Generalized relaxed elastic line on an oriented surface

We study a relaxed elastic line in the general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for a generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfac...

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Дата:	2012
Автори:	Özkan, G., Yücesan, A.
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Опубліковано:	Український математичний журнал 2012
Назва видання:	Український математичний журнал
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Цитувати:	Generalized relaxed elastic line on an oriented surface / G. Özkan A. Yücesan // Український математичний журнал. — 2012. — Т. 64, № 8. — С. 1121-1131. — Бібліогр.: 5 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	nasplib_isofts_kiev_ua-123456789-1644422025-02-09T21:21:54Z Generalized relaxed elastic line on an oriented surface Узагальнена релаксована пружна лiнiя на орiєнтованiй поверхнi Özkan, G. Yücesan, A. Статті We study a relaxed elastic line in the general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for a generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfaces. Вивчається релаксована пружна лiнiя у бiльш загальному випадку на орiєнтованiй поверхнi. Зокрема, отримано диференцiальне рiвняння з трьома граничними умовами для узагальненої релаксованої пружної лiнiї. Отриманi результати проаналiзовано на площинi, сферi, цилiндрi та на геодезичних цих поверхонь. 2012 Article Generalized relaxed elastic line on an oriented surface / G. Özkan A. Yücesan // Український математичний журнал. — 2012. — Т. 64, № 8. — С. 1121-1131. — Бібліогр.: 5 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164442 517.91 en Український математичний журнал application/pdf Український математичний журнал
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Статті Статті
spellingShingle	Статті Статті Özkan, G. Yücesan, A. Generalized relaxed elastic line on an oriented surface Український математичний журнал
description	We study a relaxed elastic line in the general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for a generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfaces.
format	Article
author	Özkan, G. Yücesan, A.
author_facet	Özkan, G. Yücesan, A.
author_sort	Özkan, G.
title	Generalized relaxed elastic line on an oriented surface
title_short	Generalized relaxed elastic line on an oriented surface
title_full	Generalized relaxed elastic line on an oriented surface
title_fullStr	Generalized relaxed elastic line on an oriented surface
title_full_unstemmed	Generalized relaxed elastic line on an oriented surface
title_sort	generalized relaxed elastic line on an oriented surface
publisher	Український математичний журнал
publishDate	2012
topic_facet	Статті
url	https://nasplib.isofts.kiev.ua/handle/123456789/164442
citation_txt	Generalized relaxed elastic line on an oriented surface / G. Özkan A. Yücesan // Український математичний журнал. — 2012. — Т. 64, № 8. — С. 1121-1131. — Бібліогр.: 5 назв. — англ.
series	Український математичний журнал
