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Lagrangian formalism and conservation laws for electrodynamics in nonlinear elastic dielectrics
ANNALS
OF PHYSICS
220, 18-39 (1992)
Lagrangian Formalism and Conservation Laws for Electrodynamics in Nonlinear Elastic Dielectrics H. SCHOELLER AND A. THELLUNC Institut jiir Theoretische Physik Schiinberggasse 9, CH-8001
der Universitiit Ziirich, Ziirich, Switzerland
Received June 9. 1992
For the continuum theory .of electromagnetic fields in interaction with an elastic dielectric medium a new Lagrangian formalism is developed. The medium may be arbitrarily nonlinear and anisotropic, but in order to have conservation of quasimomentum it has to be homogeneous. For simplicity, the frequencies of the excitations are assumed to be such that dispersion can be neglected, but it is indicated how it can be taken into account. Directly from the Lagrangian equations of motion and alternatively by means of Noether’s theorem the general conservation laws for energy, momentum, and quasimomentum are derived. They are formulated in both local (Eulerian) and material (Lagrangian) coordinate systems. Special attention is given to quasimomentum. Its density in local coordinates xkr where x = y + u (y, = material coordinates, u(y, t) = displacement vector of the elastic medium), is given by
The first two terms are the well-known expressions for the electromagnetic and the elastic quasimomentum. The third, mixed, term is new. It can only be found if the difference between local and material coordinates is taken fully into account. 0 1992 Academic PRSS, h.
1. INTRODUCTION In this paper a new Lagrangian formalism is presented which provides the phenomenological electrodynamical and mechanical equations of motion in a nonlinear elastic dielectric. The medium may be anisotropic, but is supposed to be macroscopically homogeneous. One advantage of the Lagrangian formalism is the following: It permits a clear and systematic way of deriving the conservation laws of energy, momentum, and quasimomentum ( = pseudomomentum) either directly from the Lagrangian equations of motion or, via Noether’s theorem, from the symmetry properties of the media. In this way, statements made in an earlier paper [l] about Lagrangians in nonlinear electrodynamics and about Noether’s theorem will be demonstrated in detail. Another advantage of Lagrange’s equations of motion is their suitability for the transition from one coordinate system to another, in our case from local ( = Eulerian) coordinates, as used in electrodynamics, to material ( = Lagrangian or substantial) coordinates, as usually employed for the description 18 0003-4916/92 $9.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
19
of elastic media. In some places of Ref. [l] the difference between local and material coordinates was neglected for simplicity and because in most practical cases this is a good approximation. In this paper their difference is fully taken into account. For elastic media this was already done by Kobussen [2] and Peierls 131. Some work on photoelastic media starts from a microscopic description of the material particles and then, in the long-wavelength limit, the transition to continuous media is carried out. This transition is performed in many textbooks and the limitations of the macroscopic theory are well known so that we use the continuum theary from the very beginning. Also, to keep our equations relatively lucid, we assume the frequencies and wavelengths to be such that we can neglect dispersion. This means that, e.g., in electrodynamics the fields D and H are functions of E and B (or vice versa) and not of their temporal or spatial derivatives (the case of dispersion can also be treated and will be the subject of a forthcoming paper). However, the material laws may be nonlinear and the material “constants” (like the dielectric or magnetic susceptibilities) may depend on the elastic strain. Similarly. the elastic “constants” may depend on the electrodynamic fields. Special attention is given to the conservation of quasimomentum, which in nonlinear optics and in the theory of elasticity is an especially interesting quantity. It admits a simple and very general analysis of various physical phenomena, such as, e.g., (a) the radioelectric effect (see [ 1, 211 and references given there), (b ) photomagnetism [4], (c) the acoustoelectric [S], and (d) the acoustothermal [6] effects which are caused by the transfer of quasimomentum from electromagnetic waves to conduction electrons (a), (b), or from acoustic waves to electrons (c) or thermal phonons (d). How does one define quasimomentum in electrodynamics and in the theory of elasticity? In the opinion of the authors the safest method is to couple the system under investigation to another system, where quasimomentum is a well-known concept, e.g., thermal phonons or Bloch electrons for which the expression
(k, = wave vector, N, = number of phonons or electrons in mode M) is the quasimomentum (also called pseudomomentum or crystal momentum). If now, in the system under investigation, a quantity P,, is found whose sum with expression ( 1.1) (in the absence of Umklapp processes) is conserved one obviously must identify P,, with the quasimomentum of that system. This is the method adopted in Refs. [ 1, 71. Another possibility is to quantize the theory and diagonalize the harmonic part of the Hamiltonian, which leads to photons or acoustic phonons. The quantity P,, then can be shown to take the form (1.1) where N, now means the number of photons or acoustic phonons. This is another justification of this name. The authors have verified this fact. The calculation will not be presented in this paper. For the elastic case it was done by Kobussen and Paszkiewicz [S].
20
SCHOELLERAND THELLUNG
A third possibility is to make use of the following symmetry property characteristic of homogeneous media. If all the fields (in our case electromagnetic fields and elastic deformation) are spatially displaced by a vector E, but without shifting the medium, then in a homogeneous medium the course of physical events will be the same. Via Noether’s theorem this invariance property leads to a conserved quantity which is identical with the quasimomentum found by the first two methods. Therefore, this invariance property can be used to define the quasimomenturn. This is done in Refs. [3, 91. In Section 2 a Lagrangian formalism is presented for electrodynamics in a homogeneous medium moving with a uniform velocity in. The equations are written in a relativistically covariant form, but eventually terms of relative order (G/c)’ are neglected. In Section 3 the Lagrangian formalism is extended to electrodynamics in elastic dielectric media. The difference between local and material coordinate systems is pointed out and the general form of the conservation laws is derived (a) directly from the equations of motion and (b) from Noether’s theorem. In order to discuss the energy of the system it is convenient to introduce a Hamiltonian and the canonical formalism. This is done in Section 4. Then the detailed derivation and discussion of the conservation laws for energy, momentum, and quasimomentum is given in local coordinates (Section 5) and in material coordinates (Section 6). In Section 7 the general relationship between the Lagrangian equations in local and material coordinates is analysed. This is not a trivial coordinate transformation because the two coordinate systems differ by the displacement vector of the elastic medium which is a dynamical field variable. It is shown that quasimomentum conservation in one coordinate system implies quasimomentum conservation in the other system. The same is true for momentum conservation, but it is interesting to note that the Lagrangian equations of motion for the elastic medium in material coordinates yield the momentum conservation law, whereas in local coordinates they yield the conservation law for the difference between momentum and quasimomentum. 2. LAGRANGIAN FORMALISM FOR ELECTRODYNAMICSIN A UNIFORMLY MOVING MACROSCOPICALLYHOMOGENEOUSMEDIUM
Let us consider a macroscopically homogeneous medium moving with a constant velocity 8. Maxwell’s equations read V.D=4np, VxH-ib=Fj, C
V-B=0 VxE+b=O.
c
c
(2.la) (2.lb) (2.2a) (2.2b)
LAGRANGE
FORMALISM
21
IN PHOTOELASTICITY
Later on, we shall only consider dielectric media, but in this section we can include the possibility of nonzero electric conduction current and charge densities, j,. and pc. Using a relativistically covariant notation, we can write these equations in the form (2.3) (2.4)
with j’,’ = (cp,., j,) and
*FP” =
i
0
B,
&
B,
-B,
0
E,
-B2
E,
-E, 0
-E2
El
\-B,
- El 0
The homogeneous equations (2.2) or (2.4) are equivalent to the representation the fields E and B by the electromagnetic potentials A” = (cp, A)
of
E= -IA-V+3 C
(2.5)
B=VxA.
The inhomogeneous
part (2.3) can obviously
be written
in Lagrangian
form
(A, ,,,, = dAJ8.u”) a,-+E
(2.6)
bL.1 aA,
by using a Lagrangian
density f!(A,,,
A,,) as proposed in [ 11 with (2.7)
and ai2 i W.-.---z -- cjY. 8A,
(2.8)
Considering Q = !i!(E, B, A,) as a function of the electromagnetic potentials, it is easy to show from (2.5) and (2.7) that a2
-=dE
1 4n
JX
aL! g=
-$H.
tields and the
(2.9)
22
SCHOELLERAND THELLUNG
which is also valid in the rest system of the medium, since (2.7) is written in a relativistically covariant form. We want to emphasize that all these equations are independent of the specific material laws which can have either a linear or a nonlinear form. In a homogeneous and nondispersive medium, these material laws will have the general form D’(E’, B’) and H’(E’, B’), where E’, B’, D’, H’ are the fields in the rest system. [22] Using the transformation laws [18, 191 E’=E+kB, D’=D+kH,
c c
&-B-&E C
(2.10) H’=H-&xD C
(all terms of order (C/c)’ have been neglected), we see that the material laws D(E, B, ir) and H(E, B, ir) depend on ti in the laboratory system. Thus, from (2.9), 2 will also depend on ti, i.e., I! = !Z(E, B, 8, A,). Furthermore, since (! must be an invariant (see (2.7)), the dependence on i is explicitly given by f?(E, B, 8, AJ = Q’(E’, B’, A;),
(2.11)
where 2 denotes the Lagrange function in the rest system. Using (2.10), (2.9), (2.8), and
cP’=rp-liA, A’=A-lcpi, c c
(2.12)
we obtain a!2 1 ~=~(EXH-DxB)+~p,A-;j,p. In order to be consistent with our unrelativistic approximation higher order in C/c have been neglected in (2.12) and (2.13).
