Variational Discretization of Euler s Elastica Problem

Transcription

1 Typeset with jpsj.cls <ver.1.> Full Paper Variational Discretization of Euler s Elastica Problem Kiyoshi Sogo Department of Physics, School of Science, Kitasato University, Kanagawa , Japan A discrete version of Euler s Elastica problem is formulated by a variational principle. Hirota s bilinear equations, Lax pair formalism and the exact solution are obtained explicitly. Geometrical properties are also discussed such as discrete Frenet-Serret equations. KEYWORDS: Euler s Elastica problem, discrete Sine-Gordon equation, discrete Frenet-Serret equations 1. Introduction The problem to find the equillibrium shape of an elastic rod with a constant force at both ends, usually known as Euler s Elastica, was firstly discussed and solved by L. Euler. Let us consider an elastic rod restricted in a plane with the length L under the constant force F. We write the coordinates of rod by xs), ys)) as functions of s the length along the rod 0 s L), and introduce the angle variable θ = θs) through the relation dx ds 0 = cos θ, dy ds = sin θ. 1) Then the total energy of this system is given by [ L B dθ E = ds + F cos θ], ) ds where B is an elastic constant. The equillibrium shape is then determined as a variational problem for the functional E. Therefore, requiring δe = 0, we obtain the differential equation d θ ds = F sin θ. 3) B Since this equation is the same as the equation of motion for a pendulum, where s has the meaning of time, the solution is well known, and is given for our case by sin θ = sin θ 0 K sn L s), k, k = sin θ ) 0 L where snu, k) is the Jacobi s elliptic sn function with modulus k, and K = Kk) is the complete elliptic integral of the first kind. Here the initial angle θ 0 is determined by F through the relation L = B F Kk), ) k = sin θ ) 0, 5) which is nothing but the formula of the periodicity for the corresponding pendulum problem. address: sogo@sci.kitasato-u.ac.jp 1/8

2 Although the problem is thus completely solved, it is further necessary to integrate 1) to find the shape of rod which needs some numerical integrations. And here arises the purpose of the present article, that is, the problem to discretize the whole theory preserving the integrability. Moreover we need a difference scheme with fixed L while changing the initial value θ 0, which opens up an interesting and non-trivial problem. In the following sections we formulate such discretized version of Elastica problem, and give the exact solution for such difference equation. The complete integrability of this problem will be established by relating it to the discrete Sine-Gordon equation, for which discrete version of Lax pair and Hirota equations will be also given. Finally, a geometrical interpretation such as the discrete Frenet-Serret equation is formulated. The last section is devoted to summary and concluding remarks.. Discretized Variational Principle for the Elastica Problem Let us start with the discrete version of variational principle following the work by Moser and Veselov. 1) We assume that the action is given by a summation such as I = [ ] sin θj+1 θ j ε sin θj+1 + θ j, 0 < ε < 1). 6) j Then the variation δi becomes δi = θj+1 θ j sin j which is reordered as δi = [ δθ j θj θ j 1 sin sin j ) δθ j+1 δθ j θj+1 θ j θj+1 + θ j ε sin ) ε sin θj + θ j 1 ) δθ ) j+1 + δθ j, 7) ) + sin θj+1 + θ j Therefore from the variational principle δi = 0, setting the coefficient of δθ j zero, we obtain after some calculations the difference equation θj+1 + θ j 1 tan = 1 ε 1 + ε tan ))]. 8) θj. 9) Numerically this three-term equation can be solved iteratively by assuming the first two values θ 0 and θ 1. Here θ 1 may be taken as ) 1 ε θ 1 = tan ε tan θ0, 10) which is derived by setting θ 1 = θ 1 which corresponds θ0) = 0. But how can we set the value of parameter ε? The answer is given as follows. As will be shown later, the exact solution of 9) turns out to be surprisingly simple. The solution has the same form as ), sin θ j = sin θ 0 K sn N j), k, N j = 0, 1,, N) 11) /8

3 where k = sinθ 0 /) and we interprete s = j, L = N. And the parameter ε is given by 1 ε K 1 + ε = cn N, k, 1) where cnu, k) is the Jacobi s elliptic cn function. It is lengthy but straightforward calculations to verify that 11) satisfies 9) if 1) holds. In the next section we will give another derivation. Finally the shape coordinates of the links) is given by whose similarity to 1) will be obvious. x j+1 x j = cos θ j, y j+1 y j = sin θ j, 13) Thus we can draw the shape of the Elastica for an arbitrary initial angle θ 0. The figure shows the case of N = 0 and θ 0 = 5π/6. Fig.1 Elastica with the initial value θ 0 = 5π/6. 3. Discrete Elastica as an Integrable System 3.1 Relation to discrete Sine-Gordon equation The complete integrability of difference equation 9) comes from the fact that it is the special uniform, or spatially independent) case of Sine-Gordon equation. Let us assume that the variable θ depends not only j but also k, and satisfies θj+1,k + θ j 1,k tan = 1 ε 1 + ε tan θj,k+1 + θ j,k 1. 1) Note that when θ s are independent of k, 1) reduces to 9). We can rewrite 1) as e i θ j+1,k+θ j 1,k ) = ε + e i θ j,k+1+θ j,k 1 ) 1 + εe i θ j,k+1+θ j,k 1 ), 15) which is the discrete Sine-Gordon equation according to Faddeev and Volkov. ) 3/8

