STABILITY AND OSCILLATIONS OF A CATENOID SOAP FILM SURFACE

Transcription

1 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, Edited by ANDREAS ASKER Karlstad University

2 Abstract The system of a soap lm conguration suspended between two coaxial rings is considered. By perturbing the surface for the kinetic and potential energy functional, one obtains a second variational term. This contains information about the stability and oscillations of the extremal surface. Approaching the problem as a Sturm-Liouville problem, eigenvalues and eigenfunctions 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.

3 Contents 1 Introduction 4 Background Theory 4 3 Minimal surface 7 4 Stability of a soap lm Variational problem Perturbed surface and second variation Sturm-Liouville problem formulation Oscillations of a soap lm Oscillations about an extremal surface Conclusions 16 Appendix A The Catenoid equation 17 Appendix B Second variation in terms of ξ's 18

4 1 Introduction There are some physical phenomena in classical mechanics that are variational problems. A standard example is to determine the equilibrium conguration of a soap lm suspended between two coaxial rings. When considering a variational problem one obtain information about the shape of the conguration for the given system. It however does not present any information about stability of the extremal surface or about oscillations. These issues have been studied in Loyal Durand's paper [1], where the problem has been treated as an eigenfunction problem within the frame of Sturm-Liouville theory. The author of the present paper will demonstrate the analytic approach of determining the stability and oscillations for the system treated in [1]. Some background theory and additional steps, that were not included in [1], will be conducted. The considered system is the soap lm suspended between two parallel coaxial rings. Background Theory In order to describe a catenoid soap lm there are some mathematical concepts that are required. The author recommends the reader to be familiar with some of these 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 from the calculus of variations. However, some theory about second ordered partial dierential equation and Sturm-Liouville problems will be treated in this section. Second-order partial dierential equations arise in many areas of physics. Some examples of partial dierential equations are the heat equation, the wave equation and Laplace equation [, Page: 755,768,786]. Many partial dierential equations can be solved by a method called seperation of variables [, Page: 756]. Denition.1. Consider a linear partial dierential equation of a function T. The function T = T x, y) is said to be seperable if it factors into a function of x only times a function of y only, T x, y) = Xx)Y y) [, Page: ]. Remark.. The separation of variables method is not excluded to second order partial dierential equations only, but it is also relevant for higher order dierential equations. Example.3. A brief example, consider the time-dependent Schrödinger equation 4

5 with a time-independent potential V i Ψr, t) = t m Ψr, t) + V r)ψr, t). 1) To show that Equation 1) indeed admits a solution of the form Ψr, t) = ψr)ft), one must test it in the following way. Substituting Ψr, t) = ψr)ft) into Equation 1) and rearrange it, one nds that i 1 dft) ft) dt = 1 ψr) [ ] m ψr) + V r)ψr). ) Since the left-hand side depends only on t, and the right-hand side only on r, both sides must be equal to a constant. Therefore, Equation 1) is separable. Let this constant be E, which is referred to as a separation constant. From Equation ) one then obtains the two equations and i dft) dt = Eft), 3) m ψr) + V r)ψr) = Eψr). 4) Equation 3) has the solution ft) = Cexp iet/ ), where C is an arbitrary constant and Equation 4) is called the time-independent Schrödinger equation [3]. The time-independent Schrödinger equation in Example.3 is a central part of quantum mechanics when describing important physical systems. Furthermore, the resulting dierential equations for these systems are solved as an eigenfunction problem. This is due to a mathematical formulation of quantum mechanics which originates from Sturm-Liouville theory. Denition.4. Consider a dierential equation of the form [ px)y x) ] + [ qx) + λrx) ] yx) = 0, 5) where λ is a parameter and the prime-notation means derivative with respect to x. Assume that the functions px), p x), qx) and rx) are continuous on the interval [a, b] and px) 0 and rx) 0 everywhere in [a, b]. Then this, together with the boundary conditions α 1 ya) + α y a) = 0, and β 1 yb) + β y b) = 0, 6) where the coecients are assumed real and at least one α i and one β i are non-zero, is called a Sturm-Liouville problem [, Page: 687]. 5

