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ITitle: Euler-Poincare Dynamics of Ideal Micropolar Complex Fluids

Author(s): Darryl D. Helm

Submittedto: Intended for Jerry Mamden’s Festschrift on the occasion of hissixtieth birthday. April 15, 2000 in Pasadena, CA.

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Euler-Poincar6 Dynamics ofIdeal Micropolar Complex Fluids

Darryl D. HelmTheoretical Division and Center for Nonlinear Studies

Los Alamos National Laboratory, MS B284Los Alamos, NM 87545

[email protected]

Intended for Jerry Marsden’s Festschrij%on the occasion of his sixtieth birthday

January 4, 2000

Abstract

Lagrangian reduction by stages is used to derive the Euler-Poincar6 equa-tions for the coupled motion and micromotion of ideal micropolar complexfluids. The order parameters for micropolar complex fluids are material vari-ables that may be regarded either as geometrical objects in a vector space, or astaking values in coset spaces of Lie symmetry groups. Examples include liquidcrystals, superfluids, Yang-Mills magnetofluids and spin-glasses. A Lie-PoissonHamiltonian formulation of the dynamics for ideal micropolar complex fluidsis obtained by Legendre transforming the Euler-Poincar6dynamics are also derived by using the Clebsch approach.

1

formulation. These

.

Dynamics of micropolar complex j?uids January 4, 2000

Contents

1 Introduction 2

2 The example of liquid crystals 62.1 Background for liquid crystals .,. . . . . . . . . . . . . . . . . . . . 62.2 Four action principles for liquid crystals . , . . . . . . . . . . . . . . 72.3 Hamiltonian dynamics of liquid crystals . . . . . . . . . . . . . . . . 192.4 Summary for liquid crystals . . . . . . . . . . . . . . . . . . . . . . . 25

3 Action principles and Lagrangian reduction 263.1 Lagrangian reduction by stages . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Lagrange-Poincar6 equations . . . . . . . . . . . . . . . . . . 293.1.2 Euler-Poincar6 equations . . . . . . . . . . . . . . . . . . . . 31

3.2 Hamiltonian dynamics of ideal micropolar complex fluids . . . . . . 353.3 Clebsch approach for micropolar complex fluids . . . . . . . . . . . . 39

4 Conclusions 42 ‘

1 Introduction

The motion of a complex fluid depends on additional material variables called orderparameters that describe the macroscopic variations of the internal structure of thefluid parcels. These macroscopic variations of internal structure may form observablepatterns, as seen, for example, via the gradients of optical scattering properties inliquid tryst als arising due to the spatially varying orientations of their molecules,as discussed in, e.g., Chandrasekhar [1992], de Gennes and Prost [1993]. Thus, themicro- order of a complex fluid is described by an auxiliary macroscopic cent inuumfield of geometrical objects associated with each fluid element and taking values ina vector space (or a manifold) called the order pammeter space. The canonicalexample is the description of the local directional asymmetries of nematic liquidcrystal molecules by a spatially and temporally varying macroscopic continuumfield of unsigned unit vectors called “directors,” see, e.g., Chandrasekhar [1992], deGennes and Prost [1993].

Typically, the presence of micro-order breaks the symmetry group O of theuniform fluid state to a subgroup P. The subgroup P c 0 is the isotropy groupof whatever geometrical object it is (e.g., a vector, a spin, a director, etc.) thatbreaks the symmetry of the uniform fluid state to produce the micro-order. That is,the subgroup 7 is the symmetry group of the micro-order. Equivalently, the orderparameter may also be regarded as taking its values in the coset space C = O/P and

Dynamics of micropolar complex j?uids January 4, 2000 3

its space and time variations may be represented by a space and time dependentcurve in the Lie symmetry group 0 through the action of 0 on its coset space C.Thus, associating a geometrical object in a vector space (say, a director) with anorder parameter may be regarded as simply a way of visualizing the coset spaceC, see, e.g., Mermin [1979] for physical examples. The symmetry group 0 of theuniform fluid state that acts on the order parameter coset space C = 0/7 is calledthe order parameter group, or the broken sgmmetry.

Spatial and temporal variations in the micro-order are measured relative to areference configuration. Let 0 act transitively from the right on a manifold M.Suppose the subgroup P c O leaves invariant an arbitrarily chosen reference pointm. ● M, i.e., mop = no, Vp E P. The reference point rno then corresponds to thecoset [e] = ep = T of 0/7, where e is the identity element of 0. For another choiceof reference point, say W&= rnoh with h ~ 0, the isotropy subgroup becomes theconjugate subgroup P + P’ = h– 1Ph. and the corresponding coset space O/P’ is,thus, isomorphic to the original coset space O/P. Hence, the conjugacy equivalenceclasses of the order parameter coset spaces account for the arbitrariness in the choiceof reference point m.. One may think equally well of the order parameter group asact ing either on a manifold M with “origin” no, or on a coset space O/P, where Tis the stabilizer of the reference point no.

For example, let O be the group of orthogonal transformations O(3) actingtransitively on directors in R3 (unit vectors with ends identified) and choose m.to be the vertical director in an arbitrary reference frame; so riaois invariant underrotations around the vertical axis in that frame and under reflections across itshorizontal plane. This example applies to cylindrically symmetric nematic liquidcrystal properties. The isotropy subgroup P is the O(2) subgroup of the groupO(3) of rotations and reflections in three-dimensional space and the liquid crystaldirector may be represented in the coset space 0(3)/0(2), which is isomorphic toS2/22, the unit sphere S 2 with diametrically opposite points identified, i.e., theprojective plane P2. Thus, after a reference configuration has been chosen, themicro-order of nematic liquid crystals may be represented equivalently as a spaceand time dependent curve in the broken symmetry group 0 acting on either theorder parameter manifold M = P2, or on the coset space 0(3)/0(2) of the brokensymmetry O(3). The order parameter dynamics represented by this curve in thebroken symmetry group for a complex fluid is called its micrornotion, although itrefers to continuum properties at the coarse-grained macroscopic scale.

In addition to its micromotion, the motion of a complex fluid also possesses theusual properties of classical fluid dynamics. In particular, the motion of a complexfluid involves the advection of thermodynamic state variables such as heat andmass, regarded as fluid properties taking values in a vector space V*. The completedynamical equations for complex fluids must describe both their motion and their

+

Dynamics of micropolar complex fluids January 4, 2000 4

micromotion. In general, these two types of motion will be nonlinearly coupled toeach other.

This definition of complex fluid motion in terms of the continuum dynamics ofboth its order parameter and its usual fluid properties encompasses a wide range ofmodels for complex fluid motion, including binary fluids, multi-phase fluids, poly-meric materials, spin glasses, various other types of magnetic materials, superfluidsand, of course, liquid tryst als. The order parameter for each of these models pro-vides a continuum (i.e., coarse-grained) description of the complex fluid’s internaldegrees of freedom, or micro-order. Here we will discuss only the case in which theseorder parameters are material variables, that is, they are carried along with thefluid parcels. 1

Once the equations for the nonlinear dynamics of their ideal (nondissipative)motion and micro-motion are established, dissipative processes must be includedfor most physical applications of complex fluids. By tradition, this is accomplishedphenomenologically in these models, by introducing kinetic coefficients, such as vis-cosity, mobility, thermal diffusivity, etc., so as to obey the requirements of theClausius-Duhem relation that the entropy production rate be positive when the dy-namics of all thermodynamic variables (including order parameters) are included,as in Dunn and Serrin [1985], Hohenberg and Halperin [1977]. In this paper, weshall ignore dissipation entirely, trusting that it can be added later by using thestandard phenomenological methods. Instcad, we shall concentrate on deriving thenonlinear dynamical equations for the ideal continuum motion and micromotion ofcomplex fluids whose order parameters are material variables. Here such continuummaterials shall be called micropolar complex fhids.

For the case that the order parameter group 0 is the proper orthogonal groupSO(3), a version of micropolar fihid dynamics already exists as part of the ratio-nal theory of cent inuum dynamics for materials with orient ational internal degreesof freedom, such as liquid. tryst als. Rational theories of micropolar media beganwith E. and F. Cosserat [1909]. The Cosserat theories were recapitulated at varioustimes by many different people. See Eringen and Kafadar [1976], and Kleman [1983]for reviews. See Eringen [1997] for recent developments and proposed applicationsof this version of micropolar fluid dynamics for treating liquid crystal dynamics inthe tradition of the rational theory of continuum media.

The present paper starts with the Ericksen-Leslie theory of nematic liquid crys-tals and develops the geometrical framework for continuum theories of micropolarcomplex fluids. In this geometrical framework, the motion and micromotion are

1In some~we~,~r+r ~arameter~aredetermined from constraint relations that are Eulerian in

nature, e.g., the volume fraction in two-phase interpenetrating flow as in Helm and Kupershmidt

[1986a]. Such cases will not be discussed here.

*

Dynamics of mkropolar complex jluzds January 4, 2000 5

nonlinearly and self-consistently coupled to one another by the composite actions~ of the diffeomorphisms and the order parameter group. The micromot ion follows a

curve in the order parameter group depending on time and material coordinate, andthe motion is a time-dependent curve in the group of diffeomorphisms, which actson the material coordinates of the fluid parcels to carry them from their referenceconfiguration to their current positions. A feedback develops between the motionand micromotion, because the stress tensor affecting the velocity of the diffeomor-phisms depends on the gradient of the order parameter. The mathematical basis forour development is the method of Lagrangian reduction by stages, due to Cendra,Marsden and Ratiu [1999], Obtaining the Euler-Poincar6 equations for micropolarcomplex fluids requires two stages of reduction, first by the order parameter groupand then by the diffeomorphism group.

