿ
3
1
The duality product between a vector v and a covector ω is denoted 〈v ; ω〉.
2
Locality means they act on covectors at a point, so that ε(x)e(x) makes sense, as a
2-covector, almost everywhere. Properties evoked here are somewhat redundant.
The "momentum" p will soon be recognized as, up to sign, the "displacement
current" d of (1).
4
No need for a measure: q ∧ p is a twisted 3-form, whose integral over the 3D
manifold A3 is well defined, even though A3 is not oriented.
this level does not spoil the metric-free character of the duality pairing,
since L 2-topologies induced by different Hodge operators coincide, under
mild assumptions.
So the "manifold" Q will be F1(A3), a Hilbert space. Tangent
vectors can be defined the usual way. Tangent spaces are isomorphic to
F1(A3). But for cotangent space we select F 2(A3), refraining from using
the Riesz isomorphism. T*Q is the (trivial) fiber bundle based on F 1(A3),
with fiber F 2(A3).
So the Hamiltonian dynamics on T*Q associated with H is the
one described by the Maxwell equations. As usual, the Hamiltonian is the
sum of a kinetic energy term, here 1/2 ∫ ε–1 d ∧ d ≡ 1/2 ∫ ε e ∧ e, and a
potential energy one, 1/2 ∫ µ–1 b ∧ b – ∫ j ∧ a, where we have reverted to
electromagnetics nomenclature. One hardly escapes a feeling of arbitrariness,
and the whole procedure is anything but limpid.
Ÿ
Ÿ
We can now set up a Hamiltonian:
(5)
H(q, p) = 1/2 ∫ ε–1 p ∧ p + 1/2 ∫ µ–1 dq ∧ dq – ∫ j ∧ q.
(read q as a, the "vector potential", and p as d, the "displacement
current", up to signs, soon to be fixed). All integrands being twisted
3-forms, this is a well-formed expression. The partial derivative ∂qH is,
by definition, the twisted 2-form ω such that H(q + δq, p) – H(q, p) =
∫ ω ∧ δq, up to higher-order terms, that is to say, as readily computed 5,
ω = d(µ–1 dq) – j, an element of the cotangent space Tq*Q. The other
partial derivative, ∂pH, is the straight 1-form ε–1 p, an element of the
tangent space TqQ. Since ∂tq and ∂tp are in TqQ and Tq*Q
respectively, the following equations are well formed:
∂tq = ∂pH,
but again, no provision is made for non-constant ε and µ (and besides,
the confusion between B and H is patent). The canonical Hamiltonian
formalism one could imagine from that, namely
∂te = ∂hH,
∂th = – ∂eH
with H(e, h) = 1/2 ∫ h ∧ dh + 1/2 ∫ e ∧ de – ∫ j ∧ h, doesn't foot the bill:
the types of the different entities do not match, and trying to insert Hodge
operators wreaks havoc.
References
∂tp = – dµ–1 dq + j.
Setting b = dq, h = µ–1b, and d = –p, we recover (1) and (4). Setting e
= –∂ tq yields (3), and time-differentiation of b = dq then provides (2).
One checks that d(εe), which is – dp, is indeed the charge ρ evoked
earlier. Note the implicit "gauging" of q, alias a: the vector potential
selected by this procedure is minus the primitive in time of the electric
field, a popular choice in computational electromagnetism [Pl, B1].
5
L(E, H) = 2 ∫ B · ∂tE – ∫ E · rot E – ∫ B · rot B + ∫ J · B,
∂tp = –∂qH,
and develop as
ε∂tq = p,
Could one proceed in more direct fashion, in terms of fields only,
without the intermediation of a vector potential? There is indeed a variational
formulation using helicity [AA], which consists in stationarizing the functional
(expressed here in terms of proxy vector fields, not differential forms)
Use symmetry of µ– 1 and notice that d(µ–1 dq ∧ δq) = d(µ–1 dq) ∧ δq
– µ– 1 dq ∧ dδq, hence ∫ µ–1 dq ∧ dδq = ∫ d(µ–1 dq) ∧ δq, integrating by parts.
[AA] N. Anderson, A.M. Arthurs: "Helicity and a variational principle for Maxwell's equations",
Int. J. Electronics, 54, 6 (1983), pp. 861-4.
[Br] H. Brezis: "Équations et inéquations non linéaires dans les espaces vectoriels en dualité",
Ann. Inst. Fourier, 18, 1 (1968), pp. 115-75.
[Pl] R.D. Pillsbury, Jr.: "A three dimensional eddy current formulation using two potentials:
The magnetic vector potential and total magnetic scalar potential", IEEE Trans.,
MAG-19, 6 (1983), pp. 2284-7.
[B1] A. Bossavit: "Two dual formulations of the 3-D eddy-currents problem", COMPEL, 4,
2 (1985), pp. 103-16.
[Be] G. Belot: "Symmetry and Gauge Freedom", Studies in Hist. & Phil. Mod. Phys., 34
(2003), pp. 189-225.