6 By general conditions on px), qx) and rx), it turns out that there are special values of λ for which non-trivial solutions are obtained. These values of λ are called eigenvalues and the corresponding solutions are called eigenfunctions [, page:688]. By introducing the dierential operator L Equation 5) becomes Lyx) = [ px)y x) ] qx)yx), 7) Lyx) = λrx)yx). 8) Equation 8) is called a generalized eigenvalue problem and rx) is called a weight [, Page: 678, 688]. Lemma.5. Let px), qx) and rx) be real functions. Then L is a Hermitian operator. Proof. Allow λ and yx) to be complex. From Equation 8) one obtains for y n x) and y m x) Ly n x) = λ n rx)y n x), and Ly mx) = λ mrx)y mx), 9) where the equation for y m x) have been complex conjugated. By multiplying the rst equation by y mx) and the second equation by y n x), one then integrates both from a to b. Subtracting these two results yields b a y mx)ly n x) dx b a y n x)lymx) dx = ) b λ n λ m rx)ymx)y n x) dx. Using the denition of L given by Equation 7) and the boundary conditions in Equation 6), one can show that the left-hand side of Equation 10) equals zero for a Sturm-Liouville problem. Equation 10) becomes b a y mx)ly n x) dx = b a a 10) y n x)ly mx) dx. 11) An operator that satises this equation is said to be a hermitian operator. Theorem.6. The eigenvalues of a Sturm-Liouville problem are real [, Page: 690]. 6

7 Proof. From Equation 11) and Equation 10) one has ) b λ m λ n rx)ymx)y n x) dx = 0. 1) a Let m = n and use the fact that rx) > 0 in the interval a, b). Since the integrand is everywhere greater than or equal to zero the integral cannot equal zero. This implies that λ n = λ n. Theorem.7. The eigenfunctions corresponding to dierent eigenvalues of a Sturm-Liouville system are orthogonal [, Page: 691]. Proof. If λ m = λ m λ n in Equation 1), then the integral must equal zero: b a rx)y mx)y n x) dx = 0. 13) When an equation such as Equation 13) is satised, the set of functions y n is said to be orthogonal [, Page: 678]. 3 Minimal surface In this section, the main concepts of a minimal surface are treated. Let the surface be dened by z = fx, y), where fx, y) is the height of the surface above the point x, y) in the plane. The surface area is given by the functional [4, Page: 4] A[f] = 1 + fx + fy dxdy, 14) where S is the region of the plane on which the area is dened. S Denition 3.1. A surface S R 3 is minimal if and only if its mean curvature vanishes identically [5, Page: 1]. After suitable rotation, any regular surface S R 3 can be locally expressed as the graph of a function f = fx, y). The variation of Equation 14) implies that the function fx, y) satises the Euler- Lagrange equation for two independent variables) L f x L f x y 7 L f y = 0, 15)

8 where L is L = 1 + f x + f y. 16) One obtains the following relation from Equation 15) 1 + f x)f yy f x f y f xy f y )f xx = 0, 17) which is a second order nonlinear elliptic partial dierential equation [5, Page: 1]. It has been shown that the condition that the mean curvature vanishes for a minimal S is satised if the graph fx, y) satises Equation 17). Denition 3.. A surface S R 3 is minimal if and only if it can be locally expressed as the graph of a solution of Equation 17) [5, Page: 1]. An example of a minimal surface that satises 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 coaxial rings 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 may be determined. 8

9 4.1 Variational problem Let S be the surface of the soap lm, σ the surface tension and neglect gravitational eects to the surface. The potential energy functional for an ideal soap lm is given by [1, Page: 1] V [S] = σ ds. 18) S The possible equilibrium shapes of the lm are determined by evaluating those surfaces where the quantity 18) has a local minimum. For the system described in Figure 1, Equation 18) becomes V [S] = 4πσ r 1 + rz dz, 19) where r z dr. It should be noted that from now on, if nothing else is explicitly dz stated, a subscript means derivative with respect to the indicated variable. The variation of Equation 19) must be zero in order for a surface S to be an extremal, that is δv [S] = 0. The integrand is identied to be the Lagrangian and must satisfy the Euler-Lagrange equation ) d L = L dz r z r. 0) The calculation of δv [S] will lead to the result ) z b rz) = a cosh, 1) a where a and b are constants. Equation 1) is called the catenoid equation. The derivation of the catenoid equation is shown in Appendix A. Restricting the attention to two coaxial rings of equal radii r0) = rh) r 0, the constant of integration b becomes b = h. Thus z rz) = a cosh a h ), ) a and the boundary value problem reduces to determining a from Equation ) for the case ) h r0) r 0 = a cosh a > 0. 3) a 9

