STABILITY AND OSCILLATIONS OF A

CATENOID SOAP FILM SURFACE

Interpretable quantities and equations of motion

derived from Sturm-Liouville theory

for a minimized surface

Sweden, 2015-2016

Edited by

ANDREAS ASKER

Karlstad University

Abstract

The system of a soap �lm con�guration suspended between two coaxial

rings is considered. By perturbing the surface for the kinetic and poten-

tial energy functional, one obtains a second variational term. This contains

information about the stability and oscillations of the extremal surface. Ap-

proaching the problem as a Sturm-Liouville problem, eigenvalues and ei-

genfunctions are obtained. These in turn provide interpretable result for

the stability and oscillation of the extremal surface. One should therefore

investigate further whether this mathematical formulation for a variational

problem is applicable for more complicated problems in physics.

Contents

1 Introduction 4

2 Background Theory 4

3 Minimal surface 7

4 Stability of a soap �lm 8

4.1 Variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Perturbed surface and second variation . . . . . . . . . . . . . . . . 104.3 Sturm-Liouville problem formulation . . . . . . . . . . . . . . . . . 12

5 Oscillations of a soap �lm 13

5.1 Oscillations about an extremal surface . . . . . . . . . . . . . . . . 14

6 Conclusions 16

Appendix A The Catenoid equation 17

Appendix B Second variation in terms of ξ's 18

1 Introduction

There are some physical phenomena in classical mechanics that are variationalproblems. A standard example is to determine the equilibrium con�guration of asoap �lm suspended between two coaxial rings. When considering a variationalproblem one obtain information about the shape of the con�guration for the givensystem. It however does not present any information about stability of the extremalsurface or about oscillations. These issues have been studied in Loyal Durand'spaper [1], where the problem has been treated as an eigenfunction problem withinthe frame of Sturm-Liouville theory. The author of the present paper will demon-strate the analytic approach of determining the stability and oscillations for thesystem treated in [1]. Some background theory and additional steps, that werenot included in [1], will be conducted. The considered system is the soap �lmsuspended between two parallel coaxial rings.

2 Background Theory

In order to describe a catenoid soap �lm there are some mathematical conceptsthat are required. The author recommends the reader to be familiar with some ofthese concepts in order to be able to fully appreciate the content of this report.For example, the theory of variational principles and Lagrange's equations fromthe calculus of variations. However, some theory about second ordered partialdi�erential equation and Sturm-Liouville problems will be treated in this section.

Second-order partial di�erential equations arise in many areas of physics. Someexamples of partial di�erential equations are the heat equation, the wave equationand Laplace equation [2, Page: 755,768,786]. Many partial di�erential equationscan be solved by a method called seperation of variables [2, Page: 756].

De�nition 2.1. Consider a linear partial di�erential equation of a function T .The function T = T (x, y) is said to be seperable if it factors into a function of xonly times a function of y only, T (x, y) = X(x)Y (y) [2, Page: 756-757].

Remark 2.2. The separation of variables method is not excluded to second orderpartial di�erential equations only, but it is also relevant for higher order di�erentialequations.

Example 2.3. A brief example, consider the time-dependent Schrödinger equation

4

with a time-independent potential V

i~∂

∂tΨ(r, t) = − ~2

2m∇2Ψ(r, t) + V (r)Ψ(r, t). (1)

To show that Equation (1) indeed admits a solution of the form Ψ(r, t) = ψ(r)f(t),one must test it in the following way. Substituting Ψ(r, t) = ψ(r)f(t) into Equation(1) and rearrange it, one �nds that

i~1

f(t)

df(t)

dt=

1

ψ(r)

[− ~2

2m∇2ψ(r) + V (r)ψ(r)

]. (2)

Since the left-hand side depends only on t, and the right-hand side only on r, bothsides must be equal to a constant. Therefore, Equation (1) is separable. Let thisconstant be E, which is referred to as a separation constant. From Equation (2)one then obtains the two equations

i~df(t)

dt= Ef(t), (3)

and

− ~2

2m∇2ψ(r) + V (r)ψ(r) = Eψ(r). (4)

Equation (3) has the solution f(t) = Cexp(−iEt/~), where C is an arbitraryconstant and Equation (4) is called the time-independent Schrödinger equation [3].