The main results in this paper. The Euler-Poincar6 approach provides a uni-fied framework for modeling the dynamics of complex fluids that preserves andextends the mathematical structure inherent in the dynamics of ideal classical fluidsand liquid crystals in the Eulerian description. This paper provides detailed deriva-tions and discusses applications of the Euler-Lagrange equations, the Lagrange-Poincar6 equations, the Euler-Poincar6 equations, the Clebsch equations and theLie-Poisson Hamiltonian structure of ideal micropolar complex fluids.

The main new results for liquid crystals and micropolar complex fluids in thispaper are:

1.

2.

3.

4.

Four action principles and their associated dynabical equations – the Euler-Lagrange equations, the Lagrange-Poincar6 equations, the Euler-Poincar6 equa-tions, and the Clebsch equations.

The canonical and Lie-Poisson Hamiltonian formulations of these equationsand the Poisson map between them.

The momentum conservation laws and Kelvin-Noether circulation theoremsfor these equations.

The reduced equations for one-dimensional dependence on either space or time,and the relation of these reduced equations to the Euler-Poisson equations forthe dynamics of generalized tops.

Outline. In Section 2, we develop these results for the motion and micromotion ofnematic liquid tryst als, in forms that parallel the results oft he general theory derivedin the following section. The Euler-Lagrange equations from Hamilton’s principlespecialize to the Ericksen-Leslie equations, upon making the appropriate choices


of the kinetic and potential energies. The Lagrange-Poincar6 equations and Euler-Poincar6 equations that follow from applying two successive stages of Lagrangianreduction of Hamilton’s principle with respect to its symmetries provide geometricalvariants and generalizations of the Ericksen-Leslie equations.

In Section 3, we perform two successive stages of Lagrangian symmetry reductionto derive, first, the Lagrange-Poincar6 equations and, second, the Euler-Poincar6equations for the micropolar class of ideal complex fluids for an arbitrary orderparameter group. These Euler-Poincar6 equations are then Legendre-transformedto their Lie-Poisson Hamiltonian form. Finally, their derivation by the Clebschapproach is also given.

2 The example of liquid crystals

2.1 Background for liquid crystals

We begin with a hands-on example that illustrates the utility of the ideas we shalldevelop in this paper and provides a ubiquitous application that embodies theseideas and supplies a guide for developing them.

Liquid crystals are the prototype for complex fluids. For extensive reviews,see Chandrasekhar [1992] and de Gennes and Prost [1993]. An orientational orderparameter for a molecule of arbitrary shape is Chandrasekhar [1992], p.40,

S=:(3@X-Id@Id), (2.1)

where ( . ) is a statistical average and x ~ SO(3) is a rotation that specifies the localmolecular orientation relative to a fixed reference frame. Thus, in index not ation,

(2.2)

This order parameter is traceless in both pairs of its indices, since XT = X-l.For cylindrically shaped (nematic, cholesteric, or smectic2) molecules, we may

choose the 3-axis,and set

say, as the reference axis of symmetry, i.e., choose K = 3 = L

% ‘)(k3) $0 that ln12= XAkXks= 1, (2.3)

and S becomes,

(2.4)

.

2Smectics form layers, so besides their director orientation they have an additional order parsn-

eter for their broken translational symmetry.


.

Note: The order parameter S does not distinguish between n and –n. Physically,nematics are quadrupolar ferroelectrics and S may be regarded as the quadruplemoment of a local molecular charge distribution. For a clear description of the useof this order parameter in assessing nematic order-disorder phase transitions usingthe modern theory of critical phenomena; see Lammert et al. [1995]. Instead ofconsidering such phase transitions, here we shall be interested in the continuumdynamics associated with this order parameter.

In passing to a continuum mechanics description, one replaces statistical averagesby a local space and time dependent unit vector, or “director” n(x, t). Then, thecontinuum order parameter S corresponding to the statistical quantity S in (2.4) isthe symmetric traceless tensor,

(2.5)

which satisfies S(n) = S(—n) and admits n as an eigenvector, S . n = n. Thehydrodynamic tensor order parameter S represents the deviation from isotropy ofany convenient tensor property of the medium. For example, the residual dielectricand diamagnetic energy densities of a nematic liquid crystal due to anisotropy maybe expressed in terms of the tensor order parameter S as,

~E. S(n). E andAp~B. S(n). B, (2.6)

for (external) electric and magnetic fields E and B, respectively. (For simplicity, weneglect any dependence of the electric and magnetic polarizabilities of the mediumon the gradients Vn, although this is also possible to include.)

2.2 Four action principles for liquid crystals

Liquid crystal action. The standard equations for the continuum dynamics ofliquid crystals are the Ericksen-Leslie equations, due to Ericksen [1960, 1961], Leslie[1966, 1968], and reviewed, e.g., in Leslie [1979]. These equations express the dy-namics of the director n and may be derived from an action principle &S = O withaction S = ~ dt L in the following class,

1/S = dt d3X

with notation i = 8x(X, t)/8t, so that

Z(k, J,n, fi, Vn) , (2.7)

overdot denotes material time derivative,J = det(t3x/OX) and Vn(X, t)has spatial components given by

[()]8X -1 A _ 8x~ a

V~n= ~ —–— —n(X, t) e n,z .~~ ~XA – 8Xa 8XA

(2.8)

.

+


Note that the coupling between tl~e fluid dynamics x(X, t) and director dynamicsn (X, t) occurs in the Lagrangian density Z through the inverse deformation gradient,(8x/6’X)-1, via the chain rule expression above for Vi n(X, t).

Varying the action S in the fields x and n at fixed material position X and timet gives

,S= .pp3xpzp[(::j+.~~-J&(.-I&n,p)]8L “ al

+&n.[( )% m(J-’&)l}) ‘2’)

–z+J&

with natural (homogeneous) conditions, expressing continuity of

6’L

w and tim&,m

(2.10)

on a boundary, or at a material interface, whose unit normal has spatial Cartesiancomponents ?Lm. (We sum over repeated indices.)

Action principle #1 – Euler-Lagrange equations. Stationarity of the action,&S = O, i.e., Hamilton’s principle, thus yields the following Euler-Lagrange

equations in the Lagrangian fluid description,

The material volume element J = det (dx/dX) satisfies an auxiliary kinematic equa-tion, obtained from its definition,

Imposing constant J, say J = 1,gives incompressible flowWith the proper choice of Lagrangian density, namely,

L = ~li12 + ~\h12+p(J– 1) + q(ln12 – 1) – F(n, Vn) ,

(2.13)

(2.14)

where p and 1 are material constants (and we ignore anisot ropic dielectric anddiamagnetic effects for now) one finds that the Euler-Lagrange equations (2.11) and

I

Dynamics of mkropolar complex fluids

(2.12) produce the Ericksen-Leslie equations

flow, Chandrasekhar [1992], de Gennes and Prost

.. 8

(6’F

)— p&j+n,i.~ =

p ‘i + dxj ,3

Iii-2 qn+h =0, with h =

In the Lagrangian density (2.14), the Lagrangeincompressibility condition J = 1 and the director

January 4, 2000 9

for incompressible liquid crystal[1993],

o, (2.15)

(2.16)

multipliers p and q enforce thenormalization condition In12 = 1,

respectively. The standard choice for the function &’(n, Vn) is the Oseen-Zocher-Frank Helmholtz free energy density, as discussed, e.g., in Chandrasekhar [1992], deGennes and Prost [1993], name1y,3

F(n, Vn)kll

= $j(n.~ x nJ+Y(V. n)2

chirality ~ splay

kjj k33+ ~(n. Vxn)2+~(nx Vxn)2, (2.17)

~~twist bend

with k2 = Ofor nematics (but nonzero for cholesterics) and n. V x n = Ofor smect its.Since smectics form layers that break translational symmetry, their order parametergroup may be taken as the Euclidean group E(3), or perhaps as S0(3) x U(1)in simple cases. We shall concentrate our attention on nematics in this section.However, the general theory for arbitrary order parameter groups developed in thenext section will also encompass smectics.

Material angular frequency and spatial strain of rotation. The rate ofrotation of a director n in the rest frame of the fluid material element that carriesit is given by

v=nxfi. (2.18)

In terms of this material angular frequency v (which is orthogonal to the directorn), the dynamical equation (2. 16) becomes simply,

lti=hxn. (2.19)

sThisnotationtreatsSpat,ialarlcldirector components on the same (Cartesian) basis. Later, wewill distinguish between these types of components, see, e.g., equation (2.77).

.

Dynamics of micropolar complex jiuids January 4, 2000 10

Likewise, the amount by which a specified director field n(x) rotates under aninfinitesimal spatial displacement from Zi to xi + dzz is given by,

~i=nxV~n or, -y. dz-~idzx=nxdn. (2.20)

The spatial rotational strain one-form y. dx satisfies the kinematic relation,

(y. dz)” = (2v X T, +Viv)dzi, (2.21)

obtained from its definition. This kinematic relation involves only vi, v and x;,which suggests transforming variables from (n, h, Vn) to ~i and v. (Note that ~iand v do not distinguish between n and – n. ) Each component of ~i is orthogonalto the director n. That is, n . yi = O, for i = 1,2,3. Using this fact and ln12 = 1gives the relations,

V~n=–nx~i and Vxn=–ntr~+n~~~, (2.22)

or, equivalently, in index notation,

(2.23)

where lower indices are spatial and upper indices denote director components. Con-sequently, we have the transformation formulas,

Vin. Vjn=Ti. ~j, Vinx Vjn= -yix~j and n“Vxn=–tr~. (2.24)

Connection, curvature, singularities and topological indices. The Euleriancurl of the spatial rotational strain Ti yields the remarkable relation,

% = ?’i,j – 7’j,i + 27i x --/j= 07 (2.25)

where Bij vanishes, provided the vector field n is continuous. In geometrical terms,vanishing of Bij is the Maurer-Cart an relation for the flat connection one-form (or,

left-invariant Cartan one-form) Tick; = n x dn, as discussed in, e.g., Flanders [1989].That is, the curvature two-form Bi~dxi A dzj vanishes in the absence of singularities(disclinations) in the director field n(x). Thus, Bij may be regarded as the arealdensity of disclinations in a nematic liquid tryst al. In Section 3, we shall derive thedynamical equation for the disclirmtion density 13ij.

