area-constrained planar elastica - unamchryss/papers/pre_arealis.pdf · problem 15 .1 however, the...

Area-constrained planar elastica

Guillermo Arreaga* and Riccardo Capovilla†

Departamento de Fısica, CINVESTAV IPN, Apartado Postal 14740, 07000 Mexico, DF, Mexico

Chryssomalis Chryssomalakos‡ and Jemal Guven§

Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apartado Postal 70-543, 04510 Mexico, DF, Mexico�Received 27 March 2001; published 13 February 2002�

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixedenclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We findthat the area constraint gives rise to equilibria with remarkable geometrical properties; not only can theEuler-Lagrange equation be integrated to provide a quadrature for the curvature but, in addition, the embeddingitself can be expressed as a local function of the curvature. The configuration space is shown to be essentiallyone dimensional, with surprisingly rich structure. Distinct branches of integer-indexed equilibria exhibit self-intersections and bifurcations—a gallery of plots is provided to highlight these findings. Perturbations con-necting equilibria are shown to satisfy a first-order ODE which is readily solved. We also obtain analyticalexpressions for the energy as a function of the area in some limiting regimes.

DOI: 10.1103/PhysRevE.65.031801 PACS number�s�: 46.70.Hg, 87.16.Dg

I. INTRODUCTION

Our object of study in this paper is a deflated rigid loop�elastica hypoarealis�. To explain this, consider a closed loopin a plane. The loop is made of some elastic material and, iftaken thin enough, the contribution of its longitudinal defor-mations to its elastic energy can be ignored. This gives riseto a Hamiltonian density proportional to the square of itscurvature—hence elastica. It is clear that fixing only thelength L of the loop will not lead to any surprises, the onlypossible equilibrium being a circle. What will happen thoughif we, in addition, fix its area A to be less than that of thecircle? �hence hypoarealis�.

This particular Hamiltonian �without the constraints� hasa distinguished history, dating back at least to the Bernoullis.It makes a reappearance in Euler’s inspired analysis of amundane but subtle problem in mechanical engineering: thebuckling of a loaded beam �1,2�. More recently, variants ofthis problem have attracted the attention of mathematicians.In particular, the problem of determining the curves of con-stant length which minimize the bending energy, on surfacesof constant curvature, is considered in Refs. �3–5�, whileinterconnections with knot theory are explored in �6�. Addingthe constraint on the enclosed area, as we propose here, pro-vides a particularly fruitful generalization of this work.

Hamiltonians depending on extrinsic curvature have alsobeen studied in statistical physics �7,8�. For space curves, aneffective Hamiltonian which depends on twist as well asbending provides a phenomenological description of stiffpolymers, and in particular DNA �see, e.g., Refs. �9, 10��.Moving up one dimension, the leading term in the HelfrichHamiltonian, which describes the equilibrium configurationsof lipid membranes, is proportional to the square of the

traced extrinsic curvature K integrated over the surface,namely, the conformally invariant Willmore functional�11,12,13,14�. In the variant of the Helfrich model proposedby Svetina and Zeks, the so-called bilayer couple model,constraints are placed on the surface, such as constant area,constant enclosed volume, and constant integrated mean cur-vature �the latter breaking the K→�K symmetry of theproblem� �15�.1 However, the equations which determine theequilibrium are highly nontrivial higher-order partial differ-ential equations �PDE’s�. One motivation for introducing ourvariant of the elastica, as a planar analog of the above, is itspotential as a toy model for understanding these higher-dimensional membranes. At the simplest level, if a closedmembrane possesses a symmetry along the z axis, its profileat any z will be a fixed loop. The dimensionally reducedHelfrich Hamiltonian is then the bending energy of this loop.It is possible to examine the loop Hamiltonian exactly. Whilea lot is known about axially symmetrical closed configura-tions �16�, there is very little nonperturbative knowledge ofthe equilibria which exist when axial symmetry is broken;the loop configurations provide an analytic point of entry.

An additional application of the planar problem waspointed out recently by Willmore; any closed loop can beexploited to generate an axially symmetric toroidal geometry.The equilibria of the loop Hamiltonian on a surface of con-stant negative curvature can be mapped into equilibria of theHelfrich functional �17�.

We will exploit the integrability feature of our modelwhich is very well disguised when the problem is cast withrespect to the variables embedding the loop in the plane.Indeed, the determination of the curvature at equilibrium canbe reduced to the study of the motion of a particle in aone-dimensional quartic potential. The curvature can besolved as a quadrature in terms of elliptic integrals. Remark-ably, we discover that the problem possesses a second, farless obvious, level of integrability; the embedding can be

*Email address: [email protected]†Email address: [email protected]‡Email address: [email protected]§Email address: [email protected]

1We note that there is no genuine one-dimensional analog of itsrefinement, the so-called area difference model �21�.

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expressed as a local function of the curvature without anyneed of the two integrations one would have expected. As aconsequence, the equilibrium configuration is given by ageometrical construction of pythagorean simplicity.

The condition of closure of the loop results in a discretespectrum for a given constrained perimeter and area. Weshow that scaling can be exploited to fix one of these, say theperimeter. One branch of the spectrum then consists of con-figurations with an n-fold symmetry (n�2,3, . . . ) which in-flate into a circle as the enclosed area is increased. If the areais evacuated these configurations will eventually self-intersect, tending to a limiting configuration of an inverted�i.e., negative area� circle, winding n�1 times and decoratedwith n infinitesimally small circles. The latter dominate, inthis limit, the energy and give rise to a pole in the energyversus area diagram, the residue of which we evaluate ex-actly for each n.

The remaining branches of the spectrum correspond torather complicated self-intersecting pairs which bifurcatefrom a limiting configuration. While self-intersecting con-figurations are undoubtedly of interest mathematically, theyare not of primary relevance to the physical problem wepose—we determine sufficient conditions for avoiding them�nevertheless, we describe a physical situation where theymay arise�.

In general, one would expect equilibria-connecting pertur-bations to be governed by a fourth-order differential equa-tion. We show, however, that this equation can be reduced�by three orders� to a first-order one. The latter is singular atthe circular configuration, where bifurcation occurs. Analyti-cal approximations for the energy as a function of the areaare derived, in the limit of sinusoidal perturbations of thecircle of frequency n. Combined with the pole mentionedabove, these expressions provide a reliable sketch of theenergy-area relationship for each n.