4 3. Relation to Hirota s bilinear equations usual Now the difference equation 1) can be expressed in Hirota s bilinear form. If we set as then after some calculations, we obtain θ j,k = tan 1 gj,k, 16) f j,k f j+1,k g j 1,k + f j 1,k g j+1,k f j+1,k f j 1,k g j+1,k g j 1,k = 1 ε 1 + ε fj,k+1g j,k 1 + f j,k 1 g j,k+1 f j,k+1 f j,k 1 g j,k+1 g j,k 1. 17) This equation can be decoupled in bilinear forms as with a constraint f j+1,k g j 1,k + f j 1,k g j+1,k = δ f j,k+1 g j,k 1 + f j,k 1 g j,k+1 ), 18) f j+1,k f j 1,k g j+1,k g j 1,k = δ f j,k+1 f j,k 1 g j,k+1 g j,k 1 ), 19) δ δ = 1 ε 1 + ε. 0) Such equations are derived by Hirota 3) except that he took δ = δ, δ = 1 in his notations. However for our problem we cannot set δ = 1 as will be shown below. We can find a solution, which corresponds to a periodic one-soliton solution, to 17) as follows. Let us set ξ = jω + kp + ξ 0 and assume that f j,k = ϑ 3 ξ)ϑ ξ), g j,k = ϑ 1 ξ)ϑ ξ), where ϑ 1,, ϑ are Jacobi s elliptic theta functions. Then using the addition formulas for theta functions ), we obtain Therefore this is the solution if LHS of 17) = ϑ Ω) ϑ 1 ξ) ϑ Ω) ϑ 3 ξ), 1) RHS of 17) = 1 ε 1 + ε ϑp ) ϑ 1 ξ) ϑ P ) ϑ 3 ξ). ) 1 ε 1 + ε = ϑ Ω) ϑ P ) ϑ Ω) ϑ P ), 3) which gives a dispersion relation for the soliton. Furthermore if we examine 18) and 19) separately, we find that the decouplig constants are given by δ = ϑ Ω) ϑ P ), δ = ϑ Ω) ϑ P ). ) Here it should be noted that although we can have δ = 1 = δ if we put Ω = P, this implies also that the system becomes trivial case of ε = 0. This phenomenon originates from our space-time periodic situations, which is not seen in the infinite systems. The case of Elastica is recovered by setting P = 0 which gives that 1 ε 1 + ε = ϑ Ω)/ϑ ϑ Ω)/ϑ = cn πϑ 3Ω, k ), 5) where we used the abbreviations ϑ = ϑ 0) and ϑ = ϑ 0). /8

5 On the other hand, substituting tanθ j /) = g j /f j, we have sin θ j = tanθ j/) 1 + tan θ j /) = f jg j f j + g j = ϑ 1ξ)ϑ ξ)ϑ 3 ξ)ϑ ξ) ϑ 1 ξ)ϑ ξ) + ϑ 3 ξ)ϑ ξ) = ϑ ϑ 3 ϑ1ξ) ϑ ξ) = k snπϑ 3ξ, k), 6) where we used the addition formulas, equality k = ϑ /ϑ 3 and relations between theta functions and Jacobi s elliptic functions. By use of the relation K = πϑ 3 / and ξ = jω + ξ 0, finally we find that if we set ξ 0 = 1/, Ω = 1/N, k = sinθ 0 /) we have sin θ j = sin θ 0 K sn N j), k, N which are the previously stated results. 1 ε K 1 + ε = cn N, k, 7) 3.3 Relation to Lax pair formalism The discrete Sine-Gordon equation can be formulated in Lax pair form which expresses the complete integrability of the system. To make the paper self-contained we give them here without derivations. Using the auxiliary variables ψ j,k, ψj,k, the Lax equations are given by i ψj+1,k chp e θ j+1,k θ j,k 1 ) shp ψj,k 1 = ψj+1,k shp chp e i θ j+1,k θ j,k 1 ) ψj,k 1 8) ) ψj 1,k chω shωe i θ j 1,k+θ j,k 1 ) ψj,k 1 =. shωe i θ j 1,k+θ j,k 1 ) chω 9) ψ j 1,k ψ j,k 1 Then the compatibility condition for two paths j, k 1) j + 1, k) j, k + 1) and j, k 1) j 1, k) j, k + 1) gives the Sine-Gordon equation 1) if we assume that thω thp = ε. 3) It should be noted that parameters P and Ω are newly introduced here and are different from those in previous section. Now we have two parameters P and Ω with one constraint. Therefore here remains one degree of freedom which corresponds to the spectral parameter in the inverse scattering terminology. To make such spectral parameter explicit, it is useful to reparametrize P and Ω as follows. e P = ϑ 1v) ϑ v), eω = ϑ v) ϑ 3 v), 1 ε 1 + ε l = ϑ, 30) where the variable v is the spectral parameter. From the last relation l = ϑ /ϑ 3, another parameter τ of theta functions is determined through the relation 1 q 1 l n 1 8 = 1 + q n 1, q = e πiτ ) 31) n=1 ϑ 3 5/8