A second remarkable formula also stems from the curl–-y equation (2.25), whichimplies the relation,

Dynamics of micropolar complex jluids January 4, 2000 11

for a continuous director field. Contracting with n“ and taking the exterior derivativeof this relation implies

8 ~87;

& ( -)nax~dxi ~ dxj A dxk = eabcdnarl dnbA dnc = det(Vn)d3x = det(-y)d3x.

(2.27)

By ln[2 = 1, these expressions must vanish, when the director field has no singular-ities. Thus, in the absence of singularities, there is a vector v, for which

(2.28)

or, equivalently,

eabcnaVnb x Vne = curl v .

multiplicative factor, this formula is

(2.29)

the well known iWerrnin-Up to an unessentialHo relation between three-dimensional superfluid texture n and vorticity curl v indipole-locked 3_He– A, found in Mermin and Ho [1976]. Since n is a unit vector, asimple calculation in spherical polar coordinates transforms this relation to

EabCnaVnb x VnC = Vq$ x Vcos% = curl v, (2.30)

for n = (sin6cos@, sintkin~, cosd)~ in terms of the polar angle 0 and azimuthal angle@on the sphere. Hence,

curl v . dS = eabcnadnb~ dnb = d~ A dcosO,

which is the area element on the sphere. Thus, by Stokes’ theoremcurve C in R3 with unit tangent vector n(x), the integral

(2.31)

for a given closed

}v.dx

c

is the (signed) area swept out by n on the unit sphere upon traversing the curve C

in three dimensions. This is also called the writhe of the curve C’, in Fuller [1978];as cited in Kamien [1998].

Finally, the integral of curl v over the surface of the unit sphere gives a number,

IN=; curlv. dS=~

/ /’d~ A d COSO= & eabcnadnbA dnc, (2.32)

which takes integer values and is called the Hopf index or degree of mapping forthe map n(x) : R3 ~ S2. Equation (2.32) is the second remarkable formula implied


by the curl–~ equation (2.25). The Hopf degree of mapping N counts the numberof times the map n(x) : R3 ~ S2 covers the unit sphere, as discussed in, e.g.,Flanders [1989]. See Mineev [1980] for the physical interpretation of this formula interms of disclinations in nematic liquid crystal experiments. See also Trebin [1982],Kleman [1983], Kleman [1989] and references therein for additional discussions ofthe differential geometry of defects in liquid crystal physics.

The quantity ~abcnaVnb x Vnc in equation (2.30) appears in many differentialgeometric contexts in physics: in the Mermin-Ho relation for the vorticity of super-fluid 311e – A in terms of the “texture” n due to Mermin and Ho [1976]; in theSkyrmion Lagrangian, as discussed. in, e.g., Trebin [1982]; as the n–field topologicalWess-Zumino term in the O(3) nonlinear sigma model, discussed in, e.g., Yabu andKuratsuji [1999], Tsurumaru [1999]; and as the instanton number density in cosetmodels. For references to the last topic see Coquereaux and Jadcyk [1994] and fora discussion of the associated Poisson-Lie models, see Stern [1999]. This same termalso produces forces on vortices in 31fe —A(I?) as found in Hall [1985] and in fer-romagnets, discussed in Kuratsuji and Yabu [1998]. Finally, we have seen that thisterm also appears as the local “writhe” of a self-avoiding closed loop of, say, DNA,or some other polymer filament with unit tangent vector n, see, e.g., Fuller [1978],Goldstein et al. [1998], Kamien [1998]. See also Klapper [1996] and Goriely andTabor [1997], for additional recent discussions of the differential geometric writheof DNA conformations. In the case of liquid tryst als, this topological quantity andthe curvature Bij measure the presence of singularities.

Action principle #2 – Lagrange-Poincar6 equations. Intrigued by the geo-metrical properties of the variables vi and v, we shall seek to express the Ericksen-Leslie equations solely in terms of these variables, by applying symmetry reductionmethods to Hamilton’s principle for liquid crystal dynamics. The trace of the spatialrotational strain ~ gives

(2.33)

which figures in the energies of chirality and twist in equation (2.17) for J’(n, Vn).In fact, the Oseen-Frank free energy F(n, Vn) may be expressed (modulo a

divergence term that will not contribute to the Euler-Lagrange equations) entirelyas a function of a subset of the invariants of -y, namely, tr~, tr(~ + ~~)2 andtr(-y – -y~)2, with the help of the :following identities obtained using ln12 = 1. In


the notation of Eringen [1997],

.tl E (Vxn)2= [n(Vxn)]2+ [nx(Vxn)]2, (2.34)

12 = n~jn~j= (V . n)2 + (V x n)2 + ~njn~j – nin~j~ (2.35),i

()= Vxn2+n~in~j,

13 - [(n. V)n)]2 = [n x (V x n)]2 . (2.36)

Geometrically, the divergence term

(,.njn~j—n%n~j) = 2K1K2, (2.37)

,i

appearing in equation (2.35) for 12 is (twice) the Gaussian curvature of the surfaceZ whose normal is n(x). This equation follows from Rodrigues’ formula

dn=–~dx, (2.38)

for the principle directions of the surface Z. Such a surface exists globally, providedn . curl n = O, see Weatherburn [1974], as cited in Kleman [1983]. Straightforwardcalculations give the following linear relations,

1tr(7~)=–jjIl+~2+j~3, tr(~~) = – ~ (11 + 13) , (2.39)

where ~s and 7A are the symmetric and antisymmetric parts of ~,

7s=; (7+77, and ~A=&-~~). (2.40)

Hence, modulo the divergence term in (2.35), the Oseen-Frank free energy in (2,17)may be written in terms of the invariants of Y as F( tr~, tr[~~), tr(~~) ).

The reduced Lagrange-Poincar6 action. The action S for liquid crystals in(2.7) may thus be reduced, that is, rewritten in fewer variables, by defining,

H($= dt d3X ~($c, J, V, ~m) , (2.41)

where the reduced variables v and ~~ are perpendicular to the director n.A calculation using their definitions shows that the variations of J, v and ~~

satisfy

Dynamics of micropolar complex jiuids January 4, 2000 14

in terms of the variational quantity n x 6n and

This variational expression for ~~ may be rearranged into the form,

@m+-fpVm&+=2(nx h) xvm+vm(nx~n), (2.44)

in which its similarity with the kinematic relation for ~ in (2.21) is more readilyapparent, especially when the kinematic equations (2. 13) and (2.21) are rewrittenin the notation of equation (2.8) as

J= JVixi and ~~+yPV~Xp=2VXT~+V~V. (2.45)

The variation of the Lagrange-Poincar6 action S in (2.41) for liquid crystals in theLagrangian fluid description may IIOWbe rewritten as

(2.46)

Consequently, the action principle &S = Oyields the following Lagrange-Poincard

equations for liquid tryst als,

(2.47)

with natural boundary conditions, cf. equation (2.10),

6’.C8,C/8J = O and & — =0,

aym(2.48)


on a boundary, or material interface, whose unit normal has spatial Cartesian com-ponents ti~. These are the dynamical boundary conditions for liquid crystal motion.These conditions ensure that the fluid pressure 8.L/~J and the normal stress arecontinuous across a fluid interface. The term 7P . OL/dy~ contributes to the stresstensor of the complex fluid and arises from the dependence of its free energy uponthe rotational strain vi. Again J = 1 for incompressible flow, as imposed by theLagrange multiplier p, the fluid pressure. (The Lagrange multiplier q is no longernecessary. )

Upon specializing to the Oseen-Frank-type Lagrangian (2.14), stationarity ofthe action S(x, J, v, ~) in equation (2.41) recovers the Ericksen-Leslie equations(2.15) in these variables. This is an example of Lagrangian reduction - from thevariables (n, n, Vn) to (v, ~). The equations in fewer variables that result from thefirst stage of Lagrangian reduction were named the Lagrange-Poincar6 equationsin Marsden and Scheurle [1995], Cendra, Marsden and Ratiu [1999]. One of thegoals of this paper is to characterize the geometrical properties of such equationsfor rnicropolar complex fluids, using the mathematical framework of Lagrangianreduction by stages established in Cendra, Marsden and Ratiu [1999]. From thisviewpoint, the first stage of the Lagrangian reduction for liquid tryst als is finished.