The paper is organized as follows: Section II introducesthe model and the shape equation that determines equilibria.We show how it can be reduced to the motion of a fictitiousclassical particle in a quartic potential. We address the be-havior under scaling of the shape equation and we show thatthe problem has essentially one free parameter. Moreover weanalyze the loop statics; this provides a physical interpreta-tion for the shape equation itself. In Sec. III we study theconfiguration space for this model. In particular, we describethe angle �0 by which the normal to the loop gets rotated ina full oscillation of the fictitious particle in the potential.Equilibria-connecting perturbations are the subject of Sec.IV. These are determined by a complicated fourth-order dif-ferential equation, which is used to obtain the purely geo-metrical construction mentioned above. In turn, the latterpermits the reduction of the order of the original equationfrom four to one, as well as the derivation of a sufficientcondition for non-self-intersections. Finally, we obtain ana-lytical expressions for the energy as a function of the area, insome limiting regimes, which allow for a reliable sketch ofits behavior. In the appendices we collect various expres-sions, useful in the calculation of variations, we derive arecursion relation for the average of the powers of K and

comment on an interesting Legendre transform in the spaceof parameters of the model.

II. FIRST CONSIDERATIONS

A. Energy functional

A closed loop, parametrized by s��0,1� , is described bythe embedding in the plane,

x��X� �s �.

The arc length l along the loop is given by

l�s ��0

s

ds�� dX�

ds�• dX�

ds�� 1/2

. �1�

The most general expression for the energy of the configu-ration X� (s) which �i� does not depend on the parametriza-tion, �ii� involves no higher than two derivatives of X� and,�iii� is quadratic in these derivatives, is given by

F�X� �� dl K2, �2�

where K is the geodesic curvature, equal to the inverse of theradius of curvature at each point of the loop. We will takehenceforth � to be equal to 1 (� is dimensionful, unlike itstwo-dimensional analog, and this means we are measuringlength in units of ��. Let t be the unit tangent to the loop�transversed counterclockwise� and n its outwards normal.Then we have the Frenet-Serret equations for a plane curve2

�the prime denotes a derivative with respect to arclength l�,

n��Kt , t��Kn . �3�

In terms of the angle � that n makes with, say, the x axis,K��.

To implement the constraints of fixed length and enclosedarea, we introduce the constrained functional:

Fc�X��F�X�� dl�L �� int

d2x�A � . �4�

� appears as a Lagrange multiplier enforcing the constraintfixing the length of the loop to some value L. In the sameway, � is associated with the constraint fixing the enclosedarea to the value A. We note that, in general,

��Fc�L ,A �

L, ��

Fc�L ,A �

A. �5�

In particular, if the area constraint is relaxed so that �0,then Fc(L ,A)/A�0. Looking at Eq. �5�, one might be ledto identify �, with the tension and differential pressure onthe loop but this is only half true—we discuss the physical

2Notice that the opposite sign convention for K is quite commonin the literature.

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meaning of these parameters in Sec. II E. We have not in-cluded, in our energy functional, a term proportional to theintegral of K. This is because such a term is simply the totalangle by which n gets rotated in transversing the loop once,equal to 2� for a non-self-intersecting loop.

We emphasize that the bending energy is not scale invari-ant �notice that neither is the dimensionally reduced HelfrichHamiltonian�. An unconstrained circular loop will expandwithout limit. The natural scale invariant expression, �dl�K�,though not in itself a topological invariant, does have a van-ishing Euler-Lagrange derivative almost everywhere. Takenas an energy functional, its constrained configurations will bearcs of a circle �solutions of �K�� joined together at cur-vature discontinuities in such a way as to mimimize the cor-responding constrained energy function. This model will beconsidered elsewhere.

B. Shape equation

The Euler-Lagrange equations follow from extremizingFc�X� ,

Fc

X� �0. �6�

The reparametrization invariance of Fc�X� implies that thetangential projection of these equations is vacuous. The nor-mal projection gives

2K��K3��K��0 �7�

�see the Appendix for some relevant formulas�. We note thatwith the identification, �→��K0

2, Eq. �7� coincides withEq. �7� of �18� �see also �19��. Is is worth pointing out that aloop on a surface of constant Gaussian curvature G satisfiesthe above equation with �→��2G .

A plane curve is determined, up to rigid motions, by itscurvature �20�. As is well known, our one-dimensional sys-tem is completely integrable. This is obvious because wehave cast derivatives with respect to arclength. With respectto an arbitrary parametrization of the loop, first derivativeterms associated with the one-dimensional Laplacian appearwhich conceal this fact. Writing Eq. �7� in the form

K��d

dKV�K �, V�K ��

1

8K4�

�

4K2�

2K , �8�

we map our problem onto the problem of determining themotion of a fictitious particle in a quartic potential, with Kbeing the displacement of the particle, and l playing the roleof time. The total energy E of the particle is conserved,

dE

dl�0, E�

1

2K�2�V�K �, �9�

a fact that permits the expression of K� in terms of K andhence, of the arclength along the loop as an integral over K�via dl�dK/K��

l�� dK

�2�E�V�K ��. �10�

We emphasize the difference between the energy E of thefictitious particle, on the one hand, and the bending energy Fof the loop, on the other. In particular, the configuration ofleast ‘‘energy’’ in the analog is not the minimum of theHamiltonian.

The motion of the particle in the potential V is periodic.Closed loop configurations consist of an integer number ofidentical segments, each one corresponding to one full oscil-lation of the particle, and hence made up itself of two sym-metric halves. Then, the condition of closure of the loop canbe expressed as

�0�2�

n, n�2,3, . . . , �11�

where �0 is the angle by which the normal n gets rotated inone full oscillation of the particle, given by

�0�2�Kmin

Kmax K dK

�2�E�V�K ��12�

�Kmin , Kmax denote the turning points�. The equilibrium con-figurations have n-fold symmetry and a well-defined center.The value n�1 is omitted in Eq. �11� because it is special—see below. We examine in detail the resulting configurationsin Sec. III.