(2.13) in (2.10), terms of
3. LAGRANGIAN FORMALISM FOR ELECTRODYNAMICSIN HOMOGENEOUS ELASTIC DIELECTRICS. GENERAL DERIVATION OF CONSERVATIONLAWS
From now on and for the whole rest of this paper we only consider dielectric media, i.e., pC = j, = 0. Generally, the deformation state of an elastic medium is described by the displacement field v(x, t) = u(y, t) = x - y,
(3.1)
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
23
where x and y denote the local (or Eulerian) and material (or Lagrangian or substantial) coordinates of the system, respectively. Throughout this article we use the terms “local” and “material” coordinates instead of “Eulerian” and “Lagrangian” coordinates in order to avoid confusion with the term “Lagrangian” designating the formalism developed in Sections l-3. For convenience, we shall also use the four-dimensional notation U, = (0, u) and u, = (0, v) (Greek indices take the values CI= 0, 1, 2, 3 and Latin indices i = 1,2, 3). From now on all vector indices are written as subscripts; they designate contravariant components; covariant components are no longer,being used. In the same way, we write x, = (ct, x) and y1 = (ct, y) for the four-dimensional local and material coordinates. The connection between the tensors u,,8 = do,/&, and u~.~ = du,/dy,j is given by
or by the well-known
formulas [2]
The mass density in local coordinates is denoted by p(x, t ), which is connected to the mass density p0 in material coordinates by p d3x = p. d3y
(3.4a)
or
where A = det(8x,/8yp) = det(ax,/ay,) is the Jacobian of the transformation. Note that in a homogeneous medium p,, is a constant. Now, if we consider an elastic dielectric medium (with jf = 0), together with an electromagnetic field, the total Lagrangian density f! (in local coordinates) of the system no longer depends on A, (viz. (2.8)) but only on A,., (or E, B) and u~,~. It is quite possible to include higher derivatives of A, and v, as arguments in 2. (For the case of elastic media compare the remark in [7, end of Section II]). For the electromagnetic fields this means that, e.g., D and H also depend on derivatives of E and B. This gives rise to dispersion. For simplicity, we exclude this possibility in the present paper. The case of dispersive media will be treated in a future article. Furthermore, we assume that (2.9) remains valid (which is necessary for the validity of Maxwell’s equations) and that (2.13) is only supplemented by the i-derivative of the kinetic energy density $pti* of the elastic medium (3.5)
24
SCHOELLER
AND
THELLUNG
Originally, the derivation of (2.13) was based on li being the constant velocity characteristic of a Lorentz transformation. We now assume (2.13) and (3.5) to be valid as local laws for (slowly) varying ri (compare a remark in [18, p. 1011). By using (3.3), the last equation becomes
ai? ax. T=-J &+&(ExH-DxB), at+ ayj
1 .
Finally, we do not make any assumptions about the dependence of 2 on v~,~;i.e., the Lagrangian can be an arbitrary function of the strain tensor. To obtain the general conservation laws of our system, we can proceed in two different ways. The first possibility is to start from the Lagrangian equations of motion
--a aQ) ax,aA,, and
--a a2=, ax,aUa.b’
(3.8)
which yield
(3.9) or (3.10) This has the form of a continuity equation and means that there exists a conserved quantity. In Section 5 we shall see that it corresponds to the energy conservation for ,u = 0 and to the quasimomentum conservation for p = 1,2,3. The conservation of ordinary momentum, however, is given by the addition of (3.8) and (3.10), which is also demonstrated in Section 5. The second possibility is to use certain invariance properties of 2 together with Noether’s theorem. This theorem can be stated as follows [lo]: If a simultaneous infinitesimal transformation of the coordinates xP and the fields $i (i = 1, .... n), (3.1 la)
x;=x,+ax, tiltx’)
=
$iCx)
+
d$i(x)7
(3.11b)
LAGRANGE
FORMALISM
IN
PHOTOELASTICITY
25
leaves the action integral
W=ld 4xqx, $j(XLIc/;.,W) invariant (integration equation holds
over an arbitrary
(3.12)
region G), then the following continuity
(3.13) where &+!I,is defined as the field variation
at a fixed point s,
@i(X) = $:(-xl - 3,(x),
(3.14)
such that 6$j(x)=
$$itx)
+
\I/i.p
6xJ,.
(3.15)
We apply Noether’s theorem to two special cases. (a) Homogeneous media. L! only depends on the fields $, and their derivatives, not explicitly on X: f!($,, IJ+,,+).A displacement in space-time of the fields by an arbitrary infinitesimal constant displacement vector E,, together with a simultaneous shift of the integration variables (and the region of integration) by E,.,
x: - x,,= sx,,= E, I);(.~‘)- $i(x) = 6~i(x) = 0,
(3.16)
leaves f? and d4.u and, therefore, W invariant. According to (3.15) and (3.16) &hi(X) = -I),,,.&,
(3.17)
and if we take just one of the E,,‘sdifferent from zero, (3.13) takes the form (3.18) Now, in a photoelastic medium the set of fields (I,?,1 is the set {A,, v, ) and (3.18) turns into (3.10), giving conservation of energy and quasimomentum (see Section 5). In fact some authors [3,9] use this invariance property to d&e quasimomentum. (b) f? does not depend on the fields themselves (only on their derivatives, and possibly on the coordinates). L? (x, q,,,). Then L! is invariant with respect to the transformation 6x, = 0,
&hi = &hi = E,= const.
(3.19)
26
SCHOELLER ANDTHELLUNG
(and so is d4x and W). Then (3.13) yields a
ax? =o, ( wi., >
ql
(3.20)
Vi,
which are Lagrange’s equation of motion (for this particular f?). Applied to a photoelastic medium, one obtains Eqs. (3.7) and (3.8). As mentioned before, (3.8) means conservation of ordinary momentum minus quasimomentum, whereas (3.7) gives Maxwell’s equations (2.la) and (2.lb) for pC= j,=O.
4. CANONICAL Let us detine the Hamiltonian transformation of L3(E, B, u~,~)=
FORMALISM
density in local coordinates
by the Legendre
e(A, VP, B, i, u~,~) as
jjr=A.pA+i=pu-f?
(4.1)
with the canonical momenta pats=
ai?
ldL!
-cE=
1 -4ncD
(4.2)
and pa=%+
air ay
pri,+-&(ExH-DxBI,
1 ,
(4.3)
according to (2.9) and (3.6). (By the way, pD is just the difference between the momentum and quasimomentum densities, as can be seen from the results of Sections 5.2 and 5.3.) From (2.5) and (2.la) with pC=O we obtain (4.4) and, since the last term does not give any contribution H= j d3x $j’ [ 111, we define the energy density by 5j=&E.D+P.p,-2.
to the total energy
(4.5)
For its differential, we obtain from (4.2) and (4.3) (4.6)
LAGRANGE FORMALISMIN
27
PHOTOELASTICITY
so that $j has to be considered as a function $(D, B, pt., L),.~)and, in view of (2.9) its derivatives are given by
a5 -=-
asj lH -=aB 4x
lE
aD
4x
’
(4.7)
asI,. p,=v. Furthermore,
we will define a quantity
U by the relation
sj = fpic + u.
(4.8)
U is the total energy density minus the kinetic energy density of the elastic medium. Up to now, all quantities have been defined in local coordinates. For the material coordinates, in analogy to (3.4b), we define
Q(E, B, ~2,~)= AW, B, u,,,j) (4.9 1
f;CD, B, P,> u;,k) = AND, B, P, >cl,,) o(D,
B,
Pu,
l4i.k)
=
A U(D,
B,
Pt 7 l’1.k I
with -
p,=g=p@+&(E~H-DxB).
(4.10)
where we have used (3.3), (3.6), (3.4b), and the fact that A does not depend on 5. In (4.9), we have not transformed the electromagnetic fields to material coordinates since an explicit evaluation of the conservation laws (3.8) and (3.10) would lead to rather complicated expressions for the electrodynamic parts in this case. However, for the general case, we shall transform the fields as well in Section 7 and show that the Lagrangian formalism is completely equivalent in local and material coordinates.
5. CONSERVATION
LAWS IN LOCAL
COORDINATES
5.1. Energy Conservation
Equation (3.10) reads for p = 0,
= 0.