Action principle #3 – Euler-Poincar6 equations. The Lagrange-Poincar6equations (2.47) and their kinematic relations (2.45) are still expressed in the La-grangian fluid description. We shall pass to the Eulerian fluid description by ap-plying a second stage of Lagrangian reduction, now defined by the following rightactions of the diffeomorphism group,

U(Z, t) = &(x,t)g–yq ,

D(z, t)d3x = d3Xg-l(t) ,

Z/(z,t) = Z/(x, t)g–l(t) , (2.49)

dxi (x> ~)~xA g-l(t),q’~($, t)dx~ = (%(-w ~xA )where the right action denoted x(X, t) = Xg(t) defines the fluid motion as followinga time-dependent curve g(t) in the diffeomorphism group as it acts on the referenceconfiguration of the fluid with coordinate X. In the Eulerian fluid description, theaction S in Hamilton’s principle 6S = O is written as

/[s = dt d3z /(u, D, v, ~) ,

in terms of the Lagrangian density 1 given by

1(U, D, v,~)dsz = L(ig-’(t), Jg-’(t), Vg-’(t), Vg-’(t)) (dsxg-’(t))

(2.50)

(2.51)


The variations of the Eulerian fluid quantities are computed from their definitionsto be,

where X (z, t) s (n x &n)(X, t)g–l (t) and q - tigg-l (t). One may compare theseEulerian variations with the Lagrangian variations in (2.42) and (2.43). See alsoHelm, Marsden and Ratiu [1998] for more discussion of variations of this type inEulerian fluid dynamics. Here the quantities D and ~m satisfy the following Eule-

rian kinematic equations, also obtained from their definitions, cf. the Lagrangiankinematic relations (2.45),

i3D 8Duj—= ——.at axj ‘

(2.56)

Note the similarity in form between the Eulerian variations of D and ~m, and theircorresponding kinematic equations. This similarity arises because both the varia-tions and the evolution equations for these quantities are obtained as infinitesimalLie group actions.

Remark. The kinematic formula (2.57) for the evolution of ~~ immediately im-plies the following ~– circulation theorem for the spatial rotational strain,

(2.58)

Thus, the circulation of ~ around a loop c(u) moving with the fluid is conservedwhen ~n x u is a gradient. Otherwise, the curl of ~~ x v generates circulation ofT around fluid loops. By Stokes theorem and equation (2.26) we have,

L


for a surface S(u) whose boundary is the fluid loop c(u), Consequently, the y–circulationtheorem (2.58) implies that a nonvanishing curl of ~m x u generates a time-changingflux of cabcVnb x VnC for a = 1,2,3, through those surfaces whose boundaries movewith the fluid.

Euler-Poincar6 action variations. We compute the variation of the action(2.50) in Eulerian variables at fixed time t and spatial position x as,

at & a Eqm + a—— .— —— .—8U 8Xj 87m 8Xj A%)] ‘2’0)

[at 84

+Z” –:g–; (—um+—)m

m, & %rn+2ux:+27mx —

~?’m18

[ae

+ ~ %~ +x. :1

a[(

at ae—w~+ dxm “s)+ ’”(z”m+~ >‘m – D 6’” ‘~m – ‘~ 6’7m )]}

where we have substituted the variational expressions (2.52)-(2.55) and integratedby parts.

The dynamical equations resulting from Hamilton’s principle 6S = O are ob-tained by requiring the coefficients of the arbitrary variations qj and 2 to vanish.We assume these variations themselves vanish at the temporal endpoints and wedefer discussing the boundary terms for a moment. Hence, we obtain the Euler-

Poincar6 equations for liquid crystals,

a at ii?th,qj: —— .at c%+ “––- Z&&)’ “&%/%4,6’Xj

(2.61)

a?+2vxg+27mx —

a7m(2.62)

Dynamics of micropolar complex fluids January 4, 2000

Momentum conservation. In momentum conservation form, themotion equation (2.6 1) in the Eulerian fluid description becomes

18

liquid crystal

(2.63)

This equation is in agreement with the direct passage to Eulerian coordinates of themotion equation (2.47) in the Lagrangian fluid description. For this transformation,it is helpful to recognize from equation (2.51) that 1 = (LJ–l )g(t)–l implies, by thechain rule,

&)-1 =GD~, since D(z, t) = J-l (X, t)g(t)-l . (2.64)

Likewise, equation (2.62) agrees with the Lagrangian version of the micromotionequation governing v in (2.47), upon recalling that 81/8u is an Eulerian density.

Noether’s theorem. Noether’s theorem associates conservation laws to continu-ous symmetries of Hamilton’s principle. See, e.g., Olver [1993] for a clear discussionof the classical theory and Jackiw and Manton [1980] for its applications in gaugetheories. The momentum conservation equation (2.63) also emerges from Noether’stheorem, since the action S in equation (2.50) admits spatial translations, that is,since this action is invariant under the transformations,

z.j ~xj =xj+qj(x, t) with qj=cj, (2.65)

for constants Cj, with j = 1,2,3. To see how equation (2.63) emerges from Noether’stheorem, simply add the term d(@j)/&j (arising from transformations of the spatialcoordinate) to the endpoint and boundary terms in the variational formula (2.60)arising from variations at fixed time t and spatial position x, then specialize to73 = Cj.

Kelvin-Noether circulation theorem for liquid crystals. Rearranging themotion equation (2.61) and using the continuity equation for D in (2.56) gives theKelvin-Noether circulation theorem, cf. Helm, Marsden and Ratiu [1998],

Hence, stresses in the director field of a liquid crystalfrequency v and rotational strain “y can generate fluid

(2.66)

and gradients in its angularcirculation. Equivalently, by


Stokes’ theorem, these gradients of liquid crystal properties can generate vorticity,defined as w a curl (D-16’l/8u). For incompressible flows, we may set D = 1 inthese equations and write the vorticity dynamics as,

aw~~+(u”v)ui–(w” v)% = (m+ ,+V.;+--),

)(

+ .E~3~2+’”%)a‘267)axj (9Xm

Thus, spatial gradients in the director angular frequency v and rotational strain ~are sources of the fluid vorticity in a liquid crystal.

2.3 Hamiltonian dynamics of liquid crystals

The Euler-Lagrange-Poincar6 formulation of liquid crystal dynamics obtained sofar allows passage to the corresponding Hamiltonian formulation via the followingLegendre transformation of the reduced Lagrangian 1 in the velocities u and v,in the Eulerian fluid description,

Accordingly, one computes the derivatives of h as

Consequently, the Euler-Poincar6 equations (2.61) - (2.62) and the auxiliary kine-matic equations (2.56) - (2.57) for liquid crystal dynamics in the Eulerian descriptionimply the following equations, for the Legendre-transformed variables, (m, D, ~, ~),

(9D

()

8 Dbh—= —— —t% axj timj ‘

(2.71)

(2.72)

.,


These equations are Hamiltonian. That is, they may be expressed in the form

(2.74)

where z ● (m, D, a, ~) and the Hamilt onian matrix b defines the Poisson bracket

{f, h}==jinx~d):,

which is bilinear, skew symmetric and satisfies the Jacobi

(2.75)

identity,

{-f> {9, ~1}}+ q.(.f,g, h) = O.

Assembling the liquid crystal equations (2.70) - (2.73) into the Hamiltonian form(2.74) gives,

11!

m~ mjt2~+ t?jmi L@ (Ojv~– ?’j ,i) “ u.~t

II 1

6h/bmj(3D 8~D o 0 0 6h/tiDZ vi ‘– 7’j~i+7i, j o 0 –q + 27ix 6h/6y~

u dju o –dj + 27j x 20 x (Sh/&r

In components, this becomes,

(2.76)

where t & = 2.E~pfirepresents (twice) the vector cross product for liquid tryst als.In this expression, the operators act to the right on all terms in a product by thechain rule and, as usual, the summation convention is enforced on repeated indices.At this point we have switched to using both lower and upper Greek indices forthe internal degrees of freedom, so that we will agree later with the more generaltheory, in which upper Greek indices refer to a basis set in a Lie algebra, and lowerGreek indices refer to the corresponding dual basis. Lower Latin indices still denotespatial components.

Remarks about the Hamiltonian matrix. The Hamiltonian matrix in equa-tion (2.77) was discovered some time ago in the context of spin-glasses and Yang-Mills magnetohydrodynamics (YM-MHD) by using the Hamiltonian approach inHelm and Kupershmidt [1988]. There, it was shown to be a valid Hamiltonian ma-trix by associating its Poisson bracket as defined in ‘equation (2.75) with the dual

Dynamics of micropolar complex fluids

space of a certain Lie algebra of semidirect-product

January 4, 2000 21

type that has a generalized two-cocycle on it. The mathematical discussion of this Lie algebra and its generalizedtwo-cocycle is given in Helm and Kupershmidt [1988]. A related Poisson bracket forspin glass fluids was given in Volovik and Dotsenko [1980]. The Poisson bracket fornemat ic liquid tryst als given in Kats and Lebedev [1994] is a constrained Poissonbracket that in general does not satisfy the Jacobi identity.

Being dual to a Lie algebra, our matrix in equation (2.77) is in fact a Liie-

Poisson Hamiltonian matrix. See, e.g., Marsden and Ratiu [1999] and refer-ences therein for. more discussions of such Hamiltonian matrices. For our presentpurposes, its rediscovery in the micropolar complex fluid context links the physi-cal and mathematical interpret ations of the variables in the theory of micropolarcomplex fluids with earlier work in the gauge theory approach to condensed matter,see, e.g., Kleinert [1989]. These gauge theory aspects emerge upon rewriting theLie-Poisson Hamiltonian equations in terms of covariant derivatives with respect tothe space-time connection one-form given by vdt + ~~dz~, as done in Helm andKupershmidt [1988]. The gauge theory approach to liquid crystal physics is reviewedin, e.g., Trebin [1982], Kleman [1983], and Kleman [1989].