10 The variation of Equation 19) provides information about the shape of the minimized surface. Furthermore, critical ratios between h and r 0 may be obtained. However, in order to examine stability and oscillations of the minimized surface one have to consider a dierent variation. 4. Perturbed surface and second variation If V [S] has a local minimum for the surface S, then a soap lm described by Equation ) and Equation 3) will be stable by any small perturbation [1, Page: 3]. Therefore, in order to see whether this is the case the second variation must be examined. Consider a perturbed surface described by rz) = fz) + gz), 4) where rz) = fz) describes an initial surface S 0 it is not necessary for S 0 to be a minimum) and let gz) be an innitesimal twice-dierential function that satises the boundary condition g0) = gh) = 0. Assume further that the derivative of gz), g z, is innitesimal for 0 z h. Then, V [S] can be expanded in a power series of g and g z as V [S] = 4πσ = 4πσ h 0 h 0 r 1 + rz dz = f 1 + fz + g 1 + fz + ff zg z 1 + f z + + gf ) zg z fgz + ) 1 + f z 1 + f 3/ + Og 3 ) dz = z = V [S 0 ] + 4πσ h + πσ 0 h 0 1 g + f z d ) ff z dz + dz 1 + f z fg z f zz g ) 1 + f z ) 3/ dz + Og 3 ) = V [S 0 ] + δv [S 0 ] + δ V [S 0 ] + Og 3 ), 5) where integration by parts has been performed to eliminate g z and gg z. The last term in Equation 5), δ V [S 0 ], is the second variation and is calculated in the same way as an ordinary variation. V [S 0 ] in Equation 5) satises Equation 0). Therefore δv [S 0 ] = 0 and f is compare Equation )) fz) = a cosh 10 z a h a ), 6)

11 where, again, the analysis has been limited to the case of symmetrical rings. Inserting Equation 6) into the second variation yields δ V [S 0 ] = πσa = πσ h 0 g z 1 a g) cosh z a h a ) dz g u g ) cosh u) du, 7) where the substitution u = z h a, with u 0 = h, has been implemented in the a second line. From Equation 7), it is not immediately clear whether the extremal will be a minimum or a maximum of V [S] due to the indenite sign of the integrand. However, it has been shown that a sucient condition for the extremal curve to give a minimum for V [S] is that the second derivative of the integrand in Equation 19) with respect to r z all nite) is positive for all z and r in a neighborhood of the curve [1, Page: 4]. The condition is satised for this problem and it can be further relaxed because both the innitesimals g and g z, for a smooth surface S, require the coecient of g z in δ V to be positive [1, Page: 4]. There is a second condition for the extremal to be a minimum [1, Page: 4]. However, the signicance of this condition is not obvious and it is not interpreted easily. By adopting a dierent approach, both conditions will be capsuled in the formalism very naturally. Particularly, by formulating the dynamical stability of a soap lm as a Sturm-Liouville problem the rst condition will be converted into determining the sign of the lowest eigenvalue of an appropriate operator. The second condition is interchanged into the necessity of having a positive lowest eigenvalue [1, Page: 4]. For simplicity of future discussion, the radial displacement gz) will be changed into an equivalent innitesimal displacement ξz) that is perpendicular to the initial surface of the soap lm. The innitesimal displacement satises the boundary condition ξ ) = ξu 0 ) = 0. The associated vector displacement of a point r = r, z) is given by where ˆnz) is the surface normal at r, z) = fz), z): r r = ˆnz)ξz), 8) ˆn = ˆr f z ẑ ) 1 + f z. 9) 11

12 One then obtains r z ) = fz) + ξz) f, and z = z z ξz). 30) 1 + f z 1 + f z Inserting z into r z ), expanding up to rst order in ξ and compare to Equation 4) one nds that g = ξ coshu). 31) Inserting Equation 31) into Equation 7) gives the result ) δ V [S 0 ] = πσ ξ ξ uu cosh u) ξ du. 3) For the detailed derivation of Equation 3), 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 a given value u 0 one treat this as a Sturm-Liouville problem. Let L be the Sturm- Liouville operator dened as L = d du cosh u). 33) If L is a positive operator, that is, if the integral in Equation 3) is positive for any ξ, then δ V 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 dened by d ψ n u) du + Lψ n u) = λ n wu)ψ n u), 34) with the boundary conditions ψ n u 0 ) = ψ n ) = 0. The weight function in Equation 34) must be positive within the entire interval of u. As long as this condition holds the choice of weight function is arbitrary. It turns out that a natural choice is wu) = cosh u) [1, Page:5]. Equation 34) then becomes ) λ n cosh u) + ψ n u) = 0, 35) cosh u) 1