(5.11
28
SCHOELLER
AND
THELLUNG
Using (2.5), (2.9), (4.1), (4.4), and (4.5), this becomes
Due to Maxwell’s equations, all q-dependent we obtain
terms cancel each other exactly and
(5.3) which is the energy conservation law in local coordinates. Furthermore, by using (4.7), (4.8), and (4.9), we can express aQ/av].)i. by a@%,& (see the Appendix)
ax, . . ax,. L(ExH-DxB),
+y,p”ku~+dy,uk4nc
(E
x W,
(5.4)
where E and H are considered as functions of D, B, pu, and u[,,, in the last term. Multiplying this equation by tij and using (3.3), we obtain for (5.3) the final result, (fj$+&[
-~~p,(~)D,B,p”+d,t)+~(ExH),]=O.
(5.5)
Here, terms of relative order (C/C)~ with respect to (c/4n)(E x H)k have been omitted in accordance with the unrelativistic approximation already adopted in Section 2. 5.2. Quasimomentum
Conservation
For p = i, (3.10) yields
a at, (>(
- -$ A j, ; - 2, vi, i J
I
ae Acc,ia~Vj.i+6ikQ
(5.6)
LAGRANGE
Applying
FORMALISM
29
IN PHOTOELASTICITY
(2.5), (2.9), (3.6), and (3.3), we obtain a (ILt,
-~~~pri~+&(DxB)~
1
-u,,~(EXH-DxB)j+~D,A,.,
a +ig
-~qi-~(EiDk+H,B,,
+
6ik
J.h
&H-B+P
>
-&&&(HxV),A,
=O. J
(5.7)
Again, the A-dependent terms cancel each other due to Maxwell’s equations and, together with (4.5), we arrive at
a sit, (>I
1 a -gVj,i-$(EiD,+H,B,) +zk I,h -w,i~ljj+~(DxB),--u,;~(ExH-DxB);
+ 61, & (ED + HB) + 6ik ($ti’+B;&(ExH-DxB)
I
u) =O.
(5.81
Multiplying (5.4) with vi,,, using (3.3), and inserting the result in (5.8), we obtain the final form
a i z,>[
1 +~[~~u,,,(~),,.,~"-~(EiDh+H -uiipB,+~(DxB)j--,,i~~(ExH-DxB),
fh,&(ED+HB)+~(;pb’-(I)
- uj.Jni;li, - U,.,lik&(ExH-DxB), +6,,+ExH-DxB)
(5.9)
30
SCHOELLER
AND
THELLUNG
This equation corresponds to quasimomentum ( = pseudomomentum) conservation for the following reason: The density of the conserved quantity consists of three terms: a purely material term, --~~,~pti~; an electromagnetic term, (1/47rc)(D x B), and a mixed term, - uj,i(1/47rc)(E x H -D x B)i. Since - z+ptii is well known to be the quasimomentum density of an elastic medium [2, 3, 7,9] one has to interpret (1/47rc) D x B as the quasimomentum density of the macroscopic electromagnetic fields in a dielectric medium in agreement with [l] and with the interpretation of Minkowski’s result [12] by, e.g., Blount [13], Gordon [14], and Peierls [3]. The mixed term, - u,~( 1/47cc)(E x H - D x B)j is missing in the above-mentioned articles because in Refs. 12, 3, 7, 9, 12-141 electromagnetism or elastic media are studied separately, whereas [ 11 treats the coupled systems, but in a first approximation explicitly neglects the difference between local and material coordinates for the electromagnetic contributions. 5.3. Momentum
Conservation
Using the Lagrangian equations of motion (3.8), together with (4.3) and (5.4), we obtain
ax. a -Jpti,+TTt,a~i O[
~;&(ExH-DxB),]
,
axj . . axj. &ExH-DxB), + ayi p”juk + ayi uk 471c (ExH)
Since in the first term on the right-hand
C5
Ptij
=
(Sji
+
Uj,i)
1 .
(5.10)
side of (5.10)
plij
=
plii
+
pUj,itij
aYi
is the density of momentum minus quasimomentum of the elastic medium, (5.10) and (3.8) give the conservation law for this difference. In order to obtain the conservation law for the momentum alone we add (5.9) and obtain the final result
LAGRANGEFORMALISMINPHOTOELASTICITY
+ 6, & (ED + HB) -&
31
(EiDk + H;B,)
(ExH)
1
=O.
(5.11)
Since pti is the momentum density of the elastic medium, obviously (1/4rcc)(E x H) is to be interpreted as the momentum density of the macroscopic electromagnetic fields (which includes the vacuum fields as well as the polarizable and magnetizable matter). This result agrees with Abraham’s expression [15] and its interpretation by, e.g., Blount [ 133, Gordon [14], and Peierls [3]. Furthermore, using the same unrelativistic approximation as we employed in Section 2, the same has been obtained by de Groot and Suttorp [16, 171.
6. CONSERVATION
The transformation
LAWS IN MATERIAL
of an arbitrary
continuity
COORDINATES
equation in local coordinates
3, 0 -= ax,
(6.1)
to material coordinates is given by (6.2)
with
j,=&
The proof is trivial due to the Euler-PiolaaJacobi &A$)=0 . L1( 595/220/l-3
or B>
(6.3)
axflJa. identity [20]
a -2i ax = 0. ax, ( day,, >
(6.4)
32
SCHOELLER
AND
THELLUNG
From (6.3) and (3.3) it follows that (6.5)
lo = 40 and 69
(6.6)
Thus we can obtain all the conservation laws in material straightforward manner from (5.5), (5.9), and (5.11).
coordinates
in a
Energy conservation,
-a5 ( at)
+ Y
a -zii G [
1
+zA-&(ExH)k
=o.
(6.7)
Quasimomentum conservation,
-
--u ah. axk.
A kG(DxB),+$&(ExH-DxB) I
1
(ExH) Momentum
=O.
(6.8)
conservation,
+z$(ED+HB) I
-$$(E,D,CH,Bk)-$$k&(DxB)j k
+$&(ExH-DxB) I (ExH)
1
=O.
(6.9)
LAGRANGEFORMALISMIN
33
PHOTOELASTICITY
Now let us compare these results with the ones obtained in [l]. The energy conservation law (6.7) (see Eqs. (4.6)-(4.8) of Reference [ 11) has nearly the same form but the prefactor (+,/Jx,) A in front of the flux term (c/47r)(E x H) is missing in [ 11. This corresponds to the assumption that (6.10)
lUi.kl e 1
and is consistent with the neglect of the difference between local and material coordinates in [ 11. Similar correction factors can be observed in the quasimomenturn conservation law (6.8) in comparison with Eqs. (4.11 k(4.13) of Ref. [l]. Furthermore, the quasimomentum density in (6.8) has an additional term -u,,,(l/47cc)(ExH-D x B)j (see also the remarks at the end of Section (5.2)) which, in comparison to (1/47cc)(D x B), can again be neglected under the assumption (6.10). The last three terms of (6.8) are of relative order C/c and are due to the fact that we have used the correct transformation laws (2.10) for the electromagnetic fields. For B 4 c they are not important and consequently do not appear in Cl]. Finally, the conservation law (6.9) of ordinary momentum has not been considered in [ 11.
7. GENERAL LAGRANGIAN
FORMALISM
IN LOCAL
AND MATERIAL
COORDINATES
In this section we want to analyse the general relationship between the Lagrangian formalism in local and material coordinates. In the first part we will see that Lagrangian equations of motion also hold in material coordinates if the derivatives of the electromagnetic potentials (7.1) where A.LM)=hvL are transformed in the same way as the derivatives according to (3.2)
(7.2)
of the displacement
field
(7.3 1 Note that this is not a Lorentz transformation since we do not transform the electromagnetic potentials themselves but only look at the same functions (7.2) (in analogy to (3.1)) and take their derivatives in two different coordinate systems. The Lagrangian in material coordinates is then given by
34
SCHOELLER
AND
THELLUNG
in contrast to (4.9), where the electromagnetic fields are not transformed (see the remarks at the end of Section 4). To prove the equivalence of the Lagrangian equations (3.7) and (3.8) with the Lagrangian equations a
-7=o,
a2
-- a
ac-, ah aun.8
ah aA,,
we first transform the derivatives a!$/au,,p and ai?/aA,,
’
(7.5)
of (! as follows:
(7.6)
Using (7.7)
and (7.8)
which can easily be proven from (7.3) and (A.l), we obtain, together with (A.2),
-=a2 a%8
laxq;
laxajj
A ah
A ah
i ad- ax a2 aQ _+--t-Ep3aaA,,, A ah ah au,,.