work_keys_str_mv	AT ozkang generalizedrelaxedelasticlineonanorientedsurface AT yucesana generalizedrelaxedelasticlineonanorientedsurface AT ozkang uzagalʹnenarelaksovanapružnaliniânaoriêntovaniipoverhni AT yucesana uzagalʹnenarelaksovanapružnaliniânaoriêntovaniipoverhni
first_indexed	2025-11-30T23:14:44Z
last_indexed	2025-11-30T23:14:44Z
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fulltext	UDC 517.91 A. Yücesan, G. Özkan (Süleyman Demirel Univ., Isparta, Turkey) GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE УЗАГАЛЬНЕНА РЕЛАКСОВАНА ПРУЖНА ЛIНIЯ НА ОРIЄНТОВАНIЙ ПОВЕРХНI We study the relaxed elastic line in a more general case on an oriented surface. In particular, we obtain a differential equation with three boundary conditions for the generalized relaxed elastic line. Then we analyze the results in a plane, on a sphere, on a cylinder, and on the geodesics of these surfaces. Вивчається релаксована пружна лiнiя у бiльш загальному випадку на орiєнтованiй поверхнi. Зокрема, отримано диференцiальне рiвняння з трьома граничними умовами для узагальненої релаксованої пружної лiнiї. Отриманi результати проаналiзовано на площинi, сферi, цилiндрi та на геодезичних цих поверхонь. 1. Introduction. A relaxed elastic line of length ` on a connected oriented surface in three- dimensional Euclidean space E3 as defined by G. S. Manning in [2] and characterized in [3], min- imizes the total square curvature, ∫ ` 0 κ2(s)ds, in the family of all arcs of length ` having the same initial point and initial direction. In [2], he finds that whether or not the solutions are geodesic curves of the surface depends on the boundary conditions and on the surface. Because physical motivation for study of the problem on surface may be found in the nucleosome core partical of DNA molecule. In [3], H. K. Nickerson and G. S. Manning consider the relaxed elastic line model on an oriented surface and they derive an intrinsic equation with two boundary conditions for a relaxed elastic line on this surface. They give several illustrations and apply this formulation to give important results about relaxed elastic lines on various surface. They find the geodesics in a plane and on a sphere are relaxed elastic lines, but the geodesics on the other surface are not. In this paper, our purpose is to study extremal for the variational problem of minimizing the functional F(α) = `∫ 0 (κ2 + λ2τ + λ1)ds within the family of all arcs of length ` on a connected oriented surface in three-dimensional Euc- lidean space E3 having the same initial point and initial direction. The functional consist of the addition of twisting energy to bending energy. Then the functional is a generalizing of Manning’s relaxed elastic line functional. Therefore, we call as “generalized relaxed elastic line” the curve which is extremal of this functional. We obtain a differential equation with three boundary conditions for generalized relaxed elastic line on a connected oriented surface in three-dimensional Euclidean space E3. Then, we apply the results to analyze three important situations: in a plane and its geodesic, on a sphere and its geodesic and on a cylinder and its geodesics. These examples