The generalized two-cocycle on the Hamiltonian matrix in (2.77) is somewhatexotic for a classical fluid. This generalized two-cocycle consists oft he partial deriva-tives in equation (2.77) appearing with the Kronecker deltas in the u –T cross terms.The first hint of these terms comes from the exterior derivative dv appearing in thekinematic equation (2.57) for -y~. Finding such a feature in the continuum theory ofliquid crystals may provide a bridge for transferringideas and technology between theclassical and quantum fluid theories .4 See Volovick [1992] and Volovick and Vachas-pati [1996] for discussions of similar opportunities for technology transfer in liquidHelium theory. The implications of the generalized two-cocycle for the solutionsof the liquid tryst al equations can be seen by considering two special cases: staticsolutions with one-dimensional spatial dependence; and spatially homogeneous, buttime-dependent, liquid crystal dynamics.

Static ideal liquid crystal solutions with z-variation. Static (steady, zero-velocity) solutions, with one-dimensional spatial variations in, say, the z-direction

41n quantum field theory, these partial derivative operators in the Poisson bracket (or commu-tator relations) are called non-ultralocal terms, or Schwinger terms, after Schwinger [1951, 1959].These terms lead to the so-called “quantum anomalies.”


obey equations (2.70) - (2.73) specialized to

(2.78)

The sum of terms in the first equation of the set (2.78) vanishes to give zero pressuregradient, as a consequence of the latter two equations. We compare the latter twoequations with the E(3) Lie-Poisson Hamiltonian systems, given by

dIl dH 8H—dt “ EUI

—XII +-XI’,(X’

dr 8Hxr.— =:—dt 6TI “

When the Hamiltonian H(ll, I’) in these equations is specialized to

(rIJ2H=&+ Mgx J’,

i=l

(2.79)

(2.80)

where Ii, i = 1, 2, 3, and MgX are constants, then the E(3) Lie-Poisson equations(2.79) specialize to the classical Euler-Poisson equations for a heavy top, discussedin, e.g., Marsden and Ratiu [1999].

By comparing these two equation sets, one observes that the static ideal liquidtryst al flows with z-variation in equations (2.78) and the E(3) Lie-Poisson topsgoverned by equations (2.79) are Legemire duals to each other under the map,

so that

(2.81)

Hence, we arrive at the result:

The class of static ideal one dimensional jiows of a liquid crystal isLegendre-isomorphic to the class of E~3j Lie-Poisson tops.

(2.82)

These tops conserve II .17 and 11’[2,but in general they are not integrable.

,

Dynamics of micropolar complex j?uids January 4, 2000 23

Spatially homogeneous, time-dependent ideal liquid crystal flows. Spa-tially homogeneous solutions of equations (2.70) - (2.73) obey the dynamical equa-tions,

For a single value of the spatial index, say m = 3, these are nothing more than theE(3) top equations (2.79) with time re-parameterized by t e –2t. Hence, in thiscase, the Hamiltonian internal dynamics of a spatially homogeneous liquid crystalis essentially identical to the E(3) Lie-Poisson dynamics of a top. In the multi-component case, one sums over m = 1,2,3, in the second term of equation (2.83)and, tbus, the resulting dynamics is more complex than the simple top. Hence, wehave:

The class of spatially homogeneous, time-dependent ideal liquid crystalj?ows is isomorphic to the generalization (2.83) - (2.84) of the E(3) Lie-Poisson tops.

Action principle #4 – Clebsch representation. Another representation ofHamilton’s principle for liquid crystals in the Eulerian fluid description may befound by constraining the Eulerian action S in equation (2.50) by using Lagrangemultipliers to enforce the kinematic equations (2.56) and (2.57). The constrainedaction is, thus,

with Lagrange multipliers # and pm, Stationarity of S under variations in Ukandv implies the relations

(2.87)

These are the Clebsch relations for the momentum of the motion mk = 81/8u~and the director angular momentum of the micromotion o = 81/dv. Stationary

Dynamics of micropoiar complex jluids January4, 2000 24

variations of S in D and ~~ give, respectively, the dynamical equations for thecanonical momenta, TD = ~ and x~~ = ~~, as

Finally, variations in the Lagrange multipliers # and ~~ imply the kinematic equa-tions (2. 56) and (2.57), respectively. These two kinematic equations combine withthe four variational equations (2.86) through (2.89) to recover the Eulerian mo-tion and micromotion equations after a calculation using the Clebsch relations, theKelvin-Noether form of the motion equation, and the dynamical equations for theClebsch potentials.

At the end of in Section 3, we shall systematize this type of calculation and,thus, clarify its meaning as a Poisson map. For now, we simply remark that theevolutionary Clebsch relations are Hamilton’s canonical equations for the Hamilto-nian obtained from the constrained action in (2.85) by the usual Legendre trans-formation. Perhaps not unexpectedly, this Hamiltonian agrees exactly with that inequation (2.68) obtained from the Legendre transformation in u and v alone.

As we shall discuss more generally in the next section, the Clebsch representa-tions for the momentum and director angular momentum provide a Poisson mapfrom the canonical Poisson bracket in the Clebsch variables to the Lie-Poissonbracket for the Hamiltonian matrix with generalized two-cocycle found in equation(2.77).

Historically, the Hamiltonian approach has been very fruitful in modeling thehydrodynamics of complex fluids and quantum liquids, including superfluids, go-ing back to the seminal work of Khalatnikov and Lebedev [1978], Khalat nikov andLebedev [1980] and Dzyaloshinskii and Volovick [1980]. The Clebsch approach hasprovided a series of physical examples of Lie-Poisson brackets: for superfluids inHelm and Kupershmidt [1982]; superconductors in Helm and Kupershmidt [1983a];Yang-Mills plasmas (chromohydrodynamics) in Gibbons, Helm and Kupershmidt[1982], and Gibbons, Helm and Kupershmidt [1983]; magnetohydrodynamics, mul-tifluid plasmas, and elasticity, in Helm and Kupershmidt [1983b]; Yang-Mills mag-netohydrodynamics in Helm and Kupershmidt [1984]; and its relation to superfluidplasmas in Helm and Kupershmidt [1986b] and spin-glasses in Helm and Kupersh-midt [1988]. Many, but not all, of these Lie-Poisson brackets fit into the presentEuler-Poincar6 framework for micropolar complex fluids.

The Euler-Poincar6 framework also accommodatesmany of the various types ofPoisson brackets (such as ‘(rigid body fluids” ) studied over the years by Grmela,

I


Edwards, Beris, and others, assummarized in Beris and Edwards [1994]. Forliq-uid tryst als, these authors develop a bracket description both for Ericksen-Leslieequations and the Doi-Edwards theory based on the conformation tensor C, whichis related to the director theory by C = n 8 n. The extension of the present re-sults to this case may be accomplished, e.g., by following the Clebsch approach ofHelm and Kupershmidt [1983b] who treated the corresponding case of Lie-Poissonbrackets for nonlinear elasticity. The treatment in Beris and Edwards [1994] ignoresthe geometrical content of the Li~Poisson formulation in preference for its tensorproperties alone.

2.4 Summary for liquid crystals

We now recapitulate the steps in the procedure we have followed in deriving theEuler-LagrangePaincar6-Clebsch equations and the Lie-Poisson Hamiltonian for-mulat ions of the dynamics of liquid tryst als.

1.

2.

3.

4.

Define the order parameter group and its coset space.

Write Hamilton’s principle in the Lagrangian fluid description.

Make 2 stages of reduction:

I’t, to introduce the reduced set of variables in the Lagrangian fluid descrip-tion; and

2n~, to pass to the Eulerian fluid description.

Legendre transform to obtain the Hamiltonian formulation.

The alternative Clebsch procedure starts directly with an action for Hamilton’sprinciple that is defined in the Eulerian fluid description and constrained by theEulerian kinematic equations. It’s Hamiltonian formulation is canonical and passesto a Lie-Poisson formulation via the Poisson map that is defined by the Clebschrepresentations of the momentum and internal angular momentum in equations(2.86) and (2.87), respectively.

Many physical extensions of these results for liquid crystals are available, e.g., toinclude MHD, compressibility, anisotropic dielectric and diamagnetic effects, linearwave excitation properties, etc. However, we wish to spend the rest of this papersetting the formulations we have established here for liquid crystal dynamics intothe geometrical framework of Lagrangian reduction by stages developed in Cendra,Marsden and Ratiu [1999]. This geometrical setting will take advantage of theunifying interpretation of order parameters as coset spaces of broken symmetrygroups. (The coset interpretation of order parameters for liquid crystals, superfluidsand spin glasses is reviewed, e.g., in Mermin [1979].) The present formulations are


geometrical variants of the Ericksen-Leslie equations for liquid tryst al dynamics thatilluminate their gauge theoretical aspects.

The passage to reduced variables v and ~~ for liquid crystals restricts the vari-ables n, h, Vn to the coset space O (3)/0(2) of rotations that properly affect thedirector n and imposes invariance of the theory under the reflections n -+ –n.The reduced variables transform properly under 0(3), because the 0(2) isotropysubgroup of n has been factored out of them. Thus, we “mod out” or “reduce”the symmetry-associated degrees of freedom by passing to variables that transformproperly under rot ations in SO(3) and admit the 22 reflections n ~ –n. The re-moval of degrees of freedom associated with symmetries is the essential idea behindMarsden-Weinstein group reduction in Marsden and Weinstein [1974]. Marsden-Weinstein reduction first appeared in the Hamiltonian setting. However, this sort ofreduction by symmetry groups has been recently extended to the Lagrangian setting,see Cendra, Marsden and Ratiu [1999] and Marsden, Ratiu and Scheurle [1999]. Theremainder of the paper applies the mathematical framework of Lagrangian reduc-tion by stages due to Cendra, Marsden and Ratiu [1999] to express the properties ofideal micropolar complex fluid dynamics in the Eulerian description for an arbitraryorder parameter group.