(7.9)
and a2 i ax, a!2 -=--7. a4, A ah a-4,”
(7.10)
For the derivative of (7.9) we obtain
where we have used the Euler-Piola-Jacobi (a/ax,). Since ae_, ay,-
&+u
Ir,v,aaA,,,
identity (6.4) and alay, = (axs/ayd) x ai2
p,v,a aup,v’
LAGRANGE
FORMALISM
IN
35
PHOTOELASTICITY
(7.11) can be brought to the final form a a2 ---=-a
1
axgav,,8 A
a ai2 i ax,, a aQ ~GjpZ&+ddl?,dqlpau,,;
(7.12)
In the same way we obtain from (7.10)
--a ar! i a aQ ax, aA,,Yi~c?A,l,’
(7.13)
The last two equations are general formulas for the relationship between the derivatives (a/ax,)(aC/av,,), (d/ax,)(a~/dA,,), and the corresponding derivatives in material coordinates. They were obtained without the use of the equations of motion. Obviously, they manifest the equivalence of the Lagrangian equations of motion in both coordinate systems. We want to emphasize that our result is not simply a consequence of the invariance of the Lagrangian equation of motions under ordinary coordinate transformations, since here the transformation matrix ax,/ayl, = 6,, + u~,~ is a field variable. In the second part of this section we want to demonstrate the equivalence of the conservation laws in the two coordinate systems. Let us multiply (7.12) by -u,.,, and (7.13) by -A,,, and take the sum of the two expressions. Then, with the aid of (3.3) and the inverse of Eq. (7.3),
we arrive at the relation
I
(7.14)
Remembering that (3.10) means conservation of energy (p = 0) and quasimomenturn (p = 1,2,3), Eq. (7.14) shows that these conservation laws take the same form in local and material coordinates. In Section 5.3 it was shown that (3.8) expresses the conservation law for the difference between momentum and quasimomentum. In order to obtain the conservation law for the momentum alone we take Eq. (7.12) for r = i and add Eq. (7.14) with p = i. After some straightforward manipulations we arrive at the relation
-+b,f?--
ait a2 i a a2 A,,;-- a0 ‘=,i 7 iiy, au,,s’ W., 2.It I
(7.15)
36
SCHOELLER
AND
THELLUNG
This shows that the momentum conservation law takes a particularly simple form in material coordinates. It is just expressed by the Lagrangian equation of motion for the field u. This can easily be checked by looking at the term containing the time derivative on the right-hand side of (7.15), (8/8t)(dQ/&,). Since the n-dependence of j? is given by the kinetic energy &,8* + linear terms in ti, we have that &/atii = pOlii + terms independent of 8, which contains the momentum density of the matter without any admixtures of quasimomentum. Finally, let us investigate the conservation laws from the point of view of Noether’s theorem. In Section 3 we saw that the energy-quasimomentum conservation law (3.10) is the result of the invariance of 2 under the variation in x-space, (7.16a)
6x, = E,, ihi = 6A, = 0,
which implies &Ii= -Vi*“&“, Since y = x - v, x = ye+ u, A,(x(y))
= A,(y),
&Y, = Ev,
(7.16b)
c?Ar = - Ap,“&“.
it follows from (7.16a) that
6Ui = sa, = 0,
(7.17a)
which implies &li= -Ui,“&,,
q
= -ap,“&,.
(7.17b)
The variations in x-space and y-space are thus completely analogous and consequently the energy-quasimomentum conservation law has exactly the same form, as already put in evidence by (7.14). For the elastic variables the equivalence of the variations (7.16b) and.(7.17b) was already demonstrated by Kobussen [2] and Peierls [3] (compare the remark in [ 1, end of Section 33). As a last application of Noether’s theorem let us look at momentum conservation. Its simple form in y-space,
--a a-c, ah
(7.18) ’
aui,p
is obtained from the invariance of !G with respect to the variations 6Ui = &di = E,,
6y,=O,
The corresponding 6t=O,
sa,
= 22,
= 0.
(7.19)
variations in x-space are easily found to be 6Xi = Ei,
6A,=O,
Now, (7.20) is a superposition
&Ii
= Ei,
hi
JA, = - A,,iei.
of two variations,
= Ei -
V+Ek
(7.20)
LAGRANGE
FORMALISM
1N PHOTOELASTIClTY
37
and (7.22 I
Invariance with respect to the first variation leads to (3.7), i.e., conservation of momentum minus quasimomentum, invariance with respect to the second variation gives quasimomentum conservation, the sum yields momentum conservation. The proof of the equivalence of all conservation laws in this section is a generalization of the work by Kobussen [2]. Here we have included electrodynamics. Furthermore, we note that the relations (7.12))(7.15) have been obtained without the use of the equations of motion. APPENDIX
In this Appendix we derive Eq. (5.4). To begin, let us mention elementary properties:
the following
(A.1 1
--a i = ---ax, i aua.S0A ay, A’
(7 au,.,
QA A = ax,
(A.21
The first one can be obtained by differentiating 6 Jw@v --==((s,,7-t~,,,,)
I’? ayvax, a?,
with respect to v~,~, and the second one follows directly from l/A = det(6,, - r,,,{) and A = det(6,, -t usB). Now, from (4.7) and (4.9) we have
i a2dlmas --AA au,, ah,’
(A.3 1
The first and third terms of this expression can be evaluated by using (A.1 ) and (A.2). For the second one we use
p.,=As ay P,, = A(6,,,I I
urn,,) PC,
38
SCHOELLER AND THELLUNG
and, observing (4.9) and { +}
= const o { u~,~}= const,
where we have used (4.7) and (3.3). Thus (A.3) reduces to
Furthermore,
from (4.8), (4.9), and (4.10) we obtain
or with (A.2) -=a.5 au,,,
al7 ay,.
A
--p.G(ExH-DxB)-l.~-
aul,,
4n~
I
a
au,m
(ExH).
(A.7)
Inserting this expression in (A.6) and using (4.3) and (4.8), we finally arrive at (5.4).
ACKNOWLEDGMENTS The authors would like to thank V. L. Gurevich for helpful remarks and fundamental discussions, L. A. Turski for an interesting comment, and the Schweizerischer Nationalfonds for financial support.
REFERENCES 1. V. L. GUREV~CHAND A. THELLUNG, Physica A 188 (1992) 654. 2. J. A. KOEJUSSEN, He/v. Phys. Acra 49 (1976), 599. 3. R. PEIERLS,in “Highlights of Condensed-Matter Theory” (F. Bassani, F. Fumi, and M. P. Tosi, Eds.), p. 237, North-Holland, Amsterdam, 1985. 4. V. L. GUREVICH, R. LAIHO, AND A. V. LASHKUL, Phys. Rev. Left. 69 (1992), 180. 5. G. WEINREICH,Phys. Rev. 107 (1957), 317. 6. Y. V. GULYAEV AND A. G. KOZOREZOV,Sov. Phys. Solid State 18 (1976), 82. 7. V. L. GUREVICH AND A. THELLUNG, Phys. Rev. B 42 (1990), 7345. 8. J. A. KOBUSSENAND T. PASZKIEWICZ,Helv. Phys. Acta 54 (1981), 395. 9. J. A. KOBUSSENAND T. PASZKIEWICZ,Helv. Phys. Acta 54 (1981), 383. 10. N. N. B~GOLIUBOV AND D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Wiley, New York, 1980. 11. L. I. SCHIST,“Quantum Mechanics,” McGraw-Hill, New York, 1968. 12. H. MINKOWSKI, Nachr. Ges. Wiss. Giittingen (1908), 53; Math. Ann. 68 (1910), 473. 13. E. I. BLIXJNT, Bell Teleph. Lab. Tech. Memo. 38139-9, 1971. 14. J. P. GORDON, Phys. Rev. A 8 (1973), 14. 15. M. ABRAHAM, Rend. Circolo Mat. Palermo 28 (1909), 1; 30 (1910), 33.
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
39
“Foundations of Electrodynamics,” North-Holland. 16. S. R. DE GROOT AND L. G. SUTTORP, Amsterdam, 1972. 17. L. SUTTORP, in “Physics in the Making” (A. Sarlemijn and M. J. Sparnaay, Eds.). North-Holland, Amsterdam, 1989. 18. W. PAULI. “Relativit5tstheorie,” Encyklopldie der mathematischen Wissenschaften, Vol. V19. Teubner. Leipzig, 1921; English translation, “Theory of Relativity.” p. 101, Pergamon, New York, 1958. 19. L. D. LANDAU AND E. M. IAFSCHITZ, “Electrodynamics of continuous media.” Pergamon, Oxford, 1960. 20. C. TRUESDELL AND R. TOUPLIN, “The Classical Field Theories,” Handbuch der Physik, Band III/I. p. 246, Berlin. 1960. 21. A. D. WIECK. H. SIGG, AND K. PLOOG, Phys. Rev. Left. 64 (1990). 463. 22. In the case of linear relations D’=sE’. H’= (l/p) B’. 2 takes the simple form, Q’ = (1/8n)(~E’~ - (l/p) B’*) - (l/c)j:“A;,.