The time-independent Schrödinger equation in Example 2.3 is a central part ofquantum mechanics when describing important physical systems. Furthermore,the resulting di�erential equations for these systems are solved as an eigenfunctionproblem. This is due to a mathematical formulation of quantum mechanics whichoriginates from Sturm-Liouville theory.

De�nition 2.4. Consider a di�erential equation of the form[p(x)y

′(x)]′

+[q(x) + λr(x)

]y(x) = 0, (5)

where λ is a parameter and the prime-notation means derivative with respect tox. Assume that the functions p(x), p

′(x), q(x) and r(x) are continuous on the

interval [a, b] and p(x) ≥ 0 and r(x) ≥ 0 everywhere in [a, b]. Then this, togetherwith the boundary conditions

α1y(a) + α2y′(a) = 0, and β1y(b) + β2y

′(b) = 0, (6)

where the coe�cients are assumed real and at least one αi and one βi are non-zero,is called a Sturm-Liouville problem [2, Page: 687].

5

By general conditions on p(x), q(x) and r(x), it turns out that there are specialvalues of λ for which non-trivial solutions are obtained. These values of λ arecalled eigenvalues and the corresponding solutions are called eigenfunctions [2,page:688]. By introducing the di�erential operator L

Ly(x) = −[p(x)y

′(x)]′− q(x)y(x), (7)

Equation (5) becomes

Ly(x) = λr(x)y(x). (8)

Equation (8) is called a generalized eigenvalue problem and r(x) is called a weight[2, Page: 678, 688].

Lemma 2.5. Let p(x), q(x) and r(x) be real functions. Then L is a Hermitianoperator.

Proof. Allow λ and y(x) to be complex. From Equation (8) one obtains for yn(x)and ym(x)

Lyn(x) = λnr(x)yn(x), and Ly∗m(x) = λ∗mr(x)y∗m(x), (9)

where the equation for ym(x) have been complex conjugated. By multiplying the�rst equation by y∗m(x) and the second equation by yn(x), one then integrates bothfrom a to b. Subtracting these two results yields∫ b

a

y∗m(x)Lyn(x) dx −∫ b

a

yn(x)Ly∗m(x) dx =(λn − λ∗m

) ∫ b

a

r(x)y∗m(x)yn(x) dx.

(10)

Using the de�nition of L given by Equation (7) and the boundary conditions inEquation (6), one can show that the left-hand side of Equation (10) equals zerofor a Sturm-Liouville problem. Equation (10) becomes∫ b

a

y∗m(x)Lyn(x) dx =

∫ b

a

yn(x)Ly∗m(x) dx. (11)

An operator that satis�es this equation is said to be a hermitian operator.

Theorem 2.6. The eigenvalues of a Sturm-Liouville problem are real [2, Page:690].

6

Proof. From Equation (11) and Equation (10) one has

(λ∗m − λn

) ∫ b

a

r(x)y∗m(x)yn(x) dx = 0. (12)

Let m = n and use the fact that r(x) > 0 in the interval (a, b). Since the integrandis everywhere greater than or equal to zero the integral cannot equal zero. Thisimplies that λ∗n = λn.

Theorem 2.7. The eigenfunctions corresponding to di�erent eigenvalues of aSturm-Liouville system are orthogonal [2, Page: 691].

Proof. If λ∗m = λm 6= λn in Equation (12), then the integral must equal zero:∫ b

a

r(x)y∗m(x)yn(x) dx = 0. (13)

When an equation such as Equation (13) is satis�ed, the set of functions yn is saidto be orthogonal [2, Page: 678].

3 Minimal surface

In this section, the main concepts of a minimal surface are treated.

Let the surface be de�ned by z = f(x, y), where f(x, y) is the height of the surfaceabove the point (x, y) in the plane. The surface area is given by the functional [4,Page: 4]

A[f ] =

∫∫S

√1 + f 2

x + f 2y dxdy, (14)

where S is the region of the plane on which the area is de�ned.

De�nition 3.1. A surface S ⊂ R3 is minimal if and only if its mean curvaturevanishes identically [5, Page: 12].