3 Action principles and Lagrangian reduction

A synthesis of the nonlinear dynamics for the ideal motion and micromotion of vari-ous complex fluid models is possible, due to their common mathematical basis. Themathematical basis common to all ideal fluid motion – both classical and complexfluids - is Hamilton’s principle, see, e.g., Serrin [1959],

J6S=6 Ldt=O.

In the Lagrangian (or material) representation for fluids,the Euler-Lagrange equations for this action principle.

In the Eulerian (or spatial) representation for fluids,

(3.1)

the motion is described by

the Euler-Lagrange equa-tions for the dynamics are replaced by the Euler-Poincar6 equation. The distinctionbetween Euler-Lagrange equations and Euler-Poincar6 equations is exemplified bythe distinction between rigid body motion expressed in terms of the Euler anglesand their time derivatives on the tangent space 7’S0 (3) of the Lie group of properrot ations SO(3), and that same motion expressed in body angular velocity variablesin its Lie algebra so(3). Poincar6 [1901] was the first to write the latter equationson an arbitrary Lie algebra; hence, the name Euler-Poincar6 equations.

Euler-Poincar6 equations may be understood and derived via the theory of La-grangian reduction as in Cendra et al. [1999], and Cendra, Marsden and Ratiu


[1999]. Euler-Poincar4 equations arise when Euler-Lagrange equations and theircorresponding Hamilton principles are mapped from a velocity phase space TQ tothe quotient space TQ/G (a vector bundle) by a Lie-group action of a symmetrygroup G on the configuration space Q. If L is a G-invariant Lagrangian on TQ,this process maps it to a reduced Lagrangian and a corresponding reduced varia-tional principle for the Euler-Poincar6 dynamics on TQ/G in which the variationsare constrained. See Weinstein [1996] and Cendra, Marsden and Ratiu [1999], forexpositions of the mathematical framework that underlies Lagrangian reduction bystages and Helm, Marsden and Rat iu [1998] for a discussion of Euler-Poincar6 equa-tions and their many applications in classical ideal fluid dynamics from the view-point of the present paper. See Marsden, Ratiu and Scheurle [1999] for additionalbackground and recent results in Lagrangian reduction.

The order parameters of micropolar complex fluids are material variables. TheLagrangian in Hamilton’s principle (3.1) for micropolar complex fluids is the map,

L: TGx V*x TO~R. (3.2)

That is, the velocity phase space for the micropolar-fluid Lagrangian L in materialvariables is the Cartesian product of three spaces:

TG, the tangent space of the Lie group G of fluid motions (the diffeomorphismsthat take the fluid parcek from their reference configuration to their currentpositions in the domain of flow),

V*, the vector space of advected quantities carried with the fluid motion, and

T(!7, the tangent space of the Lie group 0 of fluid micromotions (0 is the orderparameter Lie group).

The advected quantities in V* include the volume element or mass density andwhatever else is carried along with the fluid parcels, such as the magnetic fieldintensity in the case of magnetohydrodynamics. The new feature of micropolarcomplex fluids relative to the simple fluids with advected parameters treated inHelm, Marsden and Ratiu [1998] is the dependence of their Lagrangian on TO.The order parameter coset space at each material point is acted upon by the orderparameter Lie group. (We choose the convention of group action from the right.)Since the order parameter is a material property, the diffeomorphism group G alsoacts on the order parameter group, as 0 x G ~ 0, denoting action from the right.

In this Section, we shall use Hamilton’s principle (3.1) with Lagrangian (3.2)to obtain the dynamical equations for the motion and micromotion of micropolarcomplex fluids whose order parameters are defined as coset spaces of Lie groups.In doing so, we shall begin by assuming this Lagrangian is invariant under the


right action of the order parameter Lie group 0 on its tangent space TO. (Thisright action on the space of internal variables leaves the other components of theconfigurateion space T’G and V* fixed.) We shall also assume this Lagrangian isinvariant under the right action of the diffeomorphisms G, which relabels the fluidparcels. (This action of G does indeed affect the material variables defined on T@and V*. ) Under these symmetry assumptions we shall perform the following twogroup reductions

(TG X V* x (TO/0) )/G m g X (V* Xo)/G ,

with respect to the right actions of first 0 and then G, by applying group reductionto the velocity phase space of this Lagrangian in two stages,

1st stage, (TG x V* x ‘TO)/0 m TG x V* x o, (3.3)

2nd stage, (TG x V* x o)/G N g x (V* x o)/G. (3.4)

Here we denote isomorphisms as, e.g., o s TO/O and g H TG/G, where the Liealgebras o and g correspond, respectively, to the Lie groups O and G. The first stageis the Lagrangian reduction by the right action of the order parameter group 0 andthe second stage is the “Lagrangian reduction of the first result by the right actionof the diffeomorphisms G. Because of the assumed invariance of our Lagrangian,these two stages of reduction of the velocity phase spaces will each yield a reducedLagrangian and a corresponding reduced variational principle for the dynamics. Thegroup actions at each stage are assumed to be free and proper, so the reduced spaceswill be local principle fiber bundles .5 The mathematical formulation of the processof Lagrangian reduction by stages and the introduction of various connections onthe Lagrange-Poincar6 bundles that arise in Lagrangian reduction are discussed inCendra, Marsden and Ratiu [19991. These Lagrange-Poincar6 bundles are specialcases of Lie algebroids. See Weinstein [1996] for a fundamental description of therelation between Lagrangian mechanics and Lie algebroids.

3.1 Lagrangian reduction by stages

We are dealing with a Lagrangian defined by the map

L(g, g,ao, X,dX):TGx V*x TO~R,

where G is the diffeomorphism’ group that acts on both the vector space V* ofadvected material quantities and the order parameter group 0. We assume that L

5A natural flat connection appears on this bundle, but this bundle picture should be made

intrinsic and global, while including disclinations. A dynamical equation for the disclination density

will be discussed later in this Section. However, the global bundle picture is for future work.


has the following invariance properties,

L(g, j, ao, x, dx) = L(g, j, ao, x$, dx~) = L(gh, ~h, aoh, xtih dx@h), (3.5)

for all @ E 0 and h c G. In particular, we may choose @ = X–l (t) in the first stageand h = g–l (t) in the second stage of the reduction, so that

L(g, g,ao, X,dX) = L(g,9, aO,xx–1, dXX-l), (3.6)

after the first stage of reduction, and

L(g, g,ao, X,dX) = L(e, gg–l, a, XX–lg–l, dxx–lg–l) ~ Z(f, % v,?), (3.7)

after the second stage, with ~ a gg–l, a - aog–l, v G (Xx–l)g–l and -y . dz E(dXX-l)g-l. After the first stage of reduction, the reduced action principle yieldsthe Lagrange-Poincar6 equations, and after the second stage we shall obtain theEuler-Poincar6 equations for a micropolar complex fluid with an arbitrary orderparameter group.

3.1.1 Lagrange-Poincar& equations

The first stage

TGx V*x TC2t-TGx V*xo,

of the two-stage symmetry reduct ion in (3.3) - (3.4) affects only the internal variablesand passes from coordinates on the order parameter Lie group, 0, to coordinateson its Lie algebra, o, obtained from the tangent vectors of the order parameterLie group at the identity by the isomorphism o R TO/O. The results at this firststage consist of Euler-Lagrange equations for the fluid motion, coupled throughadditional components of the stress tensor to equations of a type called Lagrange-Poincar6 equations in Marsden and Scheurle [1995], Cendra, Marsden and Ratiu[1999]. In our case, these Lagrange-Poincar6 equations describe the micromotion inthe Lagrangian (or material) fluid description.

The first stage of reduction results in the Lagrange-Poincark action princi-

ple,

11(5dt d3X L(i, J, v,y) = 0, (3.8)

written in the material representation and denoted as follows,

L(i, J, v, ~) is the reduced Lagrangian on TG x V* x o,


J(X, t) = det(8z/~X) E V* is the volume element,

v(X, t) = Xx–l (X, t) c o is the material angular frequency and

-y . dz = dxx–l (X, t) E o with components denoted as -y~ given by

-y~ dcc~(X, t) := dxx-l (X, t) ,axm—dXd = ~~t(X, t)dXA ,= Tm ~xA (3.9)

where y. dz is the Cosserat strain one-form introduced in Cosserat [1909] andsuperposed “dot” ( )“ denotes time derivative at fixed material position X.

These material quantities satisfy auxiliary kinematic equations, obtained by dif-ferentiating their definitions,

(J-’d’z)” = (d’ X)” = O, (3.10)

(~. dx)” = (dXX-’)” = dv + adp(~ ~dz) , (3.11)

The material angular frequency v =: Xx–l and the material Cosserat strain one-form~.dx=dxx – 1 take their values in the right-invariant Lie algebra o of the orderparameter Lie group 0. The ad-operation appearing in equation (3.11) denotesmultimplication,or commutator, in the Lie algebra o.

The dynamical Lagrange-Poincar& equations determine the complex fluid’smotion with fluid trajectory &(X) := x(X, t) with @t E G and micromotion x(X, t) c0 in the material fluid description. These equations take the following forms,

The a&-operation appearing in (3.13) is defined in terms of the ad-operation andthe symmetric pairing (.,.) : o* x o ~ R between elements of the right Lie algebrao and its dual o* as, e.g.,

(3.14)

In a Lie algebra basis satisfying [e~, ep] = t~~e, and its dual basis eK satisfying(e’, e,) = 6:, we may write this formula as

(3.15)

Dynamics of micropolar complex fluids

Thus, for the sign conventions we choose in (3.14),the negative transpose of the ad-operation.