OF PHYSICS
220, 18-39 (1992)
Lagrangian Formalism and Conservation Laws for Electrodynamics in Nonlinear Elastic Dielectrics H. SCHOELLER AND A. THELLUNC Institut jiir Theoretische Physik Schiinberggasse 9, CH-8001
der Universitiit Ziirich, Ziirich, Switzerland
Received June 9. 1992
For the continuum theory .of electromagnetic fields in interaction with an elastic dielectric medium a new Lagrangian formalism is developed. The medium may be arbitrarily nonlinear and anisotropic, but in order to have conservation of quasimomentum it has to be homogeneous. For simplicity, the frequencies of the excitations are assumed to be such that dispersion can be neglected, but it is indicated how it can be taken into account. Directly from the Lagrangian equations of motion and alternatively by means of Noether’s theorem the general conservation laws for energy, momentum, and quasimomentum are derived. They are formulated in both local (Eulerian) and material (Lagrangian) coordinate systems. Special attention is given to quasimomentum. Its density in local coordinates xkr where x = y + u (y, = material coordinates, u(y, t) = displacement vector of the elastic medium), is given by
The first two terms are the well-known expressions for the electromagnetic and the elastic quasimomentum. The third, mixed, term is new. It can only be found if the difference between local and material coordinates is taken fully into account. 0 1992 Academic PRSS, h.
1. INTRODUCTION In this paper a new Lagrangian formalism is presented which provides the phenomenological electrodynamical and mechanical equations of motion in a nonlinear elastic dielectric. The medium may be anisotropic, but is supposed to be macroscopically homogeneous. One advantage of the Lagrangian formalism is the following: It permits a clear and systematic way of deriving the conservation laws of energy, momentum, and quasimomentum ( = pseudomomentum) either directly from the Lagrangian equations of motion or, via Noether’s theorem, from the symmetry properties of the media. In this way, statements made in an earlier paper [l] about Lagrangians in nonlinear electrodynamics and about Noether’s theorem will be demonstrated in detail. Another advantage of Lagrange’s equations of motion is their suitability for the transition from one coordinate system to another, in our case from local ( = Eulerian) coordinates, as used in electrodynamics, to material ( = Lagrangian or substantial) coordinates, as usually employed for the description 18 0003-4916/92 $9.00 Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
19
of elastic media. In some places of Ref. [l] the difference between local and material coordinates was neglected for simplicity and because in most practical cases this is a good approximation. In this paper their difference is fully taken into account. For elastic media this was already done by Kobussen [2] and Peierls 131. Some work on photoelastic media starts from a microscopic description of the material particles and then, in the long-wavelength limit, the transition to continuous media is carried out. This transition is performed in many textbooks and the limitations of the macroscopic theory are well known so that we use the continuum theary from the very beginning. Also, to keep our equations relatively lucid, we assume the frequencies and wavelengths to be such that we can neglect dispersion. This means that, e.g., in electrodynamics the fields D and H are functions of E and B (or vice versa) and not of their temporal or spatial derivatives (the case of dispersion can also be treated and will be the subject of a forthcoming paper). However, the material laws may be nonlinear and the material “constants” (like the dielectric or magnetic susceptibilities) may depend on the elastic strain. Similarly. the elastic “constants” may depend on the electrodynamic fields. Special attention is given to the conservation of quasimomentum, which in nonlinear optics and in the theory of elasticity is an especially interesting quantity. It admits a simple and very general analysis of various physical phenomena, such as, e.g., (a) the radioelectric effect (see [ 1, 211 and references given there), (b ) photomagnetism [4], (c) the acoustoelectric [S], and (d) the acoustothermal [6] effects which are caused by the transfer of quasimomentum from electromagnetic waves to conduction electrons (a), (b), or from acoustic waves to electrons (c) or thermal phonons (d). How does one define quasimomentum in electrodynamics and in the theory of elasticity? In the opinion of the authors the safest method is to couple the system under investigation to another system, where quasimomentum is a well-known concept, e.g., thermal phonons or Bloch electrons for which the expression
(k, = wave vector, N, = number of phonons or electrons in mode M) is the quasimomentum (also called pseudomomentum or crystal momentum). If now, in the system under investigation, a quantity P,, is found whose sum with expression ( 1.1) (in the absence of Umklapp processes) is conserved one obviously must identify P,, with the quasimomentum of that system. This is the method adopted in Refs. [ 1, 71. Another possibility is to quantize the theory and diagonalize the harmonic part of the Hamiltonian, which leads to photons or acoustic phonons. The quantity P,, then can be shown to take the form (1.1) where N, now means the number of photons or acoustic phonons. This is another justification of this name. The authors have verified this fact. The calculation will not be presented in this paper. For the elastic case it was done by Kobussen and Paszkiewicz [S].
20
SCHOELLERAND THELLUNG
A third possibility is to make use of the following symmetry property characteristic of homogeneous media. If all the fields (in our case electromagnetic fields and elastic deformation) are spatially displaced by a vector E, but without shifting the medium, then in a homogeneous medium the course of physical events will be the same. Via Noether’s theorem this invariance property leads to a conserved quantity which is identical with the quasimomentum found by the first two methods. Therefore, this invariance property can be used to define the quasimomenturn. This is done in Refs. [3, 91. In Section 2 a Lagrangian formalism is presented for electrodynamics in a homogeneous medium moving with a uniform velocity in. The equations are written in a relativistically covariant form, but eventually terms of relative order (G/c)’ are neglected. In Section 3 the Lagrangian formalism is extended to electrodynamics in elastic dielectric media. The difference between local and material coordinate systems is pointed out and the general form of the conservation laws is derived (a) directly from the equations of motion and (b) from Noether’s theorem. In order to discuss the energy of the system it is convenient to introduce a Hamiltonian and the canonical formalism. This is done in Section 4. Then the detailed derivation and discussion of the conservation laws for energy, momentum, and quasimomentum is given in local coordinates (Section 5) and in material coordinates (Section 6). In Section 7 the general relationship between the Lagrangian equations in local and material coordinates is analysed. This is not a trivial coordinate transformation because the two coordinate systems differ by the displacement vector of the elastic medium which is a dynamical field variable. It is shown that quasimomentum conservation in one coordinate system implies quasimomentum conservation in the other system. The same is true for momentum conservation, but it is interesting to note that the Lagrangian equations of motion for the elastic medium in material coordinates yield the momentum conservation law, whereas in local coordinates they yield the conservation law for the difference between momentum and quasimomentum. 2. LAGRANGIAN FORMALISM FOR ELECTRODYNAMICSIN A UNIFORMLY MOVING MACROSCOPICALLYHOMOGENEOUSMEDIUM
Let us consider a macroscopically homogeneous medium moving with a constant velocity 8. Maxwell’s equations read V.D=4np, VxH-ib=Fj, C
V-B=0 VxE+b=O.
c
c
(2.la) (2.lb) (2.2a) (2.2b)
LAGRANGE
FORMALISM
21
IN PHOTOELASTICITY
Later on, we shall only consider dielectric media, but in this section we can include the possibility of nonzero electric conduction current and charge densities, j,. and pc. Using a relativistically covariant notation, we can write these equations in the form (2.3) (2.4)
with j’,’ = (cp,., j,) and
*FP” =
i
0
B,
&
B,
-B,
0
E,
-B2
E,
-E, 0
-E2
El
\-B,
- El 0
The homogeneous equations (2.2) or (2.4) are equivalent to the representation the fields E and B by the electromagnetic potentials A” = (cp, A)
of
E= -IA-V+3 C
(2.5)
B=VxA.
The inhomogeneous
part (2.3) can obviously
be written
in Lagrangian
form
(A, ,,,, = dAJ8.u”) a,-+E
(2.6)
bL.1 aA,
by using a Lagrangian
density f!(A,,,
A,,) as proposed in [ 11 with (2.7)
and ai2 i W.-.---z -- cjY. 8A,
(2.8)
Considering Q = !i!(E, B, A,) as a function of the electromagnetic potentials, it is easy to show from (2.5) and (2.7) that a2
-=dE
1 4n
JX
aL! g=
-$H.
tields and the
(2.9)
22
SCHOELLERAND THELLUNG
which is also valid in the rest system of the medium, since (2.7) is written in a relativistically covariant form. We want to emphasize that all these equations are independent of the specific material laws which can have either a linear or a nonlinear form. In a homogeneous and nondispersive medium, these material laws will have the general form D’(E’, B’) and H’(E’, B’), where E’, B’, D’, H’ are the fields in the rest system. [22] Using the transformation laws [18, 191 E’=E+kB, D’=D+kH,
c c
&-B-&E C
(2.10) H’=H-&xD C
(all terms of order (C/c)’ have been neglected), we see that the material laws D(E, B, ir) and H(E, B, ir) depend on ti in the laboratory system. Thus, from (2.9), 2 will also depend on ti, i.e., I! = !Z(E, B, 8, A,). Furthermore, since (! must be an invariant (see (2.7)), the dependence on i is explicitly given by f?(E, B, 8, AJ = Q’(E’, B’, A;),
(2.11)
where 2 denotes the Lagrange function in the rest system. Using (2.10), (2.9), (2.8), and
cP’=rp-liA, A’=A-lcpi, c c
(2.12)
we obtain a!2 1 ~=~(EXH-DxB)+~p,A-;j,p. In order to be consistent with our unrelativistic approximation higher order in C/c have been neglected in (2.12) and (2.13).