13 where the eigenfunctions ψ n u), where n = 1,,..., are chosen to be real, assumed normalized and satisfy the orthogonality relation compare Equation 13)) ψ m u)ψ n u) cosh u) du = δ mn. 36) The eigenvalues λ n are real and discrete. Furthermore, they can be ordered in such a way that λ 1 < λ < λ 3 <... < λ n <.... This is a consequence of a theorem for regular Sturm-Liouville problems [, Page: 694]. If ξ is expanded into a complete set of eigenfunctions ψ n as ξu) = c n ψ n, 37) then, using Equation 35) and Equation 36), Equation 3) is found to be n=1 δ V [S 0 ] = πσ λ n c n. 38) From Equation 38) and the assumptions mentioned above, one can see whether a given extremal soap lm conguration is stable or unstable by examining the lowest eigenvalue λ 1. For λ 1 > 0, all the eigenvalues are positive and δ V [S 0 ] > 0 for any choice of c n. This means that for any innitesimal displacement ξu) that satises the given boundary conditions, the area of the soap lm is increasing. This is the condition for stability. If λ 1 < 0, then for c 1 0 and c n = 0 for n > 1 one obtains a ξ for which the area of the lm decreases, δ V [S 0 ] < 0. This means that the conguration is unstable. n=1 5 Oscillations of a soap lm In this section, one considers the oscillation motions of a soap lm. The motion about an extremal surface is of interest. 13

14 5.1 Oscillations about an extremal surface For an innitesimal displacement ξz, t) that is perpendicular to the surface S, the kinetic energy is given by the following functional [1, Page: 6] T [S] = 1 ρ ξt ds, 39) where ρ is the mass surface density of the soap lm. In this analysis, the subscript in Equation 39) means partial derivative with respect to time t. From now on, ρ is assumed to be a constant. Considering a symmetric lm with the radii of the coaxial rings r0) = rh) r 0, Equation 39) becomes T [S] = πρ h 0 = πρa S ξ t r 1 + r z dz = ξ t cosh u) du, 40) where r = rz) is given by Equation 6). The substitution u = z h a, with u 0 = h, has been used in the second line. a The potential energy functional is given by Equation 3). Since ξ satisfy the boundary conditions ξ ) = ξu 0 ) = 0 the integrand will be expressed in terms of ξ in second order. Therefore compare the calculation in Appendix B) V [S] = πσ ξu The Lagrangian functional is dened as cosh u) ξ ) du. 41) L[S] = T [S] V [S]. 4) The second order Lagrangian functional to the system of a displaced soap lm is given by ) 1 L[S] = π ρ a ξt cosh u) σξu σ + cosh u) ξ du. 43) The variation of Equation 43), δl[s] = 0, suggests that the expression in the integral will satisfy the Euler-Lagrange equation ) d L = L du ξ u ξ, 44) 14

15 where L is the integrand in Equation 44). After some calculation one obtains ρ a cosh u)ξ tt + σξ uu + 4σ cosh ξ = 0. 45) u) The expression in Equation 45) is a hyperbolic partial dierential equation for wavelike displacements of the surface [1, Page: 6]. To nd solutions to Equation 45) one rst implements the method of separation of variables described in section ). Inserting ξu, t) = ζu)t t) into Equation 45) one nds that the dierential equation for T t) is or T tt + σλ T = 0, 46) ρa T tt + ω T = 0, 47) where the separation constant λ has been chosen to λ = ρa ω and ω is interpreted σ as angular frequency. The solutions to Equation 47) are trigonometric functions. The dierential equation for ζu) becomes λ cosh u) + d ζu) du + ) cosh ζ = 0, 48) u) with the boundary conditions ζ ) = ζu 0 ) = 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 dened as Equation 33) and eigenfunctions ψ n. The weight function is, again, chosen to be wu) = cosh u). The angular frequencies of the normal modes may be written as ) 1/ ) σλn σ ω n = = 1/ ω ρa ρh n, n = 1,,..., 49) ) 1/ σ where the constant is the systems characteristic angular frequency and ρh ω n is a dimensionless reduced frequency which is related to λ by ω n = 4u 0λ n. Since each ξ n u, t) are solutions to Equation 45), their superposition is also a solution and is the general solution [, Page: 769]. Therefore, for a general initial displacement and velocity, ξu, 0) respectively ξ t u, 0), one obtains ξ n u, t) = a n cosω n t) + b ) n sinω n t) ψ n u), 50) ω n n=1 15