pave the way for comparing the relaxed elastic line and the generalized relaxed elastic line. 2. Intrinsic equations for a generalized relaxed elastic line on an oriented surface. Let S be a connected oriented surface in the three-dimensional Euclidean space E3 and let α : I ⊂ R→S be an arc parametrized by arc length s, 0 ≤ s ≤ `, with curvature κ(s) and torsion τ(s). The arc α is c© A. YÜCESAN, G. ÖZKAN, 2012 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 1121 1122 A. YÜCESAN, G. ÖZKAN called as a generalized relaxed elastic line if it is an extremal for variational problem of minimizing value of the functional F , F(α) = `∫ 0 (κ2 + λ2τ + λ1)ds, (2.1) which stands in the family of all arcs of length ` on S having the same initial point and initial direction as α. If λ2 = λ1 = 0 in the functional F , then the arc which is extremal of the functional is a relaxed elastic line (see [2] and [3]). Assume that the coordinate functions of S be of class C5 and that the equations of α, as functions of s, be of class C5 in these coordinates. At a point α(s) of α, let T (s) = α′(s) denote the unit tangent vector to α and let n denote the unit normal vector field of S. Then the Darboux frame T, Q, n along α on S is the orthonormal frame defined by T (s) = α′(s), Q(s) = n(s)× T (s), n(s) = n ( α(s) ) . So, the derivative equations of Darboux frame isT ′Q′ n′  =  0 κg κn −κg 0 τg −κn −τg 0 TQ n , (2.2) where κg, κn and τg are the geodesic curvature, the normal curvature and the geodesic torsion of α, respectively [1]. The square curvature κ2 and the torsion τ of α on S is given by κ2 = κ2g + κ2n (2.3) and τ = τg + κ′nκg − κ′gκn κ2g + κ2n , (2.4) respectively. Let x : D ⊂ R2 → R3, (u, v)→ x(u, v) = ( x (u, v) , y (u, v) , z (u, v) ) be a coordinate patch of S. The partial velocities of x are given by xu = ∂x ∂u , xv = ∂x ∂v . Then an arc α is expressed as α(s) = x ( u(s), v(s) ) , 0 ≤ s ≤ `, with ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE 1123 T (s) = α′(s) = du ds xu + dv ds xv and Q(s) = p(s)xu + q(s)xv for suitable scalar functions p(s) and q(s). Now we need to define a variational field for constructing a family of the curves of length `, which have the same initial point and initial direction. In order to obtain variational arcs of length `, we extend α to an arc α∗(s) defined for 0 ≤ s ≤ `, with `∗ > ` but sufficiently close to ` so that α∗ lies in the coordinate patch. Let µ(s), 0 ≤ s ≤ `∗, be a scalar function of class C3, not vanishing identically. Then it can be defined as η(s) = µ(s)p∗(s), ζ(s) = µ(s)q∗(s) and so we can write µ(s)Q(s) = η(s)xu + ζ(s)xv (2.5) along α. We also suppose that µ has only the restrictions µ(0) = 0, µ′(0) = 0. (2.6) No further restrictions may be placed on µ. By this way we define β (σ; t) = x ( u(σ), v (σ) ) + t ( η(σ), ζ(σ) ) , (2.7) for 0 ≤ σ ≤ `∗. For \|t\| < ε1 (where ε1 > 0 depends upon the choice of α∗ and µ), the point β (σ; t) lies in the coordinate patch. Because of µ has the restrictions (2.6), β(σ; t) gives an arc which is the same initial point and initial direction at fixed t. For t = 0, β (σ; t) is the