C. Scaling

Let us examine the behavior of Eq. �7� under scaling ofthe position vector, X� →�X� . We find

l→�l ,d

dl→ 1

�

d

dl, K→ 1

�K . �13�

Let X� (l) correspond to some given solution of Eq. �7�, withparameter values (1 ,�1). Then, a scaled solution withthe same shape, �X� (�l) and with (L1 ,A1 ,F1 ,E1)→(�L1 ,�2A1 ,��1F1 ,��4E1), is obtained by rescaling themultipliers as follows:

�1→1

�2 �1�� , 1→1

�3 1�� . �14�

Eliminating � we find the orbits of scaling in the �;�� plane,

��1

12/3 �

2/3 . �15�

Furthermore, an inversion in the origin, X�→�X� can beidentified with a rescaling by ��1, which maps a solutionwith a given into one with �. It follows that, for thepurpose of identifying distinct configurations at least, onecan set, e.g., �1 and scan the essentially one-dimensionalconfiguration space by varying �. For each value of �,which fixes the form of the potential V(K), one still has to

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vary the energy E of the particle in the well so as to satisfyEq. �11�, resulting, in general, in an infinite discrete spectrum‘‘above’’ the point �1, �� in the �, �� plane.

D. Windings

The closed configurations we wish to classify possess twotopological invariants, both given as the winding numbers ofmaps S1→S1. The first such map is the normal map of theloop, associating with each value of the parameter s �whichranges over the first S1 above� the corresponding value of n�which ranges over the unit circle, the second S1 above�. Foreach rotation of s, n will generally rotate m times, with m aninteger. Configurations with distinct m’s are topologicallyisolated, the physical implementation being the infinite en-ergy barrier associated with the move

Non-self-intersecting loops correspond to normal maps withindex m��1 �the converse is not true�.

The second map is from the s circle to the closed orbit(�S1) traced by the phase point in the plane (K ,K�). For aconfiguration corresponding to n complete oscillations of theparticle, the phase point goes around the closed curve 1

2 K�2

�V(K)�E n times, Fig. 1. There is a finite energy barrierseparating configurations with distinct n’s which preventstransitions between them.

E. Forces and torques

What can we learn about the loop configurations by look-ing at the equilibrium of forces and torques on an infinitesi-mal loop segment? First we ask, why would a rigid loopwant to have area less than the maximal allowed by itslength? One setup that supplies an answer is to imagine thatour loop is actually the cross section of an infinite cylinder,the interior and exterior of which are filled with an incom-pressible fluid. We start by filling the cylinder to its maxi-mum capacity, this gives a circular loop. We then take out

some fluid from the interior while making sure that the ex-ternal pressure is large enough so as to prevent the formationof bubbles in the interior. Notice that, because of its rigidity,the cylinder takes up some part of the exterior pressure andonly transmits to its interior a fraction of it, it’s the differen-tial pressure that crumbles the cylinder walls. The second ofEq. �5� then points to the identification of � with this dif-ferential pressure, taking into account that the latter pointsinwards. One has to be careful in applying the same argu-ment to �. The first of Eq. �5� seems to suggest that � is thetension of the loop but this would presuppose the possibilityof tangential deformations—there is nothing in our energyfunctional that tells us how much energy these cost and,indeed, we have already assumed that the loop cannot bestretched or compressed tangentially. The derivative in thefirst of Eq. �5� is computed by comparing distinct loops withinfinitesimally differing lengths, it does not refer to the de-formation of a single loop. With this in mind, we now turn toloop statics.

We denote by T� (l0) the total force from the segment ofthe loop with l�l0 to the one with l�l0 and by �(l0) thecorresponding torque. The latter is equal to �2K—this fol-lows from our normalization of the bending energy F. Then,balancing the torques on a segment extending from l to l�dl we get

�� l �� l�dl ��Tn� l�dl �dl

�0⇒��Tn�0, �16�

where Tn is the normal force, the reference point was takenat l, and the sign conventions are shown in Fig. 2. The equi-librium of normal forces on the segment gives

Tn� l ��Tn� l�dl ��Tt� l�dl �K� l �dl�dl

�0⇒�Tn��KTt��0, �17�

while the tangential components give

Tt� l��Tt� l�dl��Tn� l�dl�K� l�dl�0⇒Tt��KTn�0.�18�

FIG. 1. On the left: the potential V(K), for the values of � , shown. Barred quantities, in this and subsequent figures, are measured in

units of �0�q/2 , q being their length dimension. Also shown, with horizontal lines, are the energies E (n) that give rise to the n�2,3,4,5 closed

configurations, equal to 0.253, 0.278, 0.298, 0.320, respectively �in units of �02 and measured from the bottom of the well—compare with

the ��0.5 curve in Fig. 4�. On the right: the closed trajectories in the phase plane K-K� of the fictitious particle, corresponding to

oscillations in the potential V with energies E (2), E (5)—the n�3,4 trajectories lie in between the two shown.


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Notice how, in Eqs. �17� and �18�, the curvature K is respon-sible for the tangential force at l�dl contributing a normalcomponent at l and vice versa. Solving the above system ofequations we find

��2K , Tn��2K�, Tt�K2�� , �19�

where, in the third relation, the integration constant was fixedto the value � by the requirement that one recover the dif-ferential equation for K, Eq. �7�. We see from Eq. �19� that�� is the tension of the loop at its inflection points, if any.Notice also how Eq. �7� is obtained upon substitution of thelast two of Eq. �19� in Eq. �17�, thereby identifying thephysical origin of each of the terms in the former. In a forth-coming publication, where we extend our considerations tothe case of a loop in space, it is shown how the above ex-pressions for the forces and torque follow, in a model inde-pendent way, from an application of Noether’s theorem.

Our submerged cylinder model for the loop leaves noroom for self-intersecting configurations, could there be anyuse for these? Imagine the loop made of superconductingmaterial, in the presence of a uniform magnetic field perpen-dicular to its plane. In the limit where the magnetic flux dueto self-inductance is negligible compared to the one due tothe external field, the area of the loop has to be constant tokeep the flux constant. One can then adjust the area bychanging the magnetic field and, for small enough areas,self-intersecting configurations will appear.