After suitable rotation, any regular surface S ⊂ R3 can be locally expressed as thegraph of a function f = f(x, y).

The variation of Equation (14) implies that the function f(x, y) satis�es the Euler-Lagrange equation (for two independent variables)

∂L∂f− ∂

∂x

∂L∂fx− ∂

∂y

∂L∂fy

= 0, (15)

7

where L is

L =√

1 + f 2x + f 2

y . (16)

One obtains the following relation from Equation (15)

(1 + f 2x)fyy − 2fxfyfxy + (1 + f 2

y )fxx = 0, (17)

which is a second order nonlinear elliptic partial di�erential equation [5, Page:12]. It has been shown that the condition that the mean curvature vanishes for aminimal S is satis�ed if the graph f(x, y) satis�es Equation (17).

De�nition 3.2. A surface S ⊂ R3 is minimal if and only if it can be locallyexpressed as the graph of a solution of Equation (17) [5, Page: 12].

An example of a minimal surface that satis�es Equation (17) is the catenoid [4,Page: 6].

4 Stability of a soap �lm

In this section, the stability for a soap �lm suspended between two parallel coaxialrings is examined (see Figure 1).

Figure 1: Soap �lm. r, φ and z are the cylindrical coordinates.

It will be shown that the stability may be treated as Sturm-Liouville problem.From this method the range of parameters for which the solutions are stable maybe determined.

8

4.1 Variational problem

Let S be the surface of the soap �lm, σ the surface tension and neglect gravitationale�ects to the surface. The potential energy functional for an ideal soap �lm is givenby [1, Page: 1]

V [S] = 2σ

∫∫S

dS. (18)

The possible equilibrium shapes of the �lm are determined by evaluating thosesurfaces where the quantity (18) has a local minimum. For the system describedin Figure 1, Equation (18) becomes

V [S] = 4πσ

∫r√

1 + r2z dz, (19)

where rz ≡dr

dz. It should be noted that from now on, if nothing else is explicitly

stated, a subscript means derivative with respect to the indicated variable. Thevariation of Equation (19) must be zero in order for a surface S to be an extremal,that is δV [S] = 0. The integrand is identi�ed to be the Lagrangian and mustsatisfy the Euler-Lagrange equation

d

dz

(∂L

∂rz

)=∂L

∂r. (20)

The calculation of δV [S] will lead to the result

r(z) = a cosh

(z − ba

), (21)

where a and b are constants. Equation (21) is called the catenoid equation. Thederivation of the catenoid equation is shown in Appendix A.

Restricting the attention to two coaxial rings of equal radii r(0) = r(h) ≡ r0, the

constant of integration b becomes b =h

2. Thus

r(z) = a cosh

(z

a− h

2a

), (22)

and the boundary value problem reduces to determining a from Equation (22) forthe case

r(0) ≡ r0 = a cosh

(h

2a

)a > 0. (23)

9

The variation of Equation (19) provides information about the shape of the min-imized surface. Furthermore, critical ratios between h and r0 may be obtained.However, in order to examine stability and oscillations of the minimized surfaceone have to consider a di�erent variation.

4.2 Perturbed surface and second variation

If V [S] has a local minimum for the surface S, then a soap �lm described byEquation (22) and Equation (23) will be stable by any small perturbation [1,Page: 3]. Therefore, in order to see whether this is the case the second variationmust be examined.

Consider a perturbed surface described by

r(z) = f(z) + g(z), (24)

where r(z) = f(z) describes an initial surface S0 (it is not necessary for S0 to be aminimum) and let g(z) be an in�nitesimal twice-di�erential function that satis�esthe boundary condition g(0) = g(h) = 0. Assume further that the derivative ofg(z), gz, is in�nitesimal for 0 ≤ z ≤ h. Then, V [S] can be expanded in a powerseries of g and gz as

V [S] = 4πσ

∫ h

0

r√

1 + r2z dz =

= 4πσ

∫ h

0

(f√

1 + f 2z + g

√1 + f 2

z +ffzgz√1 + f 2

z

+

+gfzgz√1 + f 2

z

+fg2z(

1 + f 2z

)3/2 +O(g3)