same as α∗ and σ is arc length. For t 6= 0, the parameter σ has not arc length in general. For fixed t, \|t\| < ε1, let L∗(t) denote the length of the arc β (σ; t) , 0 ≤ σ ≤ `∗. Then L∗(t) = `∗∫ 0 √〈 ∂β ∂σ , ∂β ∂σ 〉 dσ (2.8) with L∗(0) = `∗ > `. (2.9) It is clear in (2.7) and (2.8) that L∗(t) is continuous (even differentiable) in t. In particular, it follows from (2.9) that L∗(t) > `+ `∗ 2 > ` for \|t\| < ε (2.10) for a suitable ε satisfying 0 < ε ≤ ε1. Because of (2.10) we can restrict β(σ; t), 0 ≤ \|t\| < ε, to an arc of length ` by restricting the parameter σ to an interval of 0 ≤ σ ≤ λ(t) ≤ `∗ by requiring λ(t)∫ 0 √〈 ∂β ∂σ , ∂β ∂σ 〉 dσ = `. Note that λ(0) = `. The function λ(t) need not be determined explicitly but we shall need. ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 1124 A. YÜCESAN, G. ÖZKAN Lemma 1 [3]. dλ dt ∣∣∣∣ t=0 = `∫ 0 µκgds. (2.11) Now, we will calculate some derivatives of β(σ; t). The partial derivative of (2.7) with respect to parameter σ is ∂β ∂σ ∣∣∣∣ t=0 = T, 0 ≤ s ≤ `. (2.12) By taking (2.2) into consideration, we calculate the partial derivative of (2.12) with respect to σ as ∂2β ∂σ2 ∣∣∣∣ t=0 = κgQ+ κnn (2.13) and ∂3β ∂σ3 ∣∣∣∣ t=0 = − ( κ2g + κ2n ) T + ( κ′n + κgτg ) n+ ( κ′g + κnτg ) Q. (2.14) Also we get ∂β ∂t ∣∣∣∣ t=0 = µQ (2.15) from (2.5). Further differentiation of (2.15) with respect to σ gives ∂2β ∂t∂σ ∣∣∣∣ t=0 = ∂2β ∂σ∂t ∣∣∣∣ t=0 = −µκgT + µτgn+ µ′Q (2.16) by using (2.2), and ∂3β ∂t∂σ2 ∣∣∣∣ t=0 = ( −2µ′κg − µκ′g − µκnτg ) T + ( 2µ′τg + µτ ′g − µκgκn ) n+ ( µ′′ − µκ2g − µτ2g ) Q. (2.17) Finally we get ∂4β ∂t∂σ3 ∣∣∣∣ t=0 = ( −3µ′′κg + 3µ′κ′g + µκ′′g + µκ′nτg + 2µκnτ ′ g+ +3µ′κnτg − µκ2nκg − µκ3g − µκgτ2g ) T− − ( 3µ′κgκn + 2µκ′gκn + µκ2gτg − 3µ′′τg − 3µ′τ ′g− −µτ ′′g + µκgκ ′ n + µτ3g + µκ2nτg ) n− − ( 3µ′κ2g + 3µ′τ2g + 3µκgκ ′ g + 3µτgτ ′ g − µ′′′ ) Q. (2.18) ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE 1125 Now let F (t) denote the generalized relaxed elastic line functional of arc β(σ; t), 0 ≤ σ ≤ λ (t) , \|t\| < ε. Since σ is not generally arc length for t 6= 0, the functional (2.1) calculates as F (t) = λ(t)∫ 0 {〈 ∂β ∂σ , ∂β ∂σ 〉−3/2〈∂2β ∂σ2 , ∂2β ∂σ2 〉 − 〈 ∂β ∂σ , ∂β ∂σ 〉−5/2〈∂2β ∂σ2 , ∂β ∂σ 〉2 } dσ+ +λ2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−2〈∂β ∂σ × ∂2β ∂σ2 , ∂3β ∂σ3 〉 〈 ∂β ∂σ , ∂β ∂σ 〉−1〈∂2β ∂σ2 , ∂2β ∂σ2 〉 − 〈 ∂β ∂σ , ∂β ∂σ 〉−2〈∂2β ∂σ2 , ∂β ∂σ 〉2dσ+ +λ1 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−1/2 dσ. A necessary condition that α be an extremal is that dF dt ∣∣∣∣ t=0 = 0 for arbitrary µ satisfying (2.6). In calculating dF dt , we give explicitly only those terms which do not vanish for t = 0. The omitted terms are those with a factor 〈 ∂2β ∂σ2 , ∂β ∂σ 〉 , which vanishes at t = 0 since 〈T, T ′〉 = 0. Thus, we get dF dt = dλ dt {〈 ∂β ∂σ , ∂β ∂σ 〉−3/2〈∂2β ∂σ2 , ∂2β ∂σ2 〉} σ=λ(t) − −3 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−5/2〈 ∂2β ∂t∂σ , ∂β ∂σ 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉 dσ+ +2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−3/2〈 ∂3β ∂t∂σ2 , ∂2β ∂σ2 〉 dσ+ +λ2 dλ dt {〈 ∂β ∂σ , ∂β ∂σ 〉−1〈∂β ∂σ × ∂2β ∂σ2 , ∂3β ∂σ3 