III. CONFIGURATIONS

A. Qualitative remarks

For the purposes of this section, we may set, as explainedabove, �1 in the expression for the potential V(K). Itscritical points are given by the zeros of its derivative, i.e., bythe roots Ki , i�1,2,3, of

K3��K�1�0. �20�

We find

K1��K��i)K� , K2��K��i)K� , K3�2K� ,

�21�

where

K��m

12�

�

m, K��

m

12�

�

m, m3�108�108�1�

�3

�03 ,

�22�

where �0�3/22/3. Of these, K3 is always real while the firsttwo are real only for ��0 . Notice that, for any �, theeffect of the quadratic and the linear term in the potential isrelatively important only in a neighborhood of the origin. Formotions of the particle with sufficiently high-energy E, thetime spent by the particle in this region is negligible and thequartic term dominates the motion. As a result, we may con-clude that �0 approaches zero with increasing E, due to theapproximate symmetry K→�K of the motion. The term lin-ear in K spoils this symmetry and, for �0 �as taken in thissection�, makes �0 slightly positive for large E. As �0 de-scends, with increasing E, from this positive value to 0, itwill cross all critical values 2�/n for n greater than somen0 . We expect, therefore, to encounter configurations witharbitrarily high n, for all �—see Fig. 4.

On the other extreme, when the energy E is only slightlygreater than a local minimum of the potential �at K1 or K3�,and the particle oscillates around that minimum, we mayapproximate V(K) by a quadratic expression in K and findfor �0 the limiting value

�0imax�2�

&

�3��

Ki2

, i�1,3, �23�

where the superscript �max� is used because, as we shall seeshortly, this is actually a maximum of �0i(� ,E), for fixed �.Starting from Eq. �21�, one infers the limiting values

�01max��→�0

�� , �01max��→��2� , �24�

as well as

�03max��→��0, �03

max��0 ��2�� 23 ,

�03max��→��2� . �25�

FIG. 2. Balancing of �a� torques, �b� normal forces, and �c� tangential forces on a segment of the loop. K and K� are sketched positive;the resulting directions of torques and forces are shown. For Tt this data is not enough; we have assumed additionally K2��—this puts thecenter of the loop somewhere towards the left of each picture.


031801-5

A plot of �0imax �� is given in Fig. 3. We notice that, for

��0 , there are no non-self-intersecting configurationscorresponding to oscillations in the left well.

We explain now why the value n�1 was omitted in Eq.�11�. The reasoning behind that relation was that a closedconfiguration should correspond to n complete oscillations ofthe particle. This is not necessarily so in the case of thecircle. Any number of ‘‘oscillations’’ �of zero amplitude� willfit into a circle, including irrational numbers, and indeed, Eq.�23� shows that the circle corresponding to the particle rest-ing at Ki , i�1,3, is made of �(3��/Ki

2)/& complete os-cillations.

To examine what happens for values of E that render thequadratic and linear terms important, we analyze the cases��0 and ��0 separately.

B. μËμ0

The potential possesses only one minimum, at K3�0given by Eq. �21� �see Fig. 1�. �03 starts at �03

(max) , for E�V(K3), and for � far from �0 , decreases monotonically tozero with increasing E. The configurations that do not appearare those for which 2�n�2�/�0

max as well as all with nega-tive n. This condition determines a set ��n�n�2,3, . . . � ofcritical values of �, such that, for ��n , all configurationswith n��2,3, . . . ,n are absent �the circle is, of course, al-ways there�. As � approaches �0 , the left wall of the poten-tial develops a plateau that tends to the horizontal, as �→�0 . The particle spends a relatively long time in this�negative K� region, which results in a negative bump in the�0 curve. As �→�0 , the minimum of this bump tends to��—negative n configurations appear accordingly. Directevaluation of the integral �12� gives �21�

�03�� ,E ��8g�aB�bA �

A�B ��2K�k �

��2

1��2 �� 2

�2�1,k � � , �26�

where k2�(a�b)2�(A�B)2/4AB ,

g�1

�AB, ��

A�B

A�B, �2�

bA�aB

aB�bA,

A2��a�b1�2�a1

2, B2��b�b1�2�a1

2, b1��1

2�a�b�, �27�

a12�

4

a�b�ab�

1

4�a�b �2, a�Kmax�� ,E �,

b�Kmin�� ,E �

and K, � are the complete elliptic integrals of the first andthird kind, respectively, given by

K�k ��0

�/2 d�

�1�k2 sin2 �,

��2,k ��0

�/2 d�

�1��2 sin2 ��1�k2 sin2 �. �28�

The result �26� holds for ��0 as well, with the appropriatechoice of Kmin , Kmax . A plot of �03(�;E), for various val-ues of �, appears in Fig. 4.

A couple of remarks are in order at this point. Considerthe point A in Fig. 4, which is the intersection of the ��0.5 curve with the n�2 line, corresponding to an n�2configuration. Imagine now that � is diminishedcontinuously—the corresponding curve will move more orless downwards forcing A to move to the left. The corre-sponding E then diminishes, which means that the fictitiousparticle oscillates in the potential well with smaller ampli-tude. When A hits the �03 axis, E is zero, the particle sits atthe bottom of the well, and the corresponding configurationbecomes a circle. In other words, all configurations like theone corresponding to the point A can be continuously de-formed to a circle.

FIG. 3. �0imax(�), i�1,3.

FIG. 4. �03(E), E��E�V(K3)�/�02, for various values of

��/�0 . Also shown, with horizontal dashed lines, are some ofthe values of �03 that give rise to closed configurations. The corre-

sponding values of E can be read off as in the case ��0.5,n�2 shown—the �self-intersecting� configuration itself appears inFig. 9. Notice how the ��1,�2.5 curves miss the n�2, n�2,3,4 configurations, respectively.


031801-6

This is not the case though with the points B1 , B2 in thesame figure. Consider B1—it corresponds to an n��4 con-figuration for ��1.01. Imagine now that � is diminished ina continuous way. As soon as it becomes smaller than 1, theinfinite negative pole of the corresponding curve is softenedto a negative minimum which, for � sufficiently close to 1,still intersects the n��4 line. Given that the � curves inFig. 4 become almost horizontal at high E, we give an exag-gerated sketch of the situation in Fig. 5. As the value of � islowered, this negative minimum rises and, for a critical valueof � , will just touch the n��4 line, i.e., B1 and B2 collapseto a single point B. The configuration that corresponds to thispoint is not a circle, since E is positive and the particleoscillates with a finite amplitude, i.e., K is not constant. Weconclude that configurations corresponding to points likeB1 , B2 cannot be continuously deformed to a circle. More-over, there exist bifurcation points, like B above, distinctfrom the circle. When � is lowered even more, the minimumof the � curve becomes positive and remarks similar to theabove can be made about its points of intersection B1� , B2�with sufficiently high n lines �see Fig. 5�.