(2.13) in (2.10), terms of
3. LAGRANGIAN FORMALISM FOR ELECTRODYNAMICSIN HOMOGENEOUS ELASTIC DIELECTRICS. GENERAL DERIVATION OF CONSERVATIONLAWS
From now on and for the whole rest of this paper we only consider dielectric media, i.e., pC = j, = 0. Generally, the deformation state of an elastic medium is described by the displacement field v(x, t) = u(y, t) = x - y,
(3.1)
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
23
where x and y denote the local (or Eulerian) and material (or Lagrangian or substantial) coordinates of the system, respectively. Throughout this article we use the terms “local” and “material” coordinates instead of “Eulerian” and “Lagrangian” coordinates in order to avoid confusion with the term “Lagrangian” designating the formalism developed in Sections l-3. For convenience, we shall also use the four-dimensional notation U, = (0, u) and u, = (0, v) (Greek indices take the values CI= 0, 1, 2, 3 and Latin indices i = 1,2, 3). From now on all vector indices are written as subscripts; they designate contravariant components; covariant components are no longer,being used. In the same way, we write x, = (ct, x) and y1 = (ct, y) for the four-dimensional local and material coordinates. The connection between the tensors u,,8 = do,/&, and u~.~ = du,/dy,j is given by
or by the well-known
formulas [2]
The mass density in local coordinates is denoted by p(x, t ), which is connected to the mass density p0 in material coordinates by p d3x = p. d3y
(3.4a)
or
where A = det(8x,/8yp) = det(ax,/ay,) is the Jacobian of the transformation. Note that in a homogeneous medium p,, is a constant. Now, if we consider an elastic dielectric medium (with jf = 0), together with an electromagnetic field, the total Lagrangian density f! (in local coordinates) of the system no longer depends on A, (viz. (2.8)) but only on A,., (or E, B) and u~,~. It is quite possible to include higher derivatives of A, and v, as arguments in 2. (For the case of elastic media compare the remark in [7, end of Section II]). For the electromagnetic fields this means that, e.g., D and H also depend on derivatives of E and B. This gives rise to dispersion. For simplicity, we exclude this possibility in the present paper. The case of dispersive media will be treated in a future article. Furthermore, we assume that (2.9) remains valid (which is necessary for the validity of Maxwell’s equations) and that (2.13) is only supplemented by the i-derivative of the kinetic energy density $pti* of the elastic medium (3.5)
24
SCHOELLER
AND
THELLUNG
Originally, the derivation of (2.13) was based on li being the constant velocity characteristic of a Lorentz transformation. We now assume (2.13) and (3.5) to be valid as local laws for (slowly) varying ri (compare a remark in [18, p. 1011). By using (3.3), the last equation becomes
ai? ax. T=-J &+&(ExH-DxB), at+ ayj
1 .
Finally, we do not make any assumptions about the dependence of 2 on v~,~;i.e., the Lagrangian can be an arbitrary function of the strain tensor. To obtain the general conservation laws of our system, we can proceed in two different ways. The first possibility is to start from the Lagrangian equations of motion
--a aQ) ax,aA,, and
--a a2=, ax,aUa.b’
(3.8)
which yield
(3.9) or (3.10) This has the form of a continuity equation and means that there exists a conserved quantity. In Section 5 we shall see that it corresponds to the energy conservation for ,u = 0 and to the quasimomentum conservation for p = 1,2,3. The conservation of ordinary momentum, however, is given by the addition of (3.8) and (3.10), which is also demonstrated in Section 5. The second possibility is to use certain invariance properties of 2 together with Noether’s theorem. This theorem can be stated as follows [lo]: If a simultaneous infinitesimal transformation of the coordinates xP and the fields $i (i = 1, .... n), (3.1 la)
x;=x,+ax, tiltx’)
=
$iCx)
+
d$i(x)7
(3.11b)
LAGRANGE
FORMALISM
IN
PHOTOELASTICITY
25
leaves the action integral
W=ld 4xqx, $j(XLIc/;.,W) invariant (integration equation holds
over an arbitrary
(3.12)
region G), then the following continuity
(3.13) where &+!I,is defined as the field variation
at a fixed point s,
@i(X) = $:(-xl - 3,(x),
(3.14)
such that 6$j(x)=
$$itx)
+
\I/i.p
6xJ,.
(3.15)
We apply Noether’s theorem to two special cases. (a) Homogeneous media. L! only depends on the fields $, and their derivatives, not explicitly on X: f!($,, IJ+,,+).A displacement in space-time of the fields by an arbitrary infinitesimal constant displacement vector E,, together with a simultaneous shift of the integration variables (and the region of integration) by E,.,
x: - x,,= sx,,= E, I);(.~‘)- $i(x) = 6~i(x) = 0,
(3.16)
leaves f? and d4.u and, therefore, W invariant. According to (3.15) and (3.16) &hi(X) = -I),,,.&,
(3.17)
and if we take just one of the E,,‘sdifferent from zero, (3.13) takes the form (3.18) Now, in a photoelastic medium the set of fields (I,?,1 is the set {A,, v, ) and (3.18) turns into (3.10), giving conservation of energy and quasimomentum (see Section 5). In fact some authors [3,9] use this invariance property to d&e quasimomentum. (b) f? does not depend on the fields themselves (only on their derivatives, and possibly on the coordinates). L? (x, q,,,). Then L! is invariant with respect to the transformation 6x, = 0,
&hi = &hi = E,= const.
(3.19)
26
SCHOELLER ANDTHELLUNG
(and so is d4x and W). Then (3.13) yields a
ax? =o, ( wi., >
ql
(3.20)
Vi,
which are Lagrange’s equation of motion (for this particular f?). Applied to a photoelastic medium, one obtains Eqs. (3.7) and (3.8). As mentioned before, (3.8) means conservation of ordinary momentum minus quasimomentum, whereas (3.7) gives Maxwell’s equations (2.la) and (2.lb) for pC= j,=O.
4. CANONICAL Let us detine the Hamiltonian transformation of L3(E, B, u~,~)=
FORMALISM
density in local coordinates
by the Legendre
e(A, VP, B, i, u~,~) as
jjr=A.pA+i=pu-f?
(4.1)
with the canonical momenta pats=
ai?
ldL!
-cE=
1 -4ncD
(4.2)
and pa=%+
air ay
pri,+-&(ExH-DxBI,
1 ,
(4.3)
according to (2.9) and (3.6). (By the way, pD is just the difference between the momentum and quasimomentum densities, as can be seen from the results of Sections 5.2 and 5.3.) From (2.5) and (2.la) with pC=O we obtain (4.4) and, since the last term does not give any contribution H= j d3x $j’ [ 111, we define the energy density by 5j=&E.D+P.p,-2.
to the total energy
(4.5)
For its differential, we obtain from (4.2) and (4.3) (4.6)
LAGRANGE FORMALISMIN
27
PHOTOELASTICITY
so that $j has to be considered as a function $(D, B, pt., L),.~)and, in view of (2.9) its derivatives are given by
a5 -=-
asj lH -=aB 4x
lE
aD
4x
’
(4.7)
asI,. p,=v. Furthermore,
we will define a quantity
U by the relation
sj = fpic + u.
(4.8)
U is the total energy density minus the kinetic energy density of the elastic medium. Up to now, all quantities have been defined in local coordinates. For the material coordinates, in analogy to (3.4b), we define
Q(E, B, ~2,~)= AW, B, u,,,j) (4.9 1
f;CD, B, P,> u;,k) = AND, B, P, >cl,,) o(D,
B,
Pu,
l4i.k)
=
A U(D,
B,
Pt 7 l’1.k I
with -
p,=g=p@+&(E~H-DxB).
(4.10)
where we have used (3.3), (3.6), (3.4b), and the fact that A does not depend on 5. In (4.9), we have not transformed the electromagnetic fields to material coordinates since an explicit evaluation of the conservation laws (3.8) and (3.10) would lead to rather complicated expressions for the electrodynamic parts in this case. However, for the general case, we shall transform the fields as well in Section 7 and show that the Lagrangian formalism is completely equivalent in local and material coordinates.
5. CONSERVATION
LAWS IN LOCAL
COORDINATES
5.1. Energy Conservation
Equation (3.10) reads for p = 0,
= 0.