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉−1} σ=λ(t) − −2λ2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−2〈 ∂2β ∂t∂σ , ∂β ∂σ 〉〈 ∂β ∂σ × ∂2β ∂σ2 , ∂3β ∂σ3 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉−1 dσ+ +λ2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−1〈 ∂2β ∂t∂σ × ∂2β ∂σ2 + ∂β ∂σ × ∂3β ∂t∂σ2 , ∂3β ∂σ3 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉−1 dσ+ ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 1126 A. YÜCESAN, G. ÖZKAN +λ2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−1〈∂β ∂σ × ∂2β ∂σ2 , ∂4β ∂t∂σ3 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉−1 dσ− −2λ2 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−1〈∂β ∂σ × ∂2β ∂σ2 , ∂3β ∂σ3 〉〈 ∂2β ∂σ2 , ∂2β ∂σ2 〉−2〈 ∂3β ∂t∂σ2 , ∂2β ∂σ2 〉 dσ+ +λ1 dλ dt {〈 ∂β ∂σ , ∂β ∂σ 〉−1/2} σ=λ(t) − λ1 λ(t)∫ 0 〈 ∂β ∂σ , ∂β ∂σ 〉−3/2〈 ∂2β ∂t∂σ , ∂β ∂σ 〉 dσ + . . . . To make it easy, we will use left-hand sides of the equations (2.3) and (2.4). But, while we are working on an oriented surface, we have to use the geodesic curvature, the normal curvature and the geodesic torsion. So, we will use right-hand sides of the equations (2.3) and (2.4) on the surface. Then, by using (2.11), (2.3), (2.4), (2.12), (2.16), (2.13), (2.17), (2.14) and (2.18), we obtain dF dt ∣∣∣∣ t=0 = `∫ 0 µ(κgκ 2(`) + 3κgκ 2 − 2κ3g − 2κgτ 2 g + 2κnτ ′ g − 2κgκ 2 n)ds+ +λ2 `∫ 0 µ(κgτ (`) + 2κgτ + τgκg − 3κ2gκ ′ nκ −2 − 3κ3gτgκ −2)ds+ +λ2 `∫ 0 µ ( −κ′nτ2g κ−2 − 2κgτ 3 g κ −2 − κ′gτ ′gκ−2 + 4κnτgτ ′ gκ −2) ds+ +λ2 `∫ 0 µ ( 3κnκgκ ′ gκ −2 − 3κgκ 2 nτgκ −2 + κgτ ′′ g κ −2 + 2κ3gτκ −2) ds+ +λ2 `∫ 0 µ ( 2κgτ 2 g τκ −2 − 2κnτ ′ gτκ −2 + 2κ2nκgτκ −2) ds+ 2λ1 `∫ 0 µκgds+ +4 `∫ 0 µ′κnτgds+ λ2 `∫ 0 µ′ ( −κn − 2κ′gτgκ −2 + 5τ2g κnκ −2) ds+ +λ2 `∫ 0 µ′ ( 3κgτ ′ gκ −2 − 4τgκnτκ −2) ds+ 2 `∫ 0 µ′′κgds+ ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE 1127 +λ2 `∫ 0 µ′′ ( κ′nκ −2 + 4κgτgκ −2 − 2κgτκ −2) ds− λ2 `∫ 0 µ′′′κnκ −2ds. However, with integration by parts and ( 2.6), we can write dF dt ∣∣ t=0 = `∫ 0 µ ( κgκ 2(`) + 3κgκ 2 − 2κ3g − 2κgτ 2 g − 2κnτ ′ g − 2κgκ 2 n − 4κ′nτg + 2κ′′g+ +λ2(κgτ(`) + 2κgτ + κgτg − 3κ2gκ ′ nκ −2 + κ′n − 3κ3gτgκ −2 − 6κ′nτ 2 g κ −2− −2κgτ3g κ−2 − 6κ′gτ ′ gκ −2 − 6κnτgτ ′ gκ −2 + 3κgκ ′ gκnκ −2 − 3κgκ 2 nτgκ −2+ +2κgτ ′′ g κ −2 − 12κ′gτgκ −3κ′ + 2κgτ 2 g τκ −2 + 2κnτ ′ gτκ −2 + 2κ2nκgτκ −2+ +6κ′′gτgκ −2 + 2κ3gτκ −2 + 10τ2g κnκ −3κ′ − 2κgτ ′ gκ −3κ′ + 4τgκnτ ′κ−2+ +2κ′′′n κ −2 − 8τgκnτκ −3κ′ + 4κ′nτgτκ −2 − 10κ′′nκ −3κ′ + 24κ′nκ −4κ′2− −8κ′nκ−3κ′′ − 2κgτ ′′κ−2 + 8κgτ ′κ−3κ′ − 4κ′gτ ′κ−2 − 12κgτκ −4κ′2 + 4κgτκ −3κ′′+ +8κ′gτκ −3κ′ − 2κ′′gτκ −2 − 24κnκ −5κ′3 + 18κnκ −4κ′κ′′ − 2κnκ −3κ′′′) + 2λ1κg ) ds+ +µ(`)(4κn(`)τg(`)− 2κ′g(`) + λ2(−κn(`)− −6κ′g(`)τg(`)κ−2(`) + 5τ2g (`)κn(`)κ −2(`)− κg (`) τ ′g(`)κ−2 (`)− −4τg(`)κn(`)τ (`)κ−2(`)− 2κ′′n(`)κ −2(`) + 6κ′n(`)κ −3(`)κ′ (`)+ +2κg(`)τ ′(`)κ−2(`)− 4κg(`)τ (`)κ −3(`)κ′(`) + 2κ′g(`)τ(`)κ −2(`)− −6κn(`)κ−4(`)κ′2(`) + 2κn(`)κ −3(`)κ′′(`))) + µ′(`) ( 2κg(`) + λ2 ( 2κ′n(`)κ −2 (`)+ +4κg(`)τg(`)κ −2(`)− 2κg(`)τ (`)κ −2(`)− 2κn(`)κ −3(`)κ′(`) )) − −λ2κn(`)κ−2(`)µ′′(`) + λ2κn(0)κ −2(0)µ′′(0). In order that dF dt ∣∣∣∣ t=0 = 0 for all choices of the function