)dz =

= V [S0] + 4πσ

∫ h

0

g

(√1 + f 2

z −d

dz

ffz√1 + f 2

z

)dz+

+ 2πσ

∫ h

0

(fg2z − fzzg2

)(1 + f 2

z

)3/2 dz +O(g3)

= V [S0] + δV [S0] + δ2V [S0] +O(g3), (25)

where integration by parts has been performed to eliminate gz and ggz. The lastterm in Equation (25), δ2V [S0], is the second variation and is calculated in thesame way as an ordinary variation. V [S0] in Equation (25) satis�es Equation (20).Therefore δV [S0] = 0 and f is (compare Equation (22))

f(z) = a cosh

(z

a− h

2a

), (26)

10

where, again, the analysis has been limited to the case of symmetrical rings. In-serting Equation (26) into the second variation yields

δ2V [S0] = 2πσa

∫ h

0

(g2z −

1

a2g2)

cosh−2(z

a− h

2a

)dz

= 2πσ

∫ u0

−u0

(g2u − g2

)cosh−2(u) du, (27)

where the substitution u =2z − h

2a, with u0 =

h

2a, has been implemented in the

second line.

From Equation (27), it is not immediately clear whether the extremal will bea minimum or a maximum of V [S] due to the inde�nite sign of the integrand.However, it has been shown that a su�cient condition for the extremal curve togive a minimum for V [S] is that the second derivative of the integrand in Equation(19) with respect to rz (all �nite) is positive for all z and r in a neighborhood ofthe curve [1, Page: 4]. The condition is satis�ed for this problem and it can befurther relaxed because both the in�nitesimals g and gz, for a smooth surface S,require the coe�cient of g2z in δ2V to be positive [1, Page: 4].

There is a second condition for the extremal to be a minimum [1, Page: 4]. How-ever, the signi�cance of this condition is not obvious and it is not interpretedeasily.

By adopting a di�erent approach, both conditions will be capsuled in the formalismvery naturally. Particularly, by formulating the dynamical stability of a soap �lmas a Sturm-Liouville problem the �rst condition will be converted into determiningthe sign of the lowest eigenvalue of an appropriate operator. The second conditionis interchanged into the necessity of having a positive lowest eigenvalue [1, Page:4].

For simplicity of future discussion, the radial displacement g(z) will be changed intoan equivalent in�nitesimal displacement ξ(z) that is perpendicular to the initialsurface of the soap �lm. The in�nitesimal displacement satis�es the boundarycondition ξ(−u0) = ξ(u0) = 0. The associated vector displacement of a pointr = (r, z) is given by

r′ − r = n(z)ξ(z), (28)

where n(z) is the surface normal at (r, z) = (f(z), z):

n =(r − fz z

)√1 + f 2

z . (29)

11

One then obtains

r′(z

′) = f(z) +

ξ(z)√1 + f 2

z

, and z′= z − fz√

1 + f 2z

ξ(z). (30)

Inserting z′into r

′(z

′), expanding up to �rst order in ξ and compare to Equation

(24) one �nds that

g = ξ cosh(u). (31)

Inserting Equation (31) into Equation (27) gives the result

δ2V [S0] = 2πσ

∫ u0

−u0

ξ

(− ξuu −

2

cosh2(u)ξ

)du. (32)

For the detailed derivation of Equation (32), refer to Appendix B.

4.3 Sturm-Liouville problem formulation

As mentioned in the last section, to determine the stability of the soap �lm for agiven value u0 one treat this as a Sturm-Liouville problem. Let L be the Sturm-Liouville operator de�ned as

L = − d2

du2− 2

cosh2(u). (33)

If L is a positive operator, that is, if the integral in Equation (32) is positive forany ξ, then δ2V is positive for any variation and the soap �lm is stable [1, Page:5]. The operator will only be positive if the lowest eigenvalue is positive.