C. μÐμ0

The potential possesses two local minima, at K1�0 andK3�0 �with V(K1)�V(K3)� and one local maximum atK2�0 �see Fig. 6�. Depending on its energy and where it isstarted from, the particle is confined in the left well, the rightwell, or visits both during every oscillation.

We note the following, regarding the asymptotic behaviorof �0i as �→� . In this regime, the linear term becomesnegligible in V(K) and the integral giving �0i can be easilyseen to reduce �with K2→x� to an integral giving �half� theperiod of a harmonic oscillator in a potential x2/4��x/2,which is independent of its energy E, as well as the ‘‘con-stant force’’ �. Another way to see the � independence is byobserving that, with �0 �which is equivalent to sending �to infinity�, one lies on the � axis in the �, �� plane, whichis an orbit under scaling, hence changes in � in this regimeleave �0i invariant. As the energy approaches V(K2) �eitherfrom above or below�, the period of the motion tends toinfinity, most of which the particle spends at K2 . Assume westart the particle at the turning point on the right with zerovelocity and with just the right energy to reach the centralmaximum of V , E�V(K2). The corresponding curve startsin a counterclockwise sense with decreasing curvature,passes through an inflection point, and acquires infinitelength spiraling forever clockwise as it approaches asymp-totically a circle of radius 1/K2 , see Fig. 6.

�03 tends accordingly to �� , which corresponds to thenegative poles in Fig. 4. As mentioned already, for V(K1)�E�V(K2), �0 will have two branches, �01 and �03 , cor-responding to the motions confined to either well. Both ofthese branches will tend to �� as E→V(K2). For E�V(K2) there is only one branch, which starts from �� andapproaches asymptotically zero �as E→��, after havingreached a positive local maximum. A three-dimensional plotof �03(� ,E) is given in Fig. 7. We present a representativecollection of configurations in Fig. 8. We also show, for ref-erence purposes, some self-intersecting configurations in Fig.9.

IV. PERTURBATIONS

A. Equilibrium-connecting deformations

We study here the following problem: given an equilib-rium configuration X� , find an infinitesimal deformation �X�

�� n such that X� � �X� describes a nearby equilibrium. No-tice that �X� is not necessarily the displacement vector ofsome point of the loop—the latter will generally have a tan-gential component as well. Our motivation is to obtain newsolutions from known ones, with the closure condition auto-matically satisfied.

Using some of the formulas listed in the Appendix, thelinearization of the equilibrium condition �7� gives that thedeformation � must satisfy

DE4��2��5K2��10KK��

��12E� 52 K4�6�K2�10K ��

�� K� � , �29�

where the deformed configuration satisfies Eq. �7�, with�→�� , →� � . The energy will change as well,E→E� �E . This appears explicitly in the linearization ofEq. �8�, which gives

FIG. 5. Bifurcations: as � is reduced, the minimum of the curveshifts upwards and B1 , B2 collapse to a point. Further decrease of� forces B1� , B2� to collapse to a point. The corresponding configu-rations cannot be continuously deformed to a circle.

FIG. 6. On the left: the potential V(K) for ��2. On the right:spiraling configuration for E�V(K2), with the particle starting atthe turning point on the right with zero velocity.


031801-7

DE3��2K��K3��K��2K2K��

�K�12E� 12 K4�2�K2�5K ��

�� 12 K2 ��K �� E . �30�

B. The � identities

We introduce a number of functions on the loop, relevantin the study of the solutions of Eq. �29�. Any vector field a�defines the following two functions on the loop:

an�a� • n , at�a� • t . �31�

The position vector X� , in particular, supplies

h�X� • n , p�X� • t , �32�

both functions evidently dependent on the choice of origin.One finds

an��Kat , at��Kan , h��Kp , p��1�Kh ,�33�

FIG. 7. The angle �03(� ,E), for 0.5��

�1.3, in steps of 0.05 and 0�E�0.8, in variablestep. The ��1 curve has been shifted to ��1.01 in order to avoid the ambiguity in the

number of roots of V . The negative pole in thecurves for ��1 occurs at E�V(K2).

FIG. 8. Non-self-ntersecting configurations for ��0.5, �1, and n�2,3,4,5. Also shown, for each configuration, is thecircle of radius X0 �see Eq. �38��, passing through its inflectionpoints.

FIG. 9. Self-intersecting configurations for ��0.5, �1, andn�2,3,4,5. Also shown, for each configuration, is the circle of ra-dius X0 �see Eq. �38��, passing through its inflection points.


031801-8

where a� is henceforth assumed constant. We will say that afunction f on the loop is the generator of a transformation ifand only if under the latter, X� • n� f . With this definition,an , at are the generators of translations along a� and perpen-dicularly to it, respectively. p and h generate rotations anddilations, respectively �both with respect to the origin�. Oneeasily verifies that DE4(an)�DE4(at)�DE4(p)�0 whileDE4(h)�2�K�3 , in accordance with the scaling behav-ior of �, found earlier, Eq. �14�. Experimenting a little withEq. �29� we find, not without some surprise, two more solu-tions:

DE4�K��0, DE4��1�K2��2�K�3 . �34�

Noting, additionally, that when the origin is at the center ofthe loop, the zeros of p coincide with the extrema of K, aswell as the coincidence of the extrema of h and K2, we makethe ansatz

h��1�K2�� f , p�2�1K��g , �35�

with f, g to be determined. Substituting in Eq. �33�, we findf ��Kg , g��Kf , with solution

f�a cos��0�, g�a sin��0�, �36�

where a, �0 are arbitrary constants. One recognizes f, g tobe, respectively, the functions an , at corresponding to theposition vector a� of the center of the loop, then Eqs. �35�state

h��1�K2��an , p�2�1K��at , �37�

a remarkable geometrical property, the implications of whichwill occupy us shortly. Notice that neither part of Eq. �37�survives in the →0 limit. Unless otherwise stated, we willtake a� equal to zero in what follows.

One might wonder whether Eq. �29� admits other polyno-mial solutions in K, apart from the second of Eq. �34�. Toinvestigate this, we rewrite Eq. �29� in terms of derivativeswith respect to K, using d/dl�K�d/dK and find for DE(Kn)

the leading term ( 18 n4� 3

4 n3� 18 n2�3n� 5

2 )Kn�4, with onlypositive integer root n�2. We conclude that no other poly-nomial solutions of Eq. �29� exist.

