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EXPLICIT PARAMETERIZATION of EULER's ELASTICA 1. Introduction

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EXPLICIT PARAMETERIZATION of EULER's ELASTICA 1. Introduction Ninth International Conference on Geometry, Geometry, Integrability and Quantization Integrability June 8–13, 2007, Varna, Bulgaria and Ivaïlo M. Mladenov, Editor IX SOFTEX, Sofia 2008, pp 175–186 Quantization EXPLICIT PARAMETERIZATION OF EULER’S ELASTICA PETER A. DJONDJOROV, MARIANA TS. HADZHILAZOVAy, IVAÏLO M. MLADENOVy and VASSIL M. VASSILEV Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., Bl. 4, 1113 Sofia, Bulgaria yInstitute of Biophysics, Bulgarian Academy of Sciences Acad. G. Bonchev Str., Bl. 21, 1113 Sofia, Bulgaria Abstract. The consideration of some non-standard parametric Lagrangian leads to a fictitious dynamical system which turns out to be equivalent to the Euler problem for finding out all possible shapes of the lamina. Integrating the respective differential equations one arrives at novel explicit parameteri- zations of the Euler’s elastica curves. The geometry of the inflexional elastica and especially that of the figure “eight” shape is studied in some detail and the close relationship between the elastica problem and mathematical pendulum is outlined. 1. Introduction The elastic behaviour of roads and beams which attracts a continuous attention since the time of Galileo, Bernoulli and Euler has generated recently a renewed interest in plane [2, 3, 15], space [18] and space forms [1, 11]. The first elastic problem was posed by Galileo around 1638 who asked the question about the force required to break a beam set into a wall. James (or Jakob) Bernoulli raised in 1687 the question concerning the shape of the beam and had also succeeded in solving the case of the so called rectangular elastica (second case from the top in Fig. 1). Later on in 1742 Daniel Bernoulli wrote a letter to Euler in which he had suggested to him to solve the general problem of the elastica. Following closely this sugges- tion Euler cast the problem in the variational form and presented the solution in an Appendix to his book on variational calculus which appeared in 1744. Euler begins his investigation with establishing the equation of static equilibrium of the “lamina” by means of the variational techniques developed in his treatise and then rederives it from mechanical principles developed earlier by James Bernoulli. The 175 176 Peter Djondjorov, Mariana Hadzhilazova, Ivaïlo Mladenov and Vassil Vassilev Bernoulli-Euler theory which considers only bending deformations and neglects shear deformations and stretching of the center line of the beam is given in detail by Love [13]. The corresponding solutions were found by using the equivalence of the elastica with that of the mathematical pendulum (see also Section five below). Nine different classes can be distinguished and Love presents figures for the sev- enth of them (cf. Fig.1 below). Later on Birkhoff and de Boor [2] have found the fundamental equation of the Euler elastica without length constraint (i.e., the so called free elastica) 1 κ¨(s) + κ3(s) = 0: (1) 2 Here κ(s) is the curvature of the arclength parameterized smooth curve in the plane and the dots denote the derivatives with respect to its natural parameter s. In what follows we will derive the intrinsic equation of the non-free elastica by introducing a fictitious dynamical system and present its explicit solutions in terms of Jacobian elliptic functions and elliptic integrals. We hope that these dynamical considerations could be of some interest in the many other situations as well. 2. The Fictitious Dynamical System Assuming that λ is a positive real number let us consider the following system of nonlinearly coupled ordinary differential equations x¨ − λzz_ = 0 (2) z¨ + λzx_ = 0: (3) It is quite natural to consider equations (2) and (3) as equations describing the dynamics of a particle moving in XOZ plane. It is easy to check by a direct computation that equations (2) and (3) are the Euler- Lagrange equations d d F − F = 0; F − F = 0 (4) ds x_ x ds z_ z associated with an action functional whose Lagrangian F can be taken of the form 1 λ F (x; z; x;_ z;_ t) = x_ 2 +z _2 − z (zx_ − xz_) 2 3 in which λ plays the role of the Lagrange multiplier. The particle trajectories γ(s) = x (s) determined by the parametric equation x (s) = (x(s); z (s)) (5) which we will assume to be traced with unit speed, i.e., x_ 2(s) +z _2(s) = 1 (6) Explicit Parameterization of Euler’s Elastica 177 describe the plane curve we are seeking. 