The Sturm-Liouville problem is de�ned by

Lψn(u) = λnw(u)ψn(u), (34)

with the boundary conditions ψn(u0) = ψn(−u0) = 0. The weight function inEquation (34) must be positive within the entire interval of u. As long as thiscondition holds the choice of weight function is arbitrary. It turns out that anatural choice is w(u) = cosh2(u) [1, Page:5]. Equation (34) then becomes

d2ψn(u)

du2+

(λn cosh2(u) +

2

cosh2(u)

)ψn(u) = 0, (35)

12

where the eigenfunctions ψn(u), where n = 1, 2, . . ., are chosen to be real, assumednormalized and satisfy the orthogonality relation (compare Equation (13))∫ u0

−u0

ψm(u)ψn(u) cosh2(u) du = δmn. (36)

The eigenvalues λn are real and discrete. Furthermore, they can be ordered in sucha way that λ1 < λ2 < λ3 < . . . < λn < . . .. This is a consequence of a theorem forregular Sturm-Liouville problems [2, Page: 694].

If ξ is expanded into a complete set of eigenfunctions ψn as

ξ(u) =∞∑n=1

cnψn, (37)

then, using Equation (35) and Equation (36), Equation (32) is found to be

δ2V [S0] = 2πσ∞∑n=1

λnc2n. (38)

From Equation (38) and the assumptions mentioned above, one can see whethera given extremal soap �lm con�guration is stable or unstable by examining thelowest eigenvalue λ1.

For λ1 > 0, all the eigenvalues are positive and δ2V [S0] > 0 for any choice of cn.This means that for any in�nitesimal displacement ξ(u) that satis�es the givenboundary conditions, the area of the soap �lm is increasing. This is the conditionfor stability.

If λ1 < 0, then for c1 6= 0 and cn = 0 for n > 1 one obtains a ξ for which the areaof the �lm decreases, δ2V [S0] < 0. This means that the con�guration is unstable.

5 Oscillations of a soap �lm

In this section, one considers the oscillation motions of a soap �lm. The motionabout an extremal surface is of interest.

13

5.1 Oscillations about an extremal surface

For an in�nitesimal displacement ξ(z, t) that is perpendicular to the surface S, thekinetic energy is given by the following functional [1, Page: 6]

T [S] =1

2

∫∫S

ρ ξ2t dS, (39)

where ρ is the mass surface density of the soap �lm. In this analysis, the subscriptin Equation (39) means partial derivative with respect to time t. From now on, ρis assumed to be a constant. Considering a symmetric �lm with the radii of thecoaxial rings r(0) = r(h) ≡ r0, Equation (39) becomes

T [S] = πρ

∫ h

0

ξ2t r√

1 + r2z dz =

= πρa2∫ u0

−u0

ξ2t cosh2(u) du, (40)

where r = r(z) is given by Equation (26). The substitution u =2z − h

2a, with

u0 =h

2a, has been used in the second line.

The potential energy functional is given by Equation (32). Since ξ satisfy theboundary conditions ξ(−u0) = ξ(u0) = 0 the integrand will be expressed in termsof ξ in second order. Therefore (compare the calculation in Appendix B)

V [S] = 2πσ

∫ u0

−u0

(ξ2u −

2

cosh2(u)ξ2)

du. (41)

The Lagrangian functional is de�ned as

L[S] = T [S]− V [S]. (42)

The second order Lagrangian functional to the system of a displaced soap �lm isgiven by

L[S] = 2π

∫ u0

−u0

(1

2ρ a2ξ2t cosh2(u)− σξ2u +

2σ

cosh2(u)ξ2)

du. (43)

The variation of Equation (43), δL[S] = 0, suggests that the expression in theintegral will satisfy the Euler-Lagrange equation

d

du

(∂L

′

∂ξu

)=∂L

′

∂ξ, (44)

14

where L′is the integrand in Equation (44). After some calculation one obtains

− ρ a2 cosh2(u)ξtt + 2σξuu +4σ

cosh2(u)ξ = 0. (45)

The expression in Equation (45) is a hyperbolic partial di�erential equation forwavelike displacements of the surface [1, Page: 6].

To �nd solutions to Equation (45) one �rst implements the method of separationof variables (described in section 2). Inserting ξ(u, t) = ζ(u)T (t) into Equation(45) one �nds that the di�erential equation for T (t) is

Ttt +2σλ

ρa2T = 0, (46)

or

Ttt + ω2 T = 0, (47)

where the separation constant λ has been chosen to λ =ρa2ω2

2σand ω is interpreted

as angular frequency. The solutions to Equation (47) are trigonometric functions.