C. Geometrics

We are now in a position to give a purely geometricalconstruction of the equilibria. Indeed, starting from X2�h2

�p2 and using Eq. �37�, we find

X2�X02�4�1K , X0��1�8E��2, �38�

where, in the derivation, use was made of the first integral,Eq. �9�. This remarkable formula expresses the embeddingcompletely in terms of K(l). We emphasize that the shape isobtained directly from K, given ( ,� ,E), without any inte-gration. Note that both the relations �37� follow by takingderivatives with respect to the arclength of Eq. �38�. It fol-

lows from Eq. �38� that the moment of inertia I of the looparound an axis perpendicular to its plane and passing throughits center, is given by

I�LX02�8��1. �39�

To construct geometrically the equilibrium curve �forgiven , �, E�, we draw a circle with radius X0 and from apoint P outside it, bring the tangent PA to the circle �see Fig.10�. Taking as base the square of this tangent, we construct aparallellepiped with volume 4�1, the height of which is theradius of curvature of the equilibrium curve at P—this fol-lows from Eq. �38�, written in the form (X2�X0

2)��4�1, where � is the radius of curvature at P. To achieveclosure of the resulting curve, one has to start with a particu-lar slope, given by either part of Eq. �37�. In this way, oneconstructs the part of the curve lying outside the circle �thelatter intersects the curve at its inflection points, if any�—theinterior part, present only if the ficticious particle reachesinto the negative K region, is constructed in a similar man-ner.

Referring back to our expression for the force T� (l), Eq.�19�, we realize that Eq. �37� implies that T� (l) is orthogonalto X� (l) while T(l)�X(l), i.e.,

T� � l �� n� t ��X� � l �, Tn��p , Tt�h , �40�

with T� (l) the force on the part of the loop pointed to by t .Notice that T� is generally compressing but changes neverthe-less to true tension at the points were X� is tangential to theloop. Also, p has acquired a direct physical interpretation asa consequence of Eq. �40�, pdl is just the torque, withrespect to the origin, of the force due to the pressure on asegment of the loop with length dl . Then the vanishing ofthe total torque on the loop is guaranteed by the fact that p isa derivative. In fact, one may derive a compact and ratherpleasing formula for the torque due to pressure on any seg-

FIG. 10. Geometrical construction of the equilibrium curve. Theparallelepiped has a square base with side AP and volume 4�1. Itsheight p gives the radius of curvature at P. The part of the curvelying in the interior of the circle is constructed similarly.


031801-9

ment of the loop, like the one defined by A,B in the sketch ofan n�4 configuration of Fig. 11.

Suppose we take A as reference point, then Eq. �37� isvalid with a� connecting A with the center O of the loop. Weget

�AB�A ��

A

B

dl p��A

B

dl�2�1K��at�

�2�KB�KA�� •�A

B

d� l ,

i.e.,

�AB�A ��2�KB�KA��a� •AB� . �41�

Moving the reference point simply moves one endpoint of a� .For a� �0 �torque with respect to O�, �AB

(O)�2(KB�KA).

D. Self-intersections

Given that some potential applications of our model ex-clude self-intersecting configurations, we look now for suffi-cient conditions for non-self-intersection. We restrict our at-tention to self-intersections that can be reached by acontinuous deformation of non-self-intersecting configura-tions. In other words, we consider a one-parameter family ofconfigurations X� (l ,t), t��0,1� , continuous in t, such that,for every t, the corresponding curvature satisfies Eq. �7� andwe take X� (l ,0), X� (l ,1) to be non-self-intersecting and self-intersecting, respectively. Then, we observe that as t variesfrom 0 to 1, one necessarily encounters a ‘‘kiss’’:

The position vector X� A is along the axis of symmetry of thelobe and tangential to the loop at A. The third of Eq. �40�then shows that the force at A is normal to the loop, while thesecond of Eq. �40� gives its magnitude as

TA�XA . �42�

The tangency of X� at A implies that hA�0 and hence, usingEq. �37� once more, we get �assuming that K is negative at akiss3�

KA�� , �43�

for all kisses, regardless of the order of the configuration. Weconclude that a sufficient condition for non-self-intersection(of the type defined above), is ��0. Equation �43� gives KA�

2

in terms of , �, E:

KA�2�2�E� 1

8 �2� 12 ��. �44�

On the other hand, from Eq. �38� we find that

XA2 ��2�8E��2��4�1�� . �45�

The tangency of XA though means that 2�1KA��pA�XA

and the above two equations then give �setting �1�

E2� 14 ��2�4�� 1

2 �E� 164 ��2�4��2�4��1 ��0,

�46�

with roots

E1� 12 �� 1

8 �2� 18 , E2� 1

2 �� 18 �2. �47�

For the particular case n�2, KA��0 and we get E�E2 . Wehave seen in Sec. III that the relation �0�2�/n defines acurve in the ��, E� plane, consisting of all parameter pairsgiving rise to a configuration of order n. The intersection ofthat curve with the ones just written above, consists of thepoints ��, E� giving rise to a kissing configuration of order n.

Consider how Eqs. �40� and �41� guarantee equilibrium insome particular examples. First, look at an even-n configu-ration, say, n�2:

The total force due to the pressure, pushing together the twohalves in the sketch, is �AB� �. Equation �40� says that thetension at A,B �purely tangential, compressing� is �AB� �/2,thus leaving each half at rest. As a second example, considerthe lobe defined by a self-intersection:

The total force due to the pressure on the lobe is zero, andEq. �40� says that the forces from the rest of the loop, on thetwo ends of the lobe that meet at A, are opposite �with direc-tion so as to keep the lobe closed�. As a further check on ourresults, one can verify the balancing of torques using Eq.�41� on, say, the right half of a kiss.

E. Connecting equilibria

There is more to be derived from Eq. �38�. Taking � onboth sides, we find

DE2�a �� 2e��3K2��

2 �X0

2�2K�1 � ,

�48�

3This assumption is true for all configurations we have studiednumerically.

FIG. 11. Computing the torque due to pressure on the arc AB.


031801-10

a considerable improvement over Eq. �29�.4 Moreover, byvarying the first of Eq. �37� we also find

DE2�b �� 2K��2K��2K3��

��1�K2�� . �49�

Comparison with Eq. �48� leads to a first-order equation for� :

DE1��4K��4K��c2K2�c1K�c0 , �50�

where the constants ci are given by

c2��2�1 � , c1� �X02, c0�2 ��2��1 � .