3. Integration A theorem in the classical differential geometry (see e.g. [17]) claims that any plane curves is determined uniquely (up to Euclidean motion in the plane) by its curvature which in our settings can be written as κ(s) =x _z¨ − z_x:¨ (7) It is easy to see also that the equations (2) and (3) imply κ(s) = −λz(s): (8) This claim can be proved in the following manner: multiplying (3) by x_ and sub- tracting from it the result of multiplication of (2) by z_ and taking into account (6) we get exactly (8) x_z¨ − z_x¨ = −λz: Actually, the integration of (2) is immediate and produces λz2 x_ = + µ (9) 2 where µ is the integration constant. Inserting the expression for x_ from (9) into (3) leads to the equation λ2z3 z¨ + + λµz = 0: (10) 2 When rewritten in terms of the curvature the above equation becomes 1 κ¨(s) + κ3(s) + σκ(s) = 0; σ = λµ (11) 2 which is known as the intrinsic equation [3] of the elastica with tension σ and in this way we have reduced our fictitious dynamical system to the elastica problem. Continuing with the integration of (10) we get λ2z4 z_2 = − − λµz2 + C (12) 4 where C is another integration constant which however is not arbitrary but fixed by (6), i.e., C = 1 − µ2: (13) It is a trivial matter to find that the right hand side of equation (12) can be factorized in the form λ2 2(1 − µ) 2(1 + µ) − z2 z2 + (14) 4 λ λ 178 Peter Djondjorov, Mariana Hadzhilazova, Ivaïlo Mladenov and Vassil Vassilev which allows the equation itself to be rewritten as Z dz λs r = · (15) 2(1−µ) 2 2 2(1+µ) 2 λ − z z + λ Examination of the above integral leads to the conclusion that µ should be strictly smaller than one and further analysis shows that we have the following obvious possibilities: A) µ 2 (−1; 1) B) µ = −1 C) µ < −1. These possibilities will be considered below case by case: a) Introducing 2(1 − µ) 2(1 + µ) a2 = ; c2 = · (16) λ λ The integration in this case can be done with the help of the Jacobian elliptic func- tion cn(u; k) in which u is the argument and k is the so called modulus of the elliptic function (for more details on the elliptic functions, their integrals and prop- erties see e.g. [7] and [10]) p p r1 − µ z(s) = a cn(− λs, k) = a cn( λs, k); k = (17) 2 b) The integration of the second case is performed via the hyperbolic functions and one easily gets p 2 sech(s λ) z(s) = p · (18) λ c) After some preparation the integration in the last case leads also to an expression in terms of Jacobian elliptic function. For that purpose we have to change the parameter µ with ν = −µ, which means that the range of the new parameter is the ν > 1 part of the real line. Besides, it turns out convenient to introduce new parameters a^ and c^ in place of a and c used in (15), i.e., 2(ν + 1) 2(ν − 1) a^2 = ; c^2 = (19) λ λ so that this time we have to evaluate Z dz λs = · (20) p(^a2 − z2)(z2 − c^2) 2 Integrating it we obtain s s s λ(ν + 1) λ(ν + 1) 2 z(s) =a ^ dn(− s; k^) =a ^ dn( s; k^); k^ = · (21) 2 2 ν + 1 Explicit Parameterization of Euler’s Elastica 179 The above results should be completed by the integration of equation (9) in the appropriate setting and in this way we find respectively the parameterizations: 2 p p A) x(s) = p E am λs, k ; k − s; z(s) = a cn( λs, k) λ p p 2 tanh( λs) 2 sech( λs) B) x(s) = p − s; z(s) = p (22) λ λ s s λ(ν + 1) λ(ν + 1) C) x(s) =aE ^ (am( s; k^); k^) − νs; z(s) =a ^ dn ( s; k^) 2 2 in which E(u; k) and F (u; k) (which will appear later on) denote the incomplete elliptic integrals of the second respectively first kind depending on the argument u and modulus k, and am (t; k) denotes the Jacobian amplitude function (cf. [7, 10]). Actually, the integral (20) can be expressed in a slightly different form and this gives another curve p 2 2 cn(u(s); k^) sn(u(s); k^) C0) x(s) =aE ^ (am(u(s); k^); k^) − − νs pλ(ν + 1) dn(u(s); k^) (23) s c^ λ(ν + 1) z(s) = ; u(s) = s dn(u(s); k^) 2 which coincides with (22 C) (as it should be) if translated along the X axis at distance ! p rν + 1 νK(k^) ± 2 E(am(K(k^); k^); k^) − λ pλ(ν + 1) where K(k^) is the complete elliptic integral of the first kind [7, 10]. 4. Some Geometrical Remarks A few comments are in order here. E.g., already Euler had detected nine species of elastic curves while in Fig.
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