The di�erential equation for ζ(u) becomes

d2ζ(u)

du2+

(λ cosh2(u) +

2

cosh2(u)

)ζ = 0, (48)

with the boundary conditions ζ(−u0) = ζ(u0) = 0. By comparing Equation (48)and Equation (35), the normal modes of the system will be treated as a Sturm-Liouville problem with the operator L de�ned as Equation (33) and eigenfunctionsψn. The weight function is, again, chosen to be w(u) = cosh2(u).

The angular frequencies of the normal modes may be written as

ωn =

(2σλnρa2

)1/2

=

(2σ

ρh2

)1/2

ωn, n = 1, 2, . . . , (49)

where the constant

(2σ

ρh2

)1/2

is the systems characteristic angular frequency and

ωn is a dimensionless reduced frequency which is related to λ by ω2n = 4u20λn.

Since each ξn(u, t) are solutions to Equation (45), their superposition is also asolution and is the general solution [2, Page: 769]. Therefore, for a general initialdisplacement and velocity, ξ(u, 0) respectively ξt(u, 0), one obtains

ξn(u, t) =∞∑n=1

(an cos(ωnt) +

bnωn

sin(ωnt)

)ψn(u), (50)

15

where the coe�cients an and bn are determined by

an =

∫ u0

−u0

ψn(u) ξ(u, 0) cosh2(u) du, (51)

and

bn =

∫ u0

−u0

ψn(u) ξt(u, 0) cosh2(u) du. (52)

From Equation (51) and Equation (52), one may obtain interesting results whenconsidering the relation between u0 and a critical value uc. Since u0 contains aratio between the distance h and ring diameter r0, one may obtain uc for somecritical value hc from Equation (23).

If u0 < uc, then ω2n is positive and all frequencies ωn are real. This means that

the motions described by Equation (52) are bounded small-amplitude oscillationsabout a stable equilibrium con�guration of the soap �lm. For the case whenu0 > uc, ω

2n is negative and all frequencies ωn are imaginary. This means that the

trigonometric functions are replaced by hyperbolic functions. As a consequence, asmall perturbation in the n = 1 mode would grow exponentially over time. Thus,the initial surface is unstable.

6 Conclusions

By formulating the second variation of a perturbed surface as a Sturm-Liouvilleproblem one could obtain interpretable parameters and solutions. Furthermore,the derived eigenvalues and eigenfunction provides information about the stabil-ity of the surface and the nature of the motion about the given con�guration,respectively.

After this study of the catenoid soap �lm one may ask if this mathematical for-mulation, for a variational problem, may be used for di�erent problems that arisein physics. For example, if it could be used for more complicated systems of soap�lm con�gurations and for non-idealized soap �lms. Furthermore, whether it ispossible to apply the formulation for a variational problem in a di�erent area ofphysics. These are interesting ideas that should be further investigated.

16

Appendix A The Catenoid equation

Consider a cylindrical coordinate system with z-axial symmetry. Then, the surfaceS has the following parametric representation [6, Page: 6]

r =(r(z) cos(φ), r(z) sin(φ), z

), (53)

where φ is the angle around the z-axis. The surface area of revolution around thez-axis is given by the following surface integral

A =

∫∫S

dA =

∫∫s

∣∣∣∣∂r∂z × ∂r

∂φ

∣∣∣∣ dφdz. (54)

From the parametrization given in Equation (53), using the notation rz ≡dr

dz, the

length of the surface normal becomes∣∣∣∣∂r∂z × ∂r

∂φ

∣∣∣∣ =

∣∣∣∣(rz cos(φ), rz sin(φ), 1)×(− r sin(φ), r cos(φ), 0

)∣∣∣∣ =

=

∣∣∣∣(− r cos(φ),−r sin(φ), rrz)∣∣∣∣ =

= r√

1 + r2z . (55)

The surface integral in Equation (54) is evaluated to be

A =

∫∫S

dA = 2π

∫r√

1 + r2z dz. (56)

To �nd the minimal surface of revolutions the variation of the functional A[r]equals zero, δA[r] = 0 [7, Page: 38]. From Equation (56) the Lagrangian isidenti�ed to be L = r