16 where the coecients a n and b n are determined by and a n = b n = ψ n u) ξu, 0) cosh u) du, 51) ψ n u) ξ t u, 0) cosh u) du. 5) From Equation 51) and Equation 5), one may obtain interesting results when considering the relation between u 0 and a critical value u c. Since u 0 contains a ratio between the distance h and ring diameter r 0, one may obtain u c for some critical value h c from Equation 3). If u 0 < u c, then ω n is positive and all frequencies ω n are real. This means that the motions described by Equation 5) are bounded small-amplitude oscillations about a stable equilibrium conguration of the soap lm. For the case when u 0 > u c, ω n is negative and all frequencies ω n are imaginary. This means that the trigonometric functions are replaced by hyperbolic functions. As a consequence, a small 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-Liouville problem one could obtain interpretable parameters and solutions. Furthermore, the derived eigenvalues and eigenfunction provides information about the stability of the surface and the nature of the motion about the given conguration, respectively. After this study of the catenoid soap lm one may ask if this mathematical formulation, for a variational problem, may be used for dierent problems that arise in physics. For example, if it could be used for more complicated systems of soap lm congurations and for non-idealized soap lms. Furthermore, whether it is possible to apply the formulation for a variational problem in a dierent area of physics. These are interesting ideas that should be further investigated. 16

17 Appendix A The Catenoid equation Consider a cylindrical coordinate system with z-axial symmetry. Then, the surface S has the following parametric representation [6, Page: 6] r = rz) cosφ), rz) sinφ), z ), 53) where φ is the angle around the z-axis. The surface area of revolution around the z-axis is given by the following surface integral A = da = r z r φ dφdz. 54) S From the parametrization given in Equation 53), using the notation r z dr dz, the length of the surface normal becomes r z r φ = rz cosφ), r z sinφ), 1 ) r sinφ), r cosφ), 0 ) = ) = r cosφ), r sinφ), rrz = s = r 1 + r z. 55) The surface integral in Equation 54) is evaluated to be A = da = π r 1 + rz dz. 56) S 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 is identied to be L = r 1 + rz and it is required that the Lagrangian satises the Euler-Lagrange equation [7, Page: 38] ) d L = L dz r z r. 57) Inserting L into Equation 57) yields ) d rrz = 1 + r dz 1 + r z, 58) z which can be further simplied into the following form ) 1 d r = 0. 59) dz 1 + r z r z 17

18 Integrating Equation 59) and arranging the result, one have r z = dr dz = r a, 60) a where a is an integration constant. The solution to the dierential equation is ) dr r z = a r a + b = a arccosh + b, 61) a or ) z b rz) = a cosh, 6) a where b is an integration constant. Equation 6) is called the catenoid equation [6, Page: 7]. Appendix B Second variation in terms of ξ's The derivation of Equation 3) starts from Equation 7), which is δ V [S 0 ] = πσ Using the acquired expression for gu) g u in Equation 63) is determined to be and g u g ) cosh u) du. 63) g = ξ coshu), 64) g u = ξ u coshu) + ξ sinhu), 65) g u g = ξ u cosh u) + ξξ u coshu) sinhu) + ξ sinh u) cosh u) ) = = ξu cosh u) + ξξ u coshu) sinhu) ξ = ) = cosh u) ξu ξ + ξξ u tanhu) cosh, 66) u) where the identity cosh u) sinh u) = 1 has been used. Inserting Equation 66) into Equation 63) yields ) δ V [S 0 ] = πσ ξu ξ + ξξ u tanhu) cosh du. 67) u) 18

19 Since ξu) satises the boundary conditions ξ ) = ξu 0 ) = 0, the terms with ξ u and ξξ u in Equation 67) may be eliminated via integration by parts and the fact that Thus δ V [S 0 ] = πσ = πσ d ) ξξu = ξ du u + ξξ uu, 68) ξu du + 4πσ ξξ u tanhu) du πσ cosh u) du = d ) u0 [ ] u0 ξξu du πσ ξξ uu du + πσξ tanhu) du ξ πσ cosh du πσ u) ] u0 = [πσξξ u + πσ ξ ξ uu u 0 u0 = πσ ξ ξ uu ξ cosh u) du = cosh u) ξ) du + ξ [ ] u0 πσξ tanhu) = cosh u) ξ) du. 69) 19

20 References [1] Loyal Durand. Stability and oscillations of a soap lm: An analytic treatment. American Journal of Physics, 494):334343, [] Donald A. McQuarrie. Mathematical methods for scientists and engineers [3] B.H. Bransden and C.J. Joachain. Quantum Mechanics. Prentice Hall, 000. [4] Frederik Brasz. Soap lms: Statics and dynamics stonelab/teaching/fredbraszfinalpaper.pdf. [5] William Meeks III and Joaquín Pérez. The classical theory of minimal surfaces. Bulletin of the American Mathematical Society, 483):35407, 011. [6] Masato Ito and Taku Sato. In situ observation of a soap-lm catenoida simple educational physics experiment. European Journal of Physics, 31):357, 010. [7] H. Goldstein. Classical Mechanics. A-W series in advanced physics. Addison- Wesley,