January 4, 2000

the ad*-operation

31

is defined as

The dynamical equations (3. 12) and (3. 13) follow from the Lagrange-Poincar6action principle (3.8) for micropolar complex fluid dynamics in the material fluiddescription. These Lagrange-Poincar6 equations may be calculated directly, as

o=

=

——

+

H6 dt d3X L(i, J, v, ~) (3.16)

-“~+% ’’+(~’v)+(%)’’m)l/dtpx [::

kPx{’4- L%)”- ‘&#+ J&(’-’%,’p)]

+([- (~)”+ad~~- ~&(J-l#-)+ad~m~]~)}

//dt d2S tiA

a ((%)”) -(% ’p)’’p+J~”m)

Here we define E = 6XX–1 and drop endpoint terms that arise from integrating byparts in time, upon taking &zP and .X to vanish at these endpoints. The naturalboundary conditions

ensure that the fluid pressure and th’e normal stressinterface.

o, (3.17)

are continuous across a fluid

3.1.2 Euler-Poincar6 equations

The passage next from the Lagrangian fluid description of continuum mechanics tothe Eulerian fluid description will yield the Euler-Poincar6 equations. We obtainthese equations by applying to Hamilton’s principle the second stage,

TGx V*xo~gx((V*xo)/G),

of the two-stage Lagrangian reduction in (3.3) - (3.4). This second stage of reductionresults in the Euler-Poincar6 action principle,

/~ z(f, a,v, T)dt=O, (3.18)


with constrained variations

where q(t) = tg(t)g(t)–l ~ g and E(t) = 6X(t) X(t)–1 ● o both vanish at theendpoints.

The Euler-Poincar6 action principle produces the following equations defined ong x ((V* x o) /G) for the motion and micromotion, in which ~/& denotes Euleriantime derivative at fixed spatial position x,

These are the Eu2er-Poincard equations for a micropolar complex fluid. In theseequations, 1 is the reduced Lagrangian on g x (V* x o)g(t)’1. Also,

is the commutator in the Lie algebra of vector fields, g. In addition, we define thetwo operations ad; and o as

(ad’+=-(:ad”)7and

(:0”1’)‘-(:aq)

(3.23)

(3.24)

The concatenation a q denotes the right Lie algebra action of q E g on a ● V* (byLie derivative). The pairing (. , .) now includes spatial integration and, thus, allowsfor integration by parts. Similar definitions hold for (61/&y o ~) and (81/6v o v).

Eulerian kinemat ic equations. By definition, an Eulerian advected quantity

a c V*g(t)–l satisfies

~+a&=O.


This advection relation may be written equivalently as

where 3C is the Lie derivative with respect to ~ = g(t)g(t)’1, the Eulerian fluidvelocity, often denoted also as u(x, t).

The Eulerian versions of the Lagrangian kinematic equations (3.10) and (3.11)

are given in terms of the Lie derivative by

(3.25)

(3.26)

In these equations, the quantity D(z1 t) = J–l (X, t)g(t)–l is the Eulerian massdensity and the quantities V(Z, t) = v(X, t)g(t)-l and

(-y(z, t) . (h = ~(x, t)“g(m “~x)9(v

are the Eulerian counterparts of the right-invariant material quantities v(X, t) and7(X, t) in equations (3.12) - (3.11).

Remark. We note that equation (3.26) implies the ~– circulation theorem formicropolar complex fluids, cf. equation (2.58),

(3.27)Jc(O

loop c(~) moving with the fluid is conservedthe curl of this quantity generates circulation

Thus, the circulation of -y around awhen adV-y is a gradient. Otherwise,of -y around fluid loops.

The Euler-Poincar6 equations (3.20) and (3.21) may be obtained directly fromthe Euler-Poincar6 action principle (3.18), as follows.

Dynamics of micropolar complex jluids January 4, 2000

Euler-Poincar6 action variations. We compute the variation(3. 18) in Eulerian variables at fixed time t and spatial position x as,

34

of the action

where we have used the variational expressions in (3.19) and integrated by parts.Here q j a denotes substitution of the vector field q into the tensor differential forma and (. , .) denotes the natural pairing between a tensor field and its dual.

The dynamical Euler-Poincar6 equations (3.20) and (3.21) are thus obtainedfrom the Euler-Poincar6 action principle (3.18), by requiring the coefficients of thearbitrary variations q and E to vanish in the variational formula (3.28). The re-maining terms in (3.28) yield Noether’s theorem for this system, which assigns aconservation law to each symmetry of the Euler-Poincar4 variational principle.

Momentum conservation. In momentum conservation form, the micropolarmotion equation in the Eulerian fluid description (3.20) becomes

(9 a 19 ( 6’1‘— &n—% d~j = 8X~ L3f$j

+16~j–

%0+ ‘(aa)a ‘32’)

In this equation, expressed in Cartesian coordinates, there is an implied sum over thevarious types of advect ed tensor quantities, a. This momentum conservation law alsoarises from Noether’s theorem, as a consequence of the invariance of the variationalprinciple (3. 18) under spatial translations. In fact, the simplest derivation of thisequation is obtained by evaluating the variational formula (3.28) on the equationsof motion and using the translational symmetry of the Lagrangian in Noether’stheorem with qj = ~/~xj.


.

Kelvin-Noether circulation theorem for micropolar complex fluids. Re-arranging the motion equation (3.20) and using the continuity equation for D in(3.25) gives

where the circulation loop c(() moves with the fluid velocity &and we have used thefollowing relation, valid for one-form densities,

(3.31)

This relation may be checked explicitly in Cartesian coordinates, as follows,

See Helm, Marsden and Ratiu [1998] for more explanation and discussion of theKelvin-Noether circulation theorem for Euler-Poincar4 systems.

3.2 Hamiltonian dynamics of ideal micropolar complex fluids

The Legendre Transformation. One passes from Euler-Poincar4 equations ona Lie algebra g to Lie-Poisson equations on the dual g* by means of the Legen-dre transformation, see, e.g., Helm, Marsden and Ratiu [1998]. In our case, westart with the reduced Lagrangian 1 on g x (V* x o)g(t)–l and perform a Legendretransformation in the variables ~ and v only, by writing

61 61

~=@’ ‘=G’ h(p, a, a, ~) = (p,&) + (a, v) – l(f, a, v, ~). (3.33)

One then computes the variational derivatives of h as


Consequently, the Euler-Poincar6 equations (3.20) - (3.26) for micropolar dynam-ics in the Eulerian description imply the following equations, for the Legendre-transformed variables, (p, a, o, -y), cf. equations (2.70) - (2.73)

As for the case of liquid crystals discussed earier, these equations are Hamiltonian

and may be expressed in terms of a Lie-Poisson bracket.

Lie-Poisson bracket for micropolar complex fluids. Assembling the microp-olar equations (3.35) into Hamiltonian form gives, symbolically,

with boxes ❑ indicating where the matrix operations occur. More explicitly, interms of indices and differential operators, and for a = D, the mass density, thisbecomes

Here, the summation convention is enforced on repeated indices. Upper Greekindices refer to the Lie algebraic “basis set, lower Greek indices refer to the dualbasis and Latin indices refer to the spatial reference frame. The partial derivative83 = 6’/dzj, say, acts to the right on all terms in a product by the chain rule.For the case that t~K are structure constants ea~~ for the Lie algebra so(3), theLie Poisson Hamiltonian matrix (2.77) for liquid tryst als is recovered, modulo anunessential factor of 2.

As mentioned earlier in our discussion of Hamiltonian dynamics of liquid crys-tals, the Hamiltonian matrix in equation (3.37) was discovered some time ago in the

Dynamics oj micropolar complex fluids January 4, 2000 37

.

context of investigating the relation between spin-glasses and Yang-Mills magneto-hydrodynamics (YM-MHD) by using the Hamiltonian approach-in Helm and Kuper-shmidt [1988]. There, it was shown to be a valid Hamiltonian matrix by associatingits Poisson bracket with the dual space of a certain Lie algebra of semidirect-producttype that has a generalized two-cocycle on it. The mathematical discussion of thisLie algebra and its generalized two-cocycle, as well as the corresponding Lie-PoissonHamiltonian equations for spin-glass fluids and Yang-Mills magnetohydrodynamics,are given in Helm and Kupershmidt [1988]. The present work provides a rationalefor the derivation of such Lie-Poisson brackets from the Lagrangian side.

Spatially one-dimensional static solutions with z-variation. Static (steady,zero-velocity) solutions for micropolar complex fluids, with constant pressure andone-dimensional spatial variations in, say, the z-direct ion obey equations (3.37) fori = 3 = -j, rewritten as

As for the case of liquid crystals, the sum of terms in the first equation of the set(3.38) vanishes to give zero pressure gradient, as a consequence of the latter twoequations.

Under the Legendre transformation

These Legendre-transformed equations are Poincar4’s [1901] generalization of Euler’sequations for a heavy top, expressing them on an arbitrary Lie algebra with structure

constants t;fi” Thus>

The steady, spatially one-dimensional solutions for all micropolar com-plex jluids have the underlying Lie algebra structure discovered in Poincar6[1901].