√1 + r2z and it is required that the Lagrangian satis�es the

Euler-Lagrange equation [7, Page: 38]

d

dz

(∂L

∂rz

)=∂L

∂r. (57)

Inserting L into Equation (57) yields

d

dz

(rrz√1 + r2z

)=√

1 + r2z , (58)

which can be further simpli�ed into the following form

1

rz

d

dz

(r√

1 + r2z

)= 0. (59)

17

Integrating Equation (59) and arranging the result, one have

rz =dr

dz=

√r2 − a2a

, (60)

where a is an integration constant. The solution to the di�erential equation is

z = a

∫dr√r2 − a2

+ b = a arccosh

(r

a

)+ b, (61)

or

r(z) = a cosh

(z − ba

), (62)

where b is an integration constant. Equation (62) is called the catenoid equation[6, Page: 7].

Appendix B Second variation in terms of ξ's

The derivation of Equation (32) starts from Equation (27), which is

δ2V [S0] = 2πσ

∫ u0

−u0

(g2u − g2

)cosh−2(u) du. (63)

Using the acquired expression for g(u)

g = ξ cosh(u), (64)

gu in Equation (63) is determined to be

gu = ξu cosh(u) + ξ sinh(u), (65)

and

g2u − g2 = ξ2u cosh2(u) + 2ξξu cosh(u) sinh(u) + ξ2(

sinh2(u)− cosh2(u))

=

= ξ2u cosh2(u) + 2ξξu cosh(u) sinh(u)− ξ2 =

= cosh2(u)

(ξ2u + 2ξξu tanh(u)− ξ2

cosh2(u)

), (66)

where the identity cosh2(u)− sinh2(u) = 1 has been used. Inserting Equation (66)into Equation (63) yields

δ2V [S0] = 2πσ

∫ u0

−u0

(ξ2u + 2ξξu tanh(u)− ξ2

cosh2(u)

)du. (67)

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Since ξ(u) satis�es the boundary conditions ξ(−u0) = ξ(u0) = 0, the terms withξ2u and ξξu in Equation (67) may be eliminated via integration by parts and thefact that

d

du

(ξξu)

= ξ2u + ξξuu, (68)

Thus

δ2V [S0] = 2πσ

∫ u0

−u0

ξ2u du+ 4πσ

∫ u0

−u0

ξξu tanh(u) du− 2πσ

∫ u0

−u0

ξ2

cosh2(u)du =

= 2πσ

∫ u0

−u0

d

du

(ξξu)

du− 2πσ

∫ u0

−u0

ξξuu du+

[2πσξ2 tanh(u)

]u0

−u0

−

− 2πσ

∫ u0

−u0

ξ2

cosh2(u)du− 2πσ

∫ u0

−u0

ξ2

cosh2(u)du =

=

[2πσξξu

]u0

−u0

+ 2πσ

∫ u0

−u0

ξ(− ξuu −

2

cosh2(u)ξ)

du+

[2πσξ2 tanh(u)

]u0

−u0

=

= 2πσ

∫ u0

−u0

ξ(− ξuu −

2

cosh2(u)ξ)

du. (69)

19

References

[1] Loyal Durand. Stability and oscillations of a soap �lm: An analytic treatment.American Journal of Physics, 49(4):334�343, 1981.

[2] Donald A. McQuarrie. Mathematical methods for scientists and engineers.2004.

[3] B.H. Bransden and C.J. Joachain. Quantum Mechanics. Prentice Hall, 2000.

[4] Frederik Brasz. Soap �lms: Statics and dynamics. 2010.https://www.princeton.edu/�stonelab/Teaching/FredBraszFinalPaper.pdf.

[5] William Meeks III and Joaquín Pérez. The classical theory of minimal surfaces.Bulletin of the American Mathematical Society, 48(3):325�407, 2011.

[6] Masato Ito and Taku Sato. In situ observation of a soap-�lm catenoid�a simpleeducational physics experiment. European Journal of Physics, 31(2):357, 2010.

[7] H. Goldstein. Classical Mechanics. A-W series in advanced physics. Addison-Wesley, 1965.

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