µ(s) to satisfying (2.6), with arbitrary values of µ(`) and µ′(`), the given arc α must satisfy three boundary conditions 4κn(`)τg(`)− 2κ′g(`) + λ2(−κn (`) + ( κ2g(`) + κ2n (`) )−1 (−6κ′g(`)τg(`)+ +5τ2g (`)κn(`) + κg(`)τ ′ g(`)− 4τ2g (`)κn(`)− 4τg (`)κn(`)κ ′ n (`)κg(`)(κ 2 g(`)+ +κ2n(`)) −1 + 4τg(`)κ 2 n(`)κ ′ g(`) ( κ2g(`) + κ2n (`) )−1 − 2κ′′n(`) + 2κg(`)(κ ′′ n (`)κg(`)− ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 1128 A. YÜCESAN, G. ÖZKAN −κn(`)κ′′g (`))(κ2g(`)− κn(`)κ′′g(`))(κ2g (`) + κ2n(`)) −1 − 4κg (`) (κ ′ n(`)κg (`)− −κ′g(`)κn(`))(2κg(`)κ′g(`) + 2κn(`)κ ′ n(`)) ( κ2g(`) + κ2n (`) )−2 + 2κ′g(`)τg(`)+ +2κ′g(`)κ ′ n (`)κg(`) ( κ2g (`) + κ2n(`) )−1 − 2 ( κ′g(`) )2 κn(`) ( κ2g(`) + κ2n (`) )−1 ))+ + ( κn(`)κ ′ n (`) + κg(`)κ ′ g (`) ) (κ2n(`) + κ2g (`)) −2(6κ′n(`)− 4κ2g(`)κ ′ n(`)(κ 2 g(`)+ +κ2n(`)) −1 + 4κg(`)κ ′ g(`)κn(`) ( κ2g(`) + κ2n (`) )−1 )− 6κn(`)(κ 2 n(`)+ +κ2g(`)) −3 (κn (`)κ′n(`) + κg (`)κ ′ g(`) )2 + (2κn(`) ( κ′n )2 (`) + 2κ2n(`)κ ′′ n(`)+ +2κn(`) ( κ′g )2 (`) + 2κn(`)κg (`)κ ′′ g(`)) ( κ2n(`) + κ2g(`) )−2− − ( 2κ2n(`)κ ′ n (`) + 2κn(`)κg(`)κ ′ g(`) ) ( κ2n(`) + κ2g(`) )−3 ) = 0, (2.19) 2κg(`) + λ2 (( κ2g (`) + κ2n(`) )−1 (2κ′n(`) + 4κg(`)τg(`)− −2κg(`)τg(`)) + ( κ2g(`) + κ2n(`) )−2 (−2κ2g(`)κ′n(`)− κ′g(`)κn(`)− 2κn(`) )) = 0, (2.20) −λ2κn(`) ( κ2g (`) + κ2n(`) )−1 µ′′(`) + λ2κn(0) ( κ2g (0) + κ2n(0) )−1 µ′′(0) = 0 (2.21) and the differential equation κg ( κ2g(`) + κ2n (`) ) + 3κ3g + 3κgκ 2 n − 2κ3g − 2κgτ 2 g − 2κnτ ′ g − 2κgκ 2 n − 4κ′nτg + 2κ′′g+ +λ2(κgτg(`) + κgκ ′ n(`)κg(`) ( κ2g(`) + κ2n(`) )−1 − κgκ′g(`)κn(`) (κ2g(`) + κ2n(`) )−1 + +2κgτg + τgκg + ( κ2g + κ2n )−1 (κgκnκ ′ g − κ2gκ′n − κ3gτg − 6κ′nτ 2 g − 10κ′gτ ′ g − κgκ2nτg+ + ( κ2g + κ2n )−1 (2κ4gκ ′ n − 2κ3gκ ′ gκn + 2κ2gτ 2 g κ ′ n − 2κgτ 2 g κ ′ gκn + 2κnτ ′ gκ ′ nκg − 2κ2nτ ′ gκ ′ g+ +2κ2nκ 2 gκ ′ n − 2κ3nκgκ ′ g + 4τgκn ( κ′′nκg − κ′′gκn ) − 4κ′g ( κ′′nκg − κ′′gκn ) − 2κ′′gκ ′ nκg+ +2κ′′gκ ′ gκn) + 4κ′′gτg + 4κ′nτ 2 g − 8τgκn ( κ′nκg − κ′gκn ) ( κgκ ′ g + κnκ ′ n ) ( κ2g + κ2n )−2 + +(4 ( κ′n )2 τgκg − 4κ′nτgκ ′ gκn − 2κg ( κ′′′n κg + κ′′nκ ′ g − κ′′′g κn − κ′′gκ′n ) ) ( κ2g + κ2n )−1 + +2κ′′′n + ( κ2g + κ2n )−2 (8κg(κ ′′ nκg − κ′′gκn)(κgκ′g + κnκ ′ n) + 4κg ( κ′nκg − κ′gκn ) (κ′′gκg+ + ( κ′g )2 + κ′′nκn + ( κ′n )2 ) + 8κ′g ( κ′nκg − κ′gκn ) ( κgκ ′ g + κnκ ′ n ) )− 8κg(κ ′ nκg− −κ′gκn)(κgκ′g + κnκ ′ n) 2 ( κ2g + κ2n )−3 ) + κ′n + ( κnκ ′ n + κgκ ′ g ( κ2n + κ2g )−2) (−12κ′gτg+ ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE 1129 +10τ2g κn − 2κgτ ′ g − 8τ2g κn + ( κ2g + κ2n )−1 (−8τgκnκ′nκg + 8τgκ 2 nκ ′ g)− 10κ′′n+ +8κgτ ′ g + 8κg ( κ′′nκg − κ′′gκn ) ( κ2g + κ2n )−1 − 16κg ( κ′nκg − κ′gκn ) (κgκ ′ g+ +κnκ ′ n) ( κ2g + κ2n )−2 + 8κ′gτg + 8κ′gκ ′ nκg ( κ2g + κ2n )−1 − 8κ′gκ ′ gκn ( κ2g + κ2n )−1 )+ +24κ′n ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−3 − 8κ′n( ( κ′n )2 + κnκ ′′ n + ( κ′g )2 + κgκ ′′ g) ( κ2g + κ2n )−2 + +8κ′n ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−3 − 12κgτg ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−3− −12κ2gκ′n ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−4 + 12κgκ ′ gκn ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−4 + +4κgτg (( κ′n )2 + κnκ ′′ n + ( κ′g )2 + κgκ ′′ g ) ( κ2g + κ2n )−2 + 4κ2gκ ′ n( ( κ′n )2 + κnκ ′′ n+ + ( κ′g )2 + κgκ ′′ g) ( κ2g + κ2n )−3 − 4κgκ ′ gκn (( κ′n )2 + κnκ ′′ n + ( κ′g )2 + κgκ ′′ g ) ( κ2g + κ2n )−3− −4κgτg ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−3 − 4κ2gκ ′ n ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−4 + +4κgκ ′ gκn ( κnκ ′ n + κgκ ′ g )2 ( κ2g + κ2n )−4 − 24κn ( κnκ ′ n + κgκ ′ g )3 ( κ2g + κ2n )−4 + +18κn (( κ′n )2 + κnκ ′′ n + ( κ′g )2 + κgκ ′′ g ) ( κnκ ′ n + κgκ ′ g ) ( κ2g + κ2n )−3 − 18κn(κnκ ′ n+ +κgκ ′ g) 3 ( κ2g + κ2n )−4 − 2κn(3κ ′ nκ ′′ n + κnκ ′′′ n + 3κ′gκ ′′ g + κgκ ′′′ g ) ( κ2g + κ2n )−2 + +6κn( ( κ′n )2 + κnκ ′′ n + ( κ′g )2 + κgκ ′′ g) ( κnκ ′ n + κgκ ′ g ) ( κ2g + κ2n )−3− −6κn ( κnκ ′ n + κgκ ′ g ) ( κ2g + κ2n )−4 ) + 2λ1κg = 0. (2.22) Then we can give the following theorem. Theorem 1. The intrinsic equations for a generalized relaxed elastic line on a connected oriented surface in three-dimensional Euclidean space E3 are given by the differential equation (2.22) together with the boundary conditions (2.19), (2.20) and (2.21) at the free end. Here κg, κn and τg are the functions giving the geodesic curvature, the normal curvature and the geodesic torsion as functions of arc length along the line. It is clear that the solutions give us the relaxed elastic line in a special case λ2 = λ1 = 0. The following expression is a natural conclusion of this theorem. Corollary 1. The critical point of the functional F(α) = `∫ 0 κ2ds is an arc which satisfies two boundary conditions ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 1130 A. YÜCESAN, G. ÖZKAN κg(`) = 0, κ′g(`) = 2κn(`)τg(`) and differential equation 2κ′′g − 4κ′nτg − 2κnτ ′ g + κ3g + κgκ 2 n − 2κgτ 2 g + κgκ 2 n(`) = 0, for arbitrary value of the function µ(`) satisfying (2.6). Then, this differential equation with two boundary conditions is the relaxed elastic line derived by Nickerson and Manning [3]. 3. Applications. 3.1. Generalized relaxed elastic line in a plane. The critical point of the functional (2.1) is a relaxed elastic line since plane curves have identically zero torsion. Then we can give the following corollary. Corollary 2. Geodesic of a plane is a generalized relaxed elastic line. 3.2. Generalized relaxed elastic line on a sphere. The geodesic torsion τg vanishes for all curves on a sphere of radius R and normal curvature κn = − 1 R . Then (2.22) reduces to 2κ′′g + κ3g + ( 2 R2 + κ2g(`) ) κg + λ2 ( κgκ ′ g(`) ( R2κ2g (`) + 1 )−1 + +(2− 3R)κgκ ′ g ( R2κ2g + 1 )−1 + 2R2κ3gκ ′ g ( R2κ2g + 1 )−2 + +2κgκ ′ g ( R2κ2g + 1 )−2 − 2R2 (1 +R)κgκ ′′′ g ( R2κ2g + 1 )−2− −6R2 (1 +R)κ′gκ ′′ g ( R2κ2g + 1 )−2 − 18R5κgκ ′ g ( (κ′g) 2 + κgκ ′′ g ) ( R2κ2g + 1 )−2 + +2R4 (4−R)κ2gκ′gκ′′g ( R2κ2g + 1 )−3 + +2R4 ( 8 + 3R+ 9R3κ2g ) κg(κ ′ g) 3 ( R2κ2g + 1 )−3 + +2R6 (9R− 8)κ3g(κ ′ g) 3 ( R2κ2g + 1 )−4 ) + 2λ1κg = 0. (3.1) The boundary conditions (2.19), (2.20) and (2.21), which reduces to −2κ′g(`) + λ2 ( 1 R + 2R2 (1−R) (κg(`)κ′′g(`) + κ′2g (`)) ( R2κ2g(`) + 1 )−2 + +8R4(R− 1)κ2g(`)(κ ′ g) 2(`) ( R2κ2g(`) + 1 )−3) = 0, (3.2) 