Dynamics of mkropolar complex fluids January 4, 2000

Spatially homogeneous, time-dependent ideal micropolar flows.homogeneous solutions of equations (3.37) obey the dynamical equations,

38

Spatially

(3.41)

For a single spatial index, say ~i, these are again Poincar6’s [1901] equations gener-alizing Euler’s equations for a heavy top to an arbitrary Lie algebra. Of course, thecorresponding Hamiltonian matrix for this system is the lower right corner of thematrix in equation (3.37).

When t~& = cap. for the Lie algebra SO(3), Poincar6’s equations (3.41) cor-

311e — A with spin density O@ and spinrespond to the Leggett equations foranisotropy vector -yia, see Leggett [1975]. For special solutions of these and otherrelated equations in the context of 311e– A, see Golo and Monastyrskii [1977], Goloand Monastyrskii [1978], Golo et al. [1979].

Evolution of the disclination density. Helm and Kupershmidt [1988] use thechain rule and the defining relation for the disclination density, cf. equation (2.25),

Bi: = -h; – 73;+ $.7’ ~? ‘ , (3.42)

to transform the Hamiltonian matrix (3.37) to a new Hamiltonian matrix, whoseLie-Poisson Hamiltonian dynamics may be written as

rd(3.43)

The corresponding dynamics for the disclination density emerges as

~ &ht~KBi3~ . (3.44)

Of course, this equation admits the trivial solution Bi~ = O, e.g., when the directorfield is continuous. However, it also provides an evolution equation for B$ # O,


i.e., for nontrivial disclination density, when the director field is singular. Upon “recalling the Legendre tranformation formulas (3.34), this equation may be rewrittenequivalently as

This formula further implies the relation,

(3.45)

(3.46)

for any surface that moves with the fluid. Thus, the flux of disclinations through amicropolar complex fluid surface is not conserved, in general, except for the trivialsolution Bi~ = O. An intrinsic and global characterization of the vector bundleaspects of disclination density dynamics in micropolar complex fluids is for futurework.

Momentum conservation for nonzero disclination density. For a Hamilto-nian I@, D, 13ij, o) that is invariant under spatial translations, the motion equationfor Pi in (3.43) implies the following momentum conservation law in the presenceof disclinations,

(3.47)

Thus, nonzero disclination density affects the stress tensor T! to produce magneto-

striction as in magnetohydrodynamics (MHD), see Helm and Kupershmidt [1988].However, unlike MHD, the spatially homogeneous solutions for the disclination den-sity have nontrivial dynamics similar to the Euler-Poisson tops, as one sees in thelower right corner of the Hamiltonian matrix appearing in equation (3.43).

3.3 Clebsch approach for micropolar complex fluids

Following Serrin [1959], we call the auxiliary constraints imposed by the Euleriankinematic equations the Lin constraints. As we shall see, the diamond operationo defined in equation (3.24) arises naturally in imposing the Lin constraints. Takingvariations of the constrained Eulerian action,


yields the following Clebsch relations,

.6(:

(5U:

&a:

6?) :

(by :

tip :

(3.49)

We shall show that these Clebsch relations recover the Euler-Poincar4 equations(3.20) - (3.21). (In what follows, we shall ignore boundary and endpoint terms thatarise from integrating by parts.)

The diamond operation o is defined by

(ooa, q)--(v, ina)=-(v, ~q). (3.50)

This operation is antisymmetric,

(voa, q)=–(aov, q), (3.51)

as obtained from,

(v, lqa)+(~qv, a)=o, or, (v, aq)+(~q, a)=o, (3.52)

and the symmetry of the pairing (” , .). The diamond operation also satisfies thechain rule under the Lie derivative,

(<f(voa), q)=((<fv)oa, q)+(-uo(<ga), q). (3.53)

This property can be verified, as follows,

(ifvoa, q)+(vo<~a, q) = (v&q, a)–(vq(, a)

= (a, v(adtq)) =-(aov, (adtq))

= (ad~(ao v), q) = (#t(ao v), q), (3.54)

where we have used (v (, a q) + (v (q, a) = O, implied by (3.52), in the first step.Finally, we have the useful identity,

(Podv, q)=-(d,80~, q), (3.55)

Dynamics of micropolar complex jluids January 4, 2000

as obtained from (dv)q= d(vq) and

41

(p, d(zq))+(dp, vq)=o. (3.56)

These three properties of the o operation and the micropolar Clebsch relations (3.49)together imply

The term in square brackets is seen to vanish, upon pairing it with a vector field,integrating by parts and again using the properties of the o operation. This manip-ulation recovers the micropolar motion equation (3.20) as

since, as we have seen,

for one-form densities such as 6//6<.

(3.58)

(3.59)

The micromotion equation (3.21) is also recovered from the Clebsch relations(3.49). This is accomplished by taking the time derivative of the 6v–formula, sub-stituting the b~– and b~–formulas, and using linearity of ad* to find

(:+ ’J:+ ’+$ (3.60)

= - ~(wm + NCLP- ad; (MI;O) - ad~aduy)P]

——— [ad; d,f3– adf (adj ~)]

= ad* E

“/iv”

Hence, the Clebsch relations (3.49) also recover the micromotion equation (3.21).

Remarks. From the Hamiltonian viewpoint, the pairs (v, a) and (,6, ~) are canon-ically conjugate variables and the Clebsch map (w,a, ~, T) --+ (p, o), with


is a Poisson map from the canonical Poisson bracket to the Lie-Poisson Hamil-

tonian structure given in equation (3.37), in which a = D. Of course, there is no

obstruction against allowing a to be any advected quantity, as discussed in Helm,

Marsden and Ratiu [1998].

The generalized two-cocycle associated with the Hamiltonian matrix in (3.37)arises from the term d~ in the a–part of this Poisson map.

Various other applications of the Lin constraint and Clebsch representation ap-proach in formulating and analyzing ideal fluid and plasma dynamics as Hamiltoniansystems appear in Helm and Kupershmidt [1983b], Marsden and Weinstein [1983],Zakharov et al. [1985], Zakharov and Kusnetsov [1997].

4 Conclusions

Complex fluids have internal variables whose micromotion is coupled to the fluid’smotion. Examples include spin-glass fluids, superfluids and liquid crystals. Formicropolar complex fluids, these internal variables are materially advected orderparameters that may be represented equivalently as either geometrical objects, oras coset spaces of Lie groups. The new feature of micropolar complex fluids relativeto simple fluids with advected parameters treated in Helm, Marsden and Ratiu[1998] is the dependence of their Lagrangian

L: TGx V*x TOWR,

on TO, the tangent space of their order parameter group. We treat Lagrangiansthat are invariant under the right actions of both the order parameter group 0 andthe diffeomorphisms G. In this case, reaching the Euler-Poincar6 fluid descriptionrequires two stages of Lagrangian reduction,

(TG x v* x (TO/0) )/G N g x (v* x 0)/G,

rather than the single stage of reduction (with respect to the ‘relabeling transfor-mations” of G) employed for simple fluids in Helm, Marsden and Ratiu [1998].

After studying the example of nematics in Section 2, we derived the Euler-Poincar4 dynamics of micropolar complex fluids in two stages of Lagrangian reduc-tion in Section 3. The first stage produced the Lagrange-Poincar6 equations derivedfrom an action principle defined on the right invariant Lie algebra of the order pa-rameter group in the Lagrangian (m material) fluid description. The second stage ofLagrangian reduction passed from the material fluid description to the Eulerian (orspatial) fluid description and produced the Euler-Poincar6 equations for micropolar

.

Dynamics of micropolar complex fluids January 4,2000

complex fluids. We also derived these Euler-Poincar4 equations usingapproach.

43

the Clebsch

In addition, we used a Legendre transformation to obtain the Lie-Poisson Hamil-tonian formulation of ideal micropolar complex fluid dynamics in the Eulerian fluiddescription. The Lie-Poisson Hamiltonian formulation of these equations agreedwith that found earlier in Helm and Kupershmidt [1986b], Helm and Kupershmidt[1988], who treated spin-glass fluids, Yang-Mills magnetohydrodynamics and super-fluid 4He and 3He. Thus, we found that Lagrangian reduction by stages provides arationale for deriving this Lie-Poisson Hamiltonian formulation from the Lagrangianside. This approach also fits well with the gauge theoretical aspects of condensedmatter physics.

Many other potential applications of this Euler-Poincar6 framework abound inthe physics of condensed matter. For example, besides the liquid crystal dynamicstreated here explicitly, the superfluid hydrodynamics of the various phases of 311emay be treated similarly. In particular, the geometrical framework of Lagrangianreduction by stages is well-adapted to the standard identification of the phases of3He with the independent cosets of the order parameter group S0(3) x S0(3) x U(l),as discussed, e.g., in Mineev [1980] and Volovick [1992]. Magnetic materials may alsobe treated this way. The seminal papers on the geometrical properties of magneticmaterials are Dzyaloshinskii [1977], Volovik and Dotsenko [1980] and Dzyaloshinskiiand Volovick [1980]. Other recent studies of the dynamics of magnetic materialsand superfluid 3He in directions relevant to the present paper appear, e.g., in Helmand Kupershmidt [1988], Balatskii [1990], Isayev and Peletminsky [1997], Isayev,Kovalevsky and Peletminsky [1997].

Acknowledgements

I am grateful to A. Balatskii, J. Hinch, P. Hjorth, J. Louck, J. Marsden, T. Mullin,M. Perlmutter, T. Ratiu, J. Toner and A. Weinstein for constructive commentsand enlightening discussions during the course of this work. I am also grateful forhospitality at the Isaac Newton Institute for Mathematical Sciences where part ofthis work was completed. This research was supported by the U.S. Department ofEnergy under contracts W-7405-ENG-36 and the Applied Mathematical SciencesProgram KC-07-01-01.

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