�51�

Noting that d/dK�(K�)�1d/dl , Eq. �50� can be written inthe form

8�E�V ��4V��c2K2�c1K�c0 , �52�

where the dot denotes differentiation with respect to K andV�V(K). This latter equation is readily integrated to give

��1

4K�� dK

c2K2�c1K�c0

�2�E�V ��3/2 . �53�

One may add an arbitrary amount of K� �rotation� to this �butnot an or at , since these move the origin�. To get particularsolutions from Eq. �53�, we need to specify the direction ofthe deformation in the �, �� plane, i.e., the ratio ��/ � .It will prove convenient for our further analysis of Eq. �53�,to reparametrize the �, �� plane introducing new coordi-nates ��, �� via

�� ,��1/3, �� ,��2/3� . �54�

A point P with coordinates ( ,�)�(0,0) lies on a uniquescaling orbit, which can be specified by the � coordinate ofits point of intersection P� with the �1 line; this is thevalue of � for P. One can get now from P� to P by scaling by� �see Fig. 12�. The obvious advantage of these new coordi-nates is that the scaling orbits are constant-� lines. The de-pendence on � of an arbitrary quantity S(� ,�), with lengthdimension q, is S(� ,�)��qS(�), where we denote by a tildethe remaining function of �. It follows that S/��q��1S

so that �S�q��1S ��qS� �� , the prime denoting heredifferentiation with respect to �.

We verify that Eq. �53� gives ��h for scaling. Equation�51�, written in terms of ��, ��, gives

c2�6��1 �� ,

c1��2�16E�2�2� ��1�8E��2�� , �55�

c0�2��3� ��2��2 �� ,

where E(� ,�)��4E(�) is the energy that guarantees clo-sure of some particular configuration. An increment �� cor-responds to scaling by a factor of 1� ��/� , hence the cor-responding � should be ��( ��/�)h . Putting ��0 in Eq.�56� one determines the ci for pure scaling, then computingd/dK �(K2��)/K�� one finds that the K5 and K3 terms inthe numerator cancel and one recovers the integrand in Eq.�53� with just the right c’s.

Integrating the first of Eq. �37� with respect to dl , andusing

A�1

2 � d l h , �56�

we get

F��L�2A , �57�

from which

�F�L �� L�2A ��2 � A �58�

follows. On the other hand, direct variation of F, Eq. �2�,gives

�F�� L� � A . �59�

Comparing with Eq. �58� we find

L ��2A ��3 � A�2� �L�0. �60�

For a length-preserving deformation, �L�0, and Eq. �60�reduces to

L ��2A ��3 � A�0. �61�

We note in passing that the various differential operators wehave defined above are related in the following way:

�DE1��2K�DE2�a �� ,

K�DE1��2K�DE2�b �� K�DE1��,

DE3��K��DE2�a �� K�DE2

�a ��

� 14 �K2��DE1��,

K�DE4��2�DE3��.

It follows that any perturbation that satisfies Eq. �50� willnecessarily satisfy all the higher-order ones.

4Notice, however, that Eq. �48� is only valid when the center ofthe loop is at the origin.

FIG. 12. Definition of the coordinates �, �.


031801-11

F. Perturbing the circle

Referring to our cylinder model of the loop �see Sec. II E�,we consider here the following thought experiment: we startwith the loop ‘‘filled’’ to capacity, i.e., in a circular configu-ration, and start deflating it by removing fluid from its inte-rior while keeping its length fixed. We would like to followthe evolution of its shape, for distinct n’s, and derive thelimiting form of the bending energy F as a function of thearea A near the circular extreme. Going beyond the cylindermodel, we would also like to allow for self-intersections—what happens if one just keeps subtracting area?

Consider the following perturbation to a circle of radiusR:

��R��1�� sin�n�� , �1. �62�

The line element and curvature in these coordinates are

dl��2� �2 d� , K��2�2 �2��

��2� �2�3/2 , �63�

where the dot here denotes differentiation with respect to �.Substituting Eq. �62� in the first of Eq. �63�, integrating over�, and requiring the total length to be equal to the initialvalue 2�R , we find

R��R� 1�n2�2

4 ��O��3�, �64�

while for the curvature the second of Eq. �63� gives

K�1

R�1��n2�1 �sin�n��O��2�. �65�

We substitute the above expression in the differential equa-tion for K, Eq. �7�, and demand that it be a solution to O(�).The constant and O(�) terms, respectively, give

1�R2��R3�0, 2n4��R2��5 �n2�3�R2��0.

�66�

Notice that the first relation is simply the statement V(1/R)�0 while the second also follows from Eq. �29� with � as inEq. �62�. For this perturbation the differential equation can-not be satisfied to O(�2). From Eq. �66� we get

�2

R3 �n2�1 �, ��1

R2 �3�2n2�. �67�

During the deflating process, both and � will vary as func-tions of A. The point in the -� plane corresponding to theconfiguration will trace out an orbit, starting at the abovepoints �for each n�, all of which lie on the line R3�R2��1. For F and A we find

F�2�

R �1�1

2�n2�1 �2�2��O��3�,

A��R2�1�1

2�n2�1 ��2��O��3�. �68�

Notice that, near the circle,

F

A��

2

R3 �n2�1 �� ,

in agreement with Eq. �59� �for L�0�. As A keeps dimin-ishing, our numerical analysis shows that the configurationsstart self-intersecting, giving rise to regions of negativearea—this scenario is sketched in Fig. 13. Notice how thewinding numbers ��1 for the little circles, �1 for the bigone� add up correctly to match that of the circle to which thisconfiguration can be continuously deformed. This observa-tion points to a feature of the above m�1 limiting configu-rations already alluded to in the introduction; the big circle inthe configuration of order n actually winds around itself n�1 times. For example, to move from one lobe to the nextalong the loop, in the n�3 configuration, one must travel anangle of 4�/3, not 2�/3 �a glance at Fig. 9�b� reveals howthis comes about�. The limiting shape in this direction then isa circular one, with radius slightly less than R/(n�1) and nlittle circles attached to it. In this extreme, F(A) is domi-nated by the little circles and assumes the limiting form

F�A ��4�n2

R

1

1��n�1 �A

�R2

, �69�

where the area is counted with the appropriate multiplicitydue to the winding �e.g., plus three times the area of the littlecircle, minus twice that of the big one, in the n�3 case�. Asketch of F(A) that interpolates between Eqs. �68� and �69�is given in Fig. 14.