2κg(`) + λ22R 2 (R− 1)κg(`)κ ′ g(`) ( R2κ2g(`) + 1 )−2 = 0, (3.3) and λ2Rµ ′′(`) R2κ2g(`) + 1 − λ2Rµ ′′(0) R2κ2g (0) + 1 = 0. (3.4) ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8 GENERALIZED RELAXED ELASTIC LINE ON AN ORIENTED SURFACE 1131 Then, a generalized relaxed elastic line on the sphere is given by the differential equation (3.1) with three boundary conditions (3.2), (3.3) and (3.4). We know that geodesic on a sphere is a relaxed elastic line, but the following expression gives us an important case. Corollary 3. The geodesic of the sphere is a generalized relaxed elastic line in case of λ2 = 0, but it is not a generalized relaxed elastic line in case of λ2 6= 0. 3.3. Generalized relaxed elastic line on a cylinder. Let the cylinder be parametrized by x(u, v) = ( R cos u R ,R sin u R , v ) , where R is radius of the circle. Then for an arbitrary arc α on the cylinder κg = dθ ds , κn = − 1 R cos2 θ and τg = 1 R cos θ sin θ, where θ = θ(s) is the angle between the u-coordinate curve through α(s) and the arc α. The geodesics on the cylinder are characterized by θ = constant and satisfy the generalized relaxed elastic line differential equation (2.22) only if θ = 0, θ = ±π and θ = ±π 2 . But the boundary conditions (2.19) and (2.21) (the boundary condition (2.20) is already zero) − 4 R2 cos3 θ(`) sin θ (`) + λ2 R cos 2θ(`) = 0 and λ2R cos2 θ(`) ( µ′′(`)− µ′′(0) ) = 0. Then, we clearly see the following corollary. Corollary 4. If θ = 0, θ = ±π or θ = ±π 2 , the geodesics of the cylinder are generalized relaxed elastic lines in case of λ2 = 0, but they are not a generalized relaxed elastic lines in case of λ2 6= 0. Conclusion. In this work we generalize the notion of ”relaxed elastic line” by using the definition given in [2] and [3], and define the notion ”generalized relaxed elastic line”. Then we obtain the formulation to determine a generalized relaxed elastic line on an oriented surface. We apply this formulation to give results about generalized relaxed elastic line on various surfaces. We show that the geodesic of a plane is always a generalized relaxed elastic line. The geodesic of a sphere and geodesics of a cylinder are generalized relaxed elastic line only special cases, but they are not a generalized relaxed elastic lines in case of λ2 6= 0. 1. Do Carmo M. Differential geometry of curves and surfaces. – Englewood Cliffs, New Jersey: Printice-Hall, Inc., 1976. 2. Manning G. S. Relaxed elastic line on a curved surface // Quart. Appl. Math. – 1987. – 45, № 3. – P. 515 – 527. 3. Nickerson H. K., Manning G. S. Intrinsic equations for a relaxed elastic line on an oriented surface // Geom. Dedicata. – 1988. – 27, № 2. – P. 127 – 136. 4. Singer D. A. Lectures on elastic curves and rods // AIP Conf. Proc., 1002, Amer. Inst. Phys. – Melville, NY, 2008. 5. Weinstock R. Calculus of variations with applications to physics and engineering. – McGraw-Hill Book Co. Inc,. 1952. Received 30.12.11 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 8

Generalized relaxed elastic line on an oriented surface

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