6 Note that the new modulus l and theta functions) here is different from k which appeared previously. 3. Relation to discrete geometry Our Elastica problem is restricted in a plane say xy plane). Therefore it should be natural to consider relations with the theory of discrete geometry. 5) Let us introduce a moving frame whose basis t j and n j are defined by t j = cos θ j, sin θ j ) T, n j = sin θ j, cos θ j ) T, 3) where the symbol T implies the transpose of vector. Then the discrete curve of our rod is defined generally by a sequence of points of the links such that r j+1 r j = t j, 33) where is the length of linear part of discrete curve. Then we have an equality 1 κj tj+1 1 κj tj = κ j 1 κ j 1 n j+1 under the definition of the dimensionless curvature κ j by θj+1 θ j κ j = tan. 35) This equation is actually an identity, and is a two-dimensional version of Frenet-Serret equation which should hold for arbitrary discrete curves. In the next section a generalization of this identity to full three-dimensions will be given.. Summary and Concluding Remarks Discrete version of Euler s Elastica problem is derived by a discrete variational principle which preserves the integrability property. Relations to discrete Sine-Gordon equation, Hirota s bilinear equations and Lax pair formalism are examined explicitly. We derived also the discrete version of Frenet-Serret identity which holds for general curves which are not integrable. Let us now give some remarks on the obtained results. The first one is on the rather curious circumstances of the differnce equation 9). If we consider it as an equation of motion for the discrete pendulum, according to the solution 11) and 1), the parameter ε, which controls the difference equation, looks depending on k which is determined by the initial condition θ 0. If such is the case, this contradicts the philosophy of mechanics, i.e., the separation of dynamical laws from initial conditions. We may, however, resolve such circumstances as follows. Since there are four parameters ε, k, L, which are related with each other, while changing the initial condition k, we can also change with fixing L = N constant. Because this implies that we also change N, we can adjust it so that cnk/n, k) remains constant which implies a fixed ε. This resolution has however one flaw, that is, N no longer remains an integer. Such n j 3) 6/8

7 situations are similar with those encountered in the ergode theory, which is known as Weyl s argument. We can find an arbitrarily precise rational number Q/P which approximates the real number N. Thus the periodicity or non-ergodicity) of the system can be maintained. The second remark is on the Frenet-Serret identity extended to three-dimensional discrete curves. Suppose that the link position r j of a discrete curve is given again by r j+1 r j = t j. 36) The tangential vector t j and its partners, the normal vector n j and the binormal vector b j, are parametrized conveniently by the quaternion coordinates such that t j = a j + b j c j d j, a j d j + b j c j ), a j c j + b j d j )) T, 37) n j = a j d j + b j c j ), a j b j + c j d j, a j b j + c j d j )) T, 38) b j = a j c j + b j d j ), a j b j + c j d j ), a j b j c j + d j) T, 39) where we assume a j + b j + c j + d j = 1. Then it is easy to show that the properties such as t j = n j = b j = 1 and t j n j = n j b j = b j t j = 0 are fulfilled identically. Then the discrete Frenet-Serret identity is given by 1 κ j δ j κ j 1 τ j δ j τ j 1 t j+1 n j+1 b j+1 = 1 κ j δ j κ j 1 τ j δ j τ j 1 t j n j b j, 0) which holds identically if we assume that the three geometrical quantities κ j, τ j, δ j are defined by κ j = d j+1a j a j+1 d j + b j+1 c j c j+1 b j a j+1 a j + b j+1 b j + c j+1 c j + d j+1 d j 1) τ j = b j+1a j a j+1 b j + c j+1 d j d j+1 c j a j+1 a j + b j+1 b j + c j+1 c j + d j+1 d j ) δ j = c j+1a j a j+1 c j + d j+1 b j b j+1 d j a j+1 a j + b j+1 b j + c j+1 c j + d j+1 d j. 3) If we set a j = cosθ j /), d j = sinθ j /), b j = c j = 0, the two-dimensional result of the previous section can be derived. Although in the ordinary continuous version of Frenet-Serret equations the quantity δs) is set to be zero traditionally, it will be convenient to hold it for the reason of symmetry. Such expressions as 1) 3) for geometrical quantities in terms of quaternion coordinates, which are closely related to Hirota s dependent variables, have not been fully discussed in the theory of discrete geometry. 5) Their derivations and various applications will be discussed in future publications. 7/8

8 References 1) J.Moser and A.P.Veselov: Commun. Math. Phys ) 17. ) L.D.Faddeev and A.Volkov: Lett. Math. Phys ) 15. 3) R.Hirota: J. Phys. Soc. Jpn ) 079. ) E.T.Whittaker and G.N.Watson: A Course of Modern Analysis Cambridge Univ. Press 1969) ) A.I.Bobenko and R.Seiler ed.:discrete Integrable Geometry and Physics Clarendon Press, 1999). 8/8