V. CONCLUSIONS

In this paper, we have studied the equilibrium configura-tions of elastic planar loops with constant area and constantlength. The condition of closure of the loop gives a discretespectrum of configurations, lying along several branches inparameter space. We focus on non-self-intersecting loopsthat inflate to a circle when the enclosed area is increased,due to their relevance as a toy model for two-dimensionalmembranes. For this branch, starting from analytical expres-sions in the relevant limit cases, we obtain a reliable sketchof the dependence of the energy on the area. The equationthat determines equilibria-connecting perturbations led us,rather unexpectedly, to the identities, Eq. �37�, from whicha geometrical construction of the �constrained� elastic loopfollowed. In turn, these identities permit the expression of an

FIG. 13. Evolution of a n�2 configuration under deflation�sketch�. The central region in the third configuration, as well as thebig circle in the fourth one, contribute negative area.


031801-12

equilibrium-connecting normal deformation in terms of thecurvature and the appropriate perturbations of the param-eters.

In future work, we plan to address the important issue ofthe stability of the equilibrium configurations. In light of thecomplicated structure of the fourth-order differential operatorappearing in Eq. �29�, this appears to be a nontrivial task inthe general case. It is indicative of the intrinsic complexity ofthe question that even the relatively simple case of the figureeight configuration requires the elaborate analysis of �6�.

Another interesting issue is the analysis of thermal fluc-tuations in this model, which would provide an analyticalcounterpart to the study of Ref. �22� of two-dimensionalvesicles, using Monte Carlo techniques. In particular, there isa similarity between some of the configurations of Fig. 2 ofthe above work and the configuration of Fig. 8�a� in thispaper. It is also interesting to note that the shapes of Fig. 8closely resemble a top view of the starfish vesicles of Ref.�23�, which are almost planar. Finally, configurations similarto ours appear in �24�, where our problem has been ap-proached from a functional analytic point of view and impor-tant existence results have been derived, as well as in �25�.

After the completion of this paper, we set out to study itsmost natural extension: geometric models for loops in space.We became aware of a large body of literature that exploresthe interconnections among hierarchies of functionals of thegeometry of a curve, their corresponding generators of rigidmotion and integrable systems, such as the KdV equation.We have found Ref. �26� an excellent point of entry to thesubject. We also realized that our identities can be obtainedby adapting the cylindrical coordinates used by Langer andSinger �27� in the analysis of buckled rings. We expect thatthese explorations, apart from their intrinsic interest, willcontribute to a deeper understanding of two-dimensionalmembranes.

ACKNOWLEDGMENTS

G.A. wishes to thank Gilberto Tavares for technical assis-tance and CONACyT for financial support. R.C. was sup-ported by CONACyT Grant No. 32187-E.G.A. C.C. and J.G.were supported by CONACyT Grant No. 32307-E andDGAPA-UNAM Grant No. IN119792.

APPENDIX A: SOME USEFUL FORMULAS

Under an infinitesimal deformation of the loop along thenormal, X� →X� � �X� �X� ��(l) n , we find

�dl�K�dl , � t��n , �K��K2� ,

�

d

dl��K�

d

dl, �n�� t ,

�K��K2��3KK�� ,

�h��p��, �p��h ,

�K��K2��5KK��

�� 6E�11

4K4�

7

2�K2�5K � �

�we have used the equation of motion and the first integral,Eqs. �7� and �9�, to express K� and (K�)2 in terms of K�. Forthe derivatives of h,p,K we find

h��Kp ,

h��K2h�K�p�k ,

h��3KK�h�� 3

2K3�

�

2K�

2 � p�2K�,

h�� 6E�15

4K4�

7

2�K2�5K � h

�� 15

2K2K��

2K�� p�

5

2K3�

3

2�K�

3

2 ,

p��1�Kh ,

p��K�h�K2p ,

p�� 3

2K3�

�

2K�

2 � h�3KK�p�K2,

p�� 15

2K2K��

�

2K�� h

�� 15

4K4�

7

2�K2�5K�6E � p�5KK�,

K��&� E�1

8K4�

�

4K2�

2K � 1/2

,

K��1

2��K2��K��,

FIG. 14. Bending energy F vs area A/�R2 for n�2,3,4,5 �in-terpolation�. All curves touch at the point on the right, which cor-responds to a circular configuration, giving rise to bifurcation. Thevertical asymptotes are at �1/(n�1).


031801-13

K��3

2K2K��

�

2K�,

K��3

2K5�

10

4�K3�

15

4K2�� 2

4�6E �K�

�

4.

APPENDIX B: AVERAGES

Consider the quantities Wn , defined by

Wn��0

L0Kn dl�2�

Kmin

KmaxKn

dK

K�. �B1�

L0 here is the length along the loop corresponding to one fulloscillation of the particle �we will use F0 later with a similarmeaning�. Wn is then �proportional to� the average of Kn

along the loop. Starting from

�Kmin

Kmax d

dK�KnK��0, �B2�

one may derive the recursion relation �28�

�1

8�n�2 �Wn�3�

�

4�n�1 �Wn�1�

4�2n�1 �Wn

�EnWn�1�0, n�0,1,2, . . . , �B3�

which permits, in principle, the calculation of the average ofany power series in K, for a given equilibrium configuration.In particular,

W3��W1�W0 , W4� 43 W2�2W1� 8

3 EW0 . �B4�

Referring back to Fig. 1, we notice that the area S en-closed by the orbit of the phase point, for some given , �,E, is given by

S� ,� ,E ��2�Kmin

KmaxK� dK . �B5�

Differentiating with respect to and using the fact that theintegrand vanishes at the endpoints, we find

�0�2S

. �B6�

Length and bending energy per full oscillation also follow bysimple differentiation

L0�S

E, F0�4

S

�. �B7�

Notice that these quantities are defined whether or not theclosure condition is satisfied, i.e., they are functions of thethree independent variables , �, E. Combining �B5� and�B4� above, we get

S�4

3EL�

1

6�F�

1

2� . �B8�

One may regard L0 , �0 , and F0 as new coordinates in thespace of �not necessarily closed� configurations—the changeof coordinates is nonsingular, except for some special points,and is given by a Legendre transform. Then the closure con-dition, Eq. �11�, is restricted to the configurations that lie onthe L0-F0 plane, with �0�2�/n .

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