(5.11
28
SCHOELLER
AND
THELLUNG
Using (2.5), (2.9), (4.1), (4.4), and (4.5), this becomes
Due to Maxwell’s equations, all q-dependent we obtain
terms cancel each other exactly and
(5.3) which is the energy conservation law in local coordinates. Furthermore, by using (4.7), (4.8), and (4.9), we can express aQ/av].)i. by a@%,& (see the Appendix)
ax, . . ax,. L(ExH-DxB),
+y,p”ku~+dy,uk4nc
(E
x W,
(5.4)
where E and H are considered as functions of D, B, pu, and u[,,, in the last term. Multiplying this equation by tij and using (3.3), we obtain for (5.3) the final result, (fj$+&[
-~~p,(~)D,B,p”+d,t)+~(ExH),]=O.
(5.5)
Here, terms of relative order (C/C)~ with respect to (c/4n)(E x H)k have been omitted in accordance with the unrelativistic approximation already adopted in Section 2. 5.2. Quasimomentum
Conservation
For p = i, (3.10) yields
a at, (>(
- -$ A j, ; - 2, vi, i J
I
ae Acc,ia~Vj.i+6ikQ
(5.6)
LAGRANGE
Applying
FORMALISM
29
IN PHOTOELASTICITY
(2.5), (2.9), (3.6), and (3.3), we obtain a (ILt,
-~~~pri~+&(DxB)~
1
-u,,~(EXH-DxB)j+~D,A,.,
a +ig
-~qi-~(EiDk+H,B,,
+
6ik
J.h
&H-B+P
>
-&&&(HxV),A,
=O. J
(5.7)
Again, the A-dependent terms cancel each other due to Maxwell’s equations and, together with (4.5), we arrive at
a sit, (>I
1 a -gVj,i-$(EiD,+H,B,) +zk I,h -w,i~ljj+~(DxB),--u,;~(ExH-DxB);
+ 61, & (ED + HB) + 6ik ($ti’+B;&(ExH-DxB)
I
u) =O.
(5.81
Multiplying (5.4) with vi,,, using (3.3), and inserting the result in (5.8), we obtain the final form
a i z,>[
1 +~[~~u,,,(~),,.,~"-~(EiDh+H -uiipB,+~(DxB)j--,,i~~(ExH-DxB),
fh,&(ED+HB)+~(;pb’-(I)
- uj.Jni;li, - U,.,lik&(ExH-DxB), +6,,+ExH-DxB)
(5.9)
30
SCHOELLER
AND
THELLUNG
This equation corresponds to quasimomentum ( = pseudomomentum) conservation for the following reason: The density of the conserved quantity consists of three terms: a purely material term, --~~,~pti~; an electromagnetic term, (1/47rc)(D x B), and a mixed term, - uj,i(1/47rc)(E x H -D x B)i. Since - z+ptii is well known to be the quasimomentum density of an elastic medium [2, 3, 7,9] one has to interpret (1/47rc) D x B as the quasimomentum density of the macroscopic electromagnetic fields in a dielectric medium in agreement with [l] and with the interpretation of Minkowski’s result [12] by, e.g., Blount [13], Gordon [14], and Peierls [3]. The mixed term, - u,~( 1/47cc)(E x H - D x B)j is missing in the above-mentioned articles because in Refs. 12, 3, 7, 9, 12-141 electromagnetism or elastic media are studied separately, whereas [ 11 treats the coupled systems, but in a first approximation explicitly neglects the difference between local and material coordinates for the electromagnetic contributions. 5.3. Momentum
Conservation
Using the Lagrangian equations of motion (3.8), together with (4.3) and (5.4), we obtain
ax. a -Jpti,+TTt,a~i O[
~;&(ExH-DxB),]
,
axj . . axj. &ExH-DxB), + ayi p”juk + ayi uk 471c (ExH)
Since in the first term on the right-hand
C5
Ptij
=
(Sji
+
Uj,i)
1 .
(5.10)
side of (5.10)
plij
=
plii
+
pUj,itij
aYi
is the density of momentum minus quasimomentum of the elastic medium, (5.10) and (3.8) give the conservation law for this difference. In order to obtain the conservation law for the momentum alone we add (5.9) and obtain the final result
LAGRANGEFORMALISMINPHOTOELASTICITY
+ 6, & (ED + HB) -&
31
(EiDk + H;B,)
(ExH)
1
=O.
(5.11)
Since pti is the momentum density of the elastic medium, obviously (1/4rcc)(E x H) is to be interpreted as the momentum density of the macroscopic electromagnetic fields (which includes the vacuum fields as well as the polarizable and magnetizable matter). This result agrees with Abraham’s expression [15] and its interpretation by, e.g., Blount [ 133, Gordon [14], and Peierls [3]. Furthermore, using the same unrelativistic approximation as we employed in Section 2, the same has been obtained by de Groot and Suttorp [16, 171.
6. CONSERVATION
The transformation
LAWS IN MATERIAL
of an arbitrary
continuity
COORDINATES
equation in local coordinates
3, 0 -= ax,
(6.1)
to material coordinates is given by (6.2)
with
j,=&
The proof is trivial due to the Euler-PiolaaJacobi &A$)=0 . L1( 595/220/l-3
or B>
(6.3)
axflJa. identity [20]
a -2i ax = 0. ax, ( day,, >
(6.4)
32
SCHOELLER
AND
THELLUNG
From (6.3) and (3.3) it follows that (6.5)
lo = 40 and 69
(6.6)
Thus we can obtain all the conservation laws in material straightforward manner from (5.5), (5.9), and (5.11).
coordinates
in a
Energy conservation,
-a5 ( at)
+ Y
a -zii G [
1
+zA-&(ExH)k
=o.
(6.7)
Quasimomentum conservation,
-
--u ah. axk.
A kG(DxB),+$&(ExH-DxB) I
1
(ExH) Momentum
=O.
(6.8)
conservation,
+z$(ED+HB) I
-$$(E,D,CH,Bk)-$$k&(DxB)j k
+$&(ExH-DxB) I (ExH)
1
=O.
(6.9)
LAGRANGEFORMALISMIN
33
PHOTOELASTICITY
Now let us compare these results with the ones obtained in [l]. The energy conservation law (6.7) (see Eqs. (4.6)-(4.8) of Reference [ 11) has nearly the same form but the prefactor (+,/Jx,) A in front of the flux term (c/47r)(E x H) is missing in [ 11. This corresponds to the assumption that (6.10)
lUi.kl e 1
and is consistent with the neglect of the difference between local and material coordinates in [ 11. Similar correction factors can be observed in the quasimomenturn conservation law (6.8) in comparison with Eqs. (4.11 k(4.13) of Ref. [l]. Furthermore, the quasimomentum density in (6.8) has an additional term -u,,,(l/47cc)(ExH-D x B)j (see also the remarks at the end of Section (5.2)) which, in comparison to (1/47cc)(D x B), can again be neglected under the assumption (6.10). The last three terms of (6.8) are of relative order C/c and are due to the fact that we have used the correct transformation laws (2.10) for the electromagnetic fields. For B 4 c they are not important and consequently do not appear in Cl]. Finally, the conservation law (6.9) of ordinary momentum has not been considered in [ 11.
7. GENERAL LAGRANGIAN
FORMALISM
IN LOCAL
AND MATERIAL
COORDINATES
In this section we want to analyse the general relationship between the Lagrangian formalism in local and material coordinates. In the first part we will see that Lagrangian equations of motion also hold in material coordinates if the derivatives of the electromagnetic potentials (7.1) where A.LM)=hvL are transformed in the same way as the derivatives according to (3.2)
(7.2)
of the displacement
field
(7.3 1 Note that this is not a Lorentz transformation since we do not transform the electromagnetic potentials themselves but only look at the same functions (7.2) (in analogy to (3.1)) and take their derivatives in two different coordinate systems. The Lagrangian in material coordinates is then given by
34
SCHOELLER
AND
THELLUNG
in contrast to (4.9), where the electromagnetic fields are not transformed (see the remarks at the end of Section 4). To prove the equivalence of the Lagrangian equations (3.7) and (3.8) with the Lagrangian equations a
-7=o,
a2
-- a
ac-, ah aun.8
ah aA,,
we first transform the derivatives a!$/au,,p and ai?/aA,,
’
(7.5)
of (! as follows:
(7.6)
Using (7.7)
and (7.8)
which can easily be proven from (7.3) and (A.l), we obtain, together with (A.2),
-=a2 a%8
laxq;
laxajj
A ah
A ah
i ad- ax a2 aQ _+--t-Ep3aaA,,, A ah ah au,,.
(7.9)
and a2 i ax, a!2 -=--7. a4, A ah a-4,”
(7.10)
For the derivative of (7.9) we obtain
where we have used the Euler-Piola-Jacobi (a/ax,). Since ae_, ay,-
&+u
Ir,v,aaA,,,
identity (6.4) and alay, = (axs/ayd) x ai2
p,v,a aup,v’
LAGRANGE
FORMALISM
IN
35
PHOTOELASTICITY
(7.11) can be brought to the final form a a2 ---=-a
1
axgav,,8 A
a ai2 i ax,, a aQ ~GjpZ&+ddl?,dqlpau,,;
(7.12)
In the same way we obtain from (7.10)
--a ar! i a aQ ax, aA,,Yi~c?A,l,’
(7.13)
The last two equations are general formulas for the relationship between the derivatives (a/ax,)(aC/av,,), (d/ax,)(a~/dA,,), and the corresponding derivatives in material coordinates. They were obtained without the use of the equations of motion. Obviously, they manifest the equivalence of the Lagrangian equations of motion in both coordinate systems. We want to emphasize that our result is not simply a consequence of the invariance of the Lagrangian equation of motions under ordinary coordinate transformations, since here the transformation matrix ax,/ayl, = 6,, + u~,~ is a field variable. In the second part of this section we want to demonstrate the equivalence of the conservation laws in the two coordinate systems. Let us multiply (7.12) by -u,.,, and (7.13) by -A,,, and take the sum of the two expressions. Then, with the aid of (3.3) and the inverse of Eq. (7.3),
we arrive at the relation
I
(7.14)
Remembering that (3.10) means conservation of energy (p = 0) and quasimomenturn (p = 1,2,3), Eq. (7.14) shows that these conservation laws take the same form in local and material coordinates. In Section 5.3 it was shown that (3.8) expresses the conservation law for the difference between momentum and quasimomentum. In order to obtain the conservation law for the momentum alone we take Eq. (7.12) for r = i and add Eq. (7.14) with p = i. After some straightforward manipulations we arrive at the relation
-+b,f?--
ait a2 i a a2 A,,;-- a0 ‘=,i 7 iiy, au,,s’ W., 2.It I
(7.15)
36
SCHOELLER
AND
THELLUNG
This shows that the momentum conservation law takes a particularly simple form in material coordinates. It is just expressed by the Lagrangian equation of motion for the field u. This can easily be checked by looking at the term containing the time derivative on the right-hand side of (7.15), (8/8t)(dQ/&,). Since the n-dependence of j? is given by the kinetic energy &,8* + linear terms in ti, we have that &/atii = pOlii + terms independent of 8, which contains the momentum density of the matter without any admixtures of quasimomentum. Finally, let us investigate the conservation laws from the point of view of Noether’s theorem. In Section 3 we saw that the energy-quasimomentum conservation law (3.10) is the result of the invariance of 2 under the variation in x-space, (7.16a)
6x, = E,, ihi = 6A, = 0,
which implies &Ii= -Vi*“&“, Since y = x - v, x = ye+ u, A,(x(y))
= A,(y),
&Y, = Ev,
(7.16b)
c?Ar = - Ap,“&“.
it follows from (7.16a) that
6Ui = sa, = 0,
(7.17a)
which implies &li= -Ui,“&,,
q
= -ap,“&,.
(7.17b)
The variations in x-space and y-space are thus completely analogous and consequently the energy-quasimomentum conservation law has exactly the same form, as already put in evidence by (7.14). For the elastic variables the equivalence of the variations (7.16b) and.(7.17b) was already demonstrated by Kobussen [2] and Peierls [3] (compare the remark in [ 1, end of Section 33). As a last application of Noether’s theorem let us look at momentum conservation. Its simple form in y-space,
--a a-c, ah
(7.18) ’
aui,p
is obtained from the invariance of !G with respect to the variations 6Ui = &di = E,,
6y,=O,
The corresponding 6t=O,
sa,
= 22,
= 0.
(7.19)
variations in x-space are easily found to be 6Xi = Ei,
6A,=O,
Now, (7.20) is a superposition
&Ii
= Ei,
hi
JA, = - A,,iei.
of two variations,
= Ei -
V+Ek
(7.20)
LAGRANGE
FORMALISM
1N PHOTOELASTIClTY
37
and (7.22 I
Invariance with respect to the first variation leads to (3.7), i.e., conservation of momentum minus quasimomentum, invariance with respect to the second variation gives quasimomentum conservation, the sum yields momentum conservation. The proof of the equivalence of all conservation laws in this section is a generalization of the work by Kobussen [2]. Here we have included electrodynamics. Furthermore, we note that the relations (7.12))(7.15) have been obtained without the use of the equations of motion. APPENDIX
In this Appendix we derive Eq. (5.4). To begin, let us mention elementary properties:
the following
(A.1 1
--a i = ---ax, i aua.S0A ay, A’
(7 au,.,
QA A = ax,
(A.21
The first one can be obtained by differentiating 6 Jw@v --==((s,,7-t~,,,,)
I’? ayvax, a?,
with respect to v~,~, and the second one follows directly from l/A = det(6,, - r,,,{) and A = det(6,, -t usB). Now, from (4.7) and (4.9) we have
i a2dlmas --AA au,, ah,’
(A.3 1
The first and third terms of this expression can be evaluated by using (A.1 ) and (A.2). For the second one we use
p.,=As ay P,, = A(6,,,I I
urn,,) PC,
38
SCHOELLER AND THELLUNG
and, observing (4.9) and { +}
= const o { u~,~}= const,
where we have used (4.7) and (3.3). Thus (A.3) reduces to
Furthermore,
from (4.8), (4.9), and (4.10) we obtain
or with (A.2) -=a.5 au,,,
al7 ay,.
A
--p.G(ExH-DxB)-l.~-
aul,,
4n~
I
a
au,m
(ExH).
(A.7)
Inserting this expression in (A.6) and using (4.3) and (4.8), we finally arrive at (5.4).
ACKNOWLEDGMENTS The authors would like to thank V. L. Gurevich for helpful remarks and fundamental discussions, L. A. Turski for an interesting comment, and the Schweizerischer Nationalfonds for financial support.
REFERENCES 1. V. L. GUREV~CHAND A. THELLUNG, Physica A 188 (1992) 654. 2. J. A. KOEJUSSEN, He/v. Phys. Acra 49 (1976), 599. 3. R. PEIERLS,in “Highlights of Condensed-Matter Theory” (F. Bassani, F. Fumi, and M. P. Tosi, Eds.), p. 237, North-Holland, Amsterdam, 1985. 4. V. L. GUREVICH, R. LAIHO, AND A. V. LASHKUL, Phys. Rev. Left. 69 (1992), 180. 5. G. WEINREICH,Phys. Rev. 107 (1957), 317. 6. Y. V. GULYAEV AND A. G. KOZOREZOV,Sov. Phys. Solid State 18 (1976), 82. 7. V. L. GUREVICH AND A. THELLUNG, Phys. Rev. B 42 (1990), 7345. 8. J. A. KOBUSSENAND T. PASZKIEWICZ,Helv. Phys. Acta 54 (1981), 395. 9. J. A. KOBUSSENAND T. PASZKIEWICZ,Helv. Phys. Acta 54 (1981), 383. 10. N. N. B~GOLIUBOV AND D. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” Wiley, New York, 1980. 11. L. I. SCHIST,“Quantum Mechanics,” McGraw-Hill, New York, 1968. 12. H. MINKOWSKI, Nachr. Ges. Wiss. Giittingen (1908), 53; Math. Ann. 68 (1910), 473. 13. E. I. BLIXJNT, Bell Teleph. Lab. Tech. Memo. 38139-9, 1971. 14. J. P. GORDON, Phys. Rev. A 8 (1973), 14. 15. M. ABRAHAM, Rend. Circolo Mat. Palermo 28 (1909), 1; 30 (1910), 33.
LAGRANGE
FORMALISM
IN PHOTOELASTICITY
39
“Foundations of Electrodynamics,” North-Holland. 16. S. R. DE GROOT AND L. G. SUTTORP, Amsterdam, 1972. 17. L. SUTTORP, in “Physics in the Making” (A. Sarlemijn and M. J. Sparnaay, Eds.). North-Holland, Amsterdam, 1989. 18. W. PAULI. “Relativit5tstheorie,” Encyklopldie der mathematischen Wissenschaften, Vol. V19. Teubner. Leipzig, 1921; English translation, “Theory of Relativity.” p. 101, Pergamon, New York, 1958. 19. L. D. LANDAU AND E. M. IAFSCHITZ, “Electrodynamics of continuous media.” Pergamon, Oxford, 1960. 20. C. TRUESDELL AND R. TOUPLIN, “The Classical Field Theories,” Handbuch der Physik, Band III/I. p. 246, Berlin. 1960. 21. A. D. WIECK. H. SIGG, AND K. PLOOG, Phys. Rev. Left. 64 (1990). 463. 22. In the case of linear relations D’=sE’. H’= (l/p) B’. 2 takes the simple form, Q’ = (1/8n)(~E’~ - (l/p) B’*) - (l/c)j:“A;,.