We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space E₁³ and give some applications.Виведено природні рівняння для узагальненої пружної лінії, деформованої на ненульовій поверхні зовнішнім полем у тривимірному напівевклідовому просторі E₁³, та наведено деякі застосування

Görgülü, A.

Gürbüz, N.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 515.16N. Gu¨rbu¨z, A. Go¨rgu¨lu¨ (Osmangazi Univ., Turkey)GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACEBY AN EXTERNAL FIELD IN THE 3-DIMENSIONALSEMI-EUCLIDEAN SPACE E31УЗАГАЛЬНЕНА ПРУЖНА ЛIНIЯ, ДЕФОРМОВАНА НА НЕНУЛЬОВIЙПОВЕРХНI ЗОВНIШНIМ ПОЛЕМ У ТРИВИМIРНОМУНАПIВЕВКIДОВОМУ ПРОСТОРI E31We deduce intrinsic equations for a generalized elastic line deformed on the non-null surface by an external field insemi-Euclidean space E31 and give some applications.Виведено природнi рiвняння для узагальненої пружної лiнiї, деформованої на ненульовiй поверхнi зовнiшнiм полему тривимiрному напiвевклiдовому просторi E31, та наведено деякi застосування.1. Introduction. In this section we give some definitions and theorems.En with the metric, 〈v, w〉 = −∑νi=1viwi +∑nj=ν+1vjwj , v, w ∈ En, 0 ≤ ν ≤ n, is calledsemi-Euclidean space and is denoted by Enν , where is called the index of the metric. For n = 3, E31is called semi-Euclidean 3-space. Let Enν be a semi-Euclidean space furnished with a metric tensor〈 , 〉. A vector v to Enν is called, spacelike if 〈v, v〉 > 0 or v = 0 and timelike if 〈v, v〉 < 0. Spacelikeand timelike vectors are non-null vectors [1].Apart from the Frenet frame {E1, n, b}, there also exist a second frame at every point of the curveγ. At the point γ(s) of γ, let E1(s) = γ′(s) denote the unit tangent vector to γ, N denote the unitnormal of non-null surface and E2(s) = ε2N ∧ E1. Then {E1, E2, N} gives an orthonormal basisfor all vectors at γ(s) [2].The analogue of the Frenet – Serret formulas is givenE′1E′2N ′ =0 ε2cg ε3cn−ε1cg 0 −ε3τg−ε1cn ε2τg 0E1E2N, (1)where cg is the geodesic curvature, τg is the geodesic torsion, cn is the normal curvature and〈E1, E1〉 = ε1, 〈E2, E2〉 = ε2, 〈N,N〉 = ε3.Let x(u, v) be the timelike surface, having parameter curves which are perpendicular to eachother passing through point γ(s) of any curve γ. χ is angle between u = constant curve with tangentvector of timelike surface, (cg)1, (cg)2 are curvatures u = constant and v = constant curves. Thegeodesic curvature is [2]cg = (cg)1 coshχ− (cg)2 sinhχ− dχds. (2)Here (cg)1 = −12Ev|E| |G|1/2, (cg)2 =12Gu|E|1/2 |G|and the normal curvature iscn = c1 cosh2 χ− c2 sinh2 χ, (3)c© N. GU¨RBU¨Z, A. GO¨RGU¨LU¨, 2015ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3 333334 N. GU¨RBU¨Z, A. GO¨RGU¨LU¨where c1 and c2 are principal curvatures. The geodesic torsion is [2]τg = (c2 − c1) coshχ sinhχ. (4)2. Equilibrium conditions for generalized elastic line deformed. γ is called as an elasticcurve, if it is a critical point of total squared curvature energy functional∫γκ2ds. (5)The study of elastic curves have long research history. In the 1730, Bernoulli and Euler studiedthe bending energy functional (5) for in R2. In last years, elastic problem has been reconsideredmany geometers [3 – 8]. Problem play important role in the connection between the motion of curvesand integrable systems [9]: the equation describing the evolution of the torsion with respect to certainlength preserving vector fields coincides with the nonlinear Schro¨dinger equation. Vortex filamentsand patches in fluids [10 – 13], classical magnetic spin chains [14, 15], interface dynamic contextshave such curve motions [16].Generalization of the (5) is given byH =∫γh(κ, τ)ds. (6)Here, κ, τ denote curvature and torsion of non-null curve γ. The arc γ is called a generalized elasticline if it is extremal for variational problem of minimizing the value of (6) within the family of allarcs of length l on non-null surface having the same initial point and initial direction as γ in thesemi-Euclidean 3-space E31 [17, 18].In this paper, we study a special case of (6)I1 =l∫0τ2κds. (7)In [19], Manning study the intrinsic equations for elastic line deformed an external field inEuclidean 3-space. In this paper, we study the equilibrium conditions for generalized elastic linedeformed on a non-null surface in semi-Euclidean space E31.If generalized elastic line is deformed an external field, it minimizes the sum of its generalizedelastic energy and its energy of interaction with the field.The problem is to minimize the energyJ =l∫0(τ2κ− ζφ)ds = I1(t)− ζI2(t), (8)I1(t) =l∫0τ2κds, (9)ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE BY AN EXTERNAL FIELD . . . 335I2(t) =l∫0φds. (10)Here ζ is a constant measuring the strength of the external field and φ(u, v) gives its shape. For γregular curve in E31,κ =|〈γ′ ∧ γ′′, γ′ ∧ γ′′〉|1/2|〈γ′, γ′〉|3/2and τ = −ε3 〈γ′ ∧ γ′′, γ′′′〉|〈γ′ ∧ γ′′, γ′ ∧ γ′′〉| ,E1(s) and E2(s), respectively, are expressed with suitable scalar functions f(s) and g(s)E1(s) = γ′(s) =∂x∂ududs+∂x∂vdvds, E2(s) = f(s)xu + g(s)xv,f and g are expressed for spacelike surfaces and timelike surfaces with timelike arc γ as following [8]:f =u′F + v′G|EG− F 2|1/2, g = − u′E + v′F|EG− F 2|1/2.Here E = 〈xu, xu〉, G = 〈xv, xv〉 and F = 〈xu, xv〉.We defineΨ(ρ; t) = x (u(ρ) + tη(ρ), v(ρ) + tξ(ρ)) , (11)for 0 ≤ ρ ≤ l∗, Ψ(ρ; t) gives an arc with the same initial point and initial direction as γ. For t = 0,Ψ(ρ; 0) is the same as γ∗ and ρ is arc length. For t 6= 0, the parameter ρ is not non-null arc lengthin general. For fixed t, |t| < ε, let L∗(t) denote the length of the non-null arc Ψ(ρ; t), 0 ≤ ρ ≤ l∗.ThenL∗(t) =l∗∫0(∣∣∣∣〈∂Ψ∂ρ (ρ; t) , ∂Ψ∂ρ (ρ; t)〉∣∣∣∣)1/2 dρwith L∗(0) = l∗ > l.We can restrict Ψ(ρ; t), to an non-null arc of length l by restricting the parameterρ to an interval 0 ≤ ρ ≤ ω(t) ≤ l∗, ω(0) = l by requiringω(t)∫0(∣∣∣∣〈∂Ψ∂ρ , ∂Ψ∂ρ〉∣∣∣∣)1/2 dρ = l (12)anddωdt∣∣∣∣t=0= ε1l∫0δcgds. (13)The proof of (13) and of other results below will depend on calculations in (11) such as∂Ψ∂ρ∣∣∣∣t=0= E1, 0 ≤ ρ ≤ l,ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3336 N. GU¨RBU¨Z, A. GO¨RGU¨LU¨which gives∂2Ψ∂ρ2∣∣∣∣t=0= E′1 = ε2cgE2 + ε3cnN, (14)∂3Ψ∂ρ3∣∣∣∣t=0= E′1 = −ε1(ε2c2g + ε3c2n)E1 + (ε2c′g + ε2ε3cnτg)E2 − (ε2ε3cgτg + ε3c′n)N, (15)∂Ψ∂t∣∣∣∣t=0= δE2. (16)(17), (18) are obtained with aid (14) – (16)∂2Ψ∂t∂ρ∣∣∣∣t=0= −ε1δcgE1 + δ′E2 − ε3δτgN, (17)∂3Ψ∂t∂ρ2∣∣∣∣t=0=(−2ε1δ′cg − ε1δc′g + ε1ε3δτgcn)E1++(δ′′ − ε1ε2δc2g − ε2ε3δτ2g)E2 −(2ε3δ′τg + ε1ε3δcgcn + ε3δτ ′g)N. (18)To prove (13), differentiate (12) with respect to t and evaluate at t = 0,dωdt∣∣∣∣t=0(∣∣∣∣〈 ∂Ψ∂ρ∣∣∣∣t=0,∂Ψ∂ρ∣∣∣∣t=0〉∣∣∣∣)1/2 ++l∫0〈∂Ψ∂ρ∣∣∣∣t=0,∂2Ψ∂ρ∂t∣∣∣∣t=0〉 (∣∣∣∣〈 ∂Ψ∂ρ∣∣∣∣t=0,∂Ψ∂ρ∣∣∣∣t=0〉∣∣∣∣)1/2〈∂Ψ∂ρ∣∣∣∣t=0,∂Ψ∂ρ∣∣∣∣t=0〉 dρ = 0.I1 (t) is given as following from (9):I1 (t) =ω(t)∫0〈∂Ψ∂ρ∧ ∂2Ψ∂ρ2,∂3Ψ∂ρ3〉2 ∣∣∣∣〈∂Ψ∂ρ ∧ ∂2Ψ∂ρ2 , ∂Ψ∂ρ ∧ ∂2Ψ∂ρ2〉∣∣∣∣−3/2 ∣∣∣∣〈∂Ψ∂ρ , ∂Ψ∂ρ〉∣∣∣∣−1 dρ.We haveI ′1(t) =dωdt{〈∂Ψ∂ρ∧ ∂2Ψ∂ρ2,∂3Ψ∂ρ3〉2 ∣∣∣∣〈∂Ψ∂ρ ∧ ∂2Ψ∂ρ2 , ∂Ψ∂ρ ∧ ∂2Ψ∂ρ2〉∣∣∣∣−3/2 ∣∣∣∣〈∂Ψ∂ρ , ∂Ψ∂ρ〉∣∣∣∣−1}ρ=ω(t)++ω(t)∫0∂∂t(〈∂Ψ∂ρ∧ ∂2Ψ∂ρ2,∂3Ψ∂ρ3〉2)∣∣∣∣〈∂Ψ∂ρ ∧ ∂2Ψ∂ρ2 , ∂Ψ∂ρ ∧ ∂2Ψ∂ρ2〉∣∣∣∣−3/2 ∣∣∣∣〈∂Ψ∂ρ , ∂Ψ∂ρ〉∣∣∣∣−1 dρ +ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE BY AN EXTERNAL FIELD . . . 337+ω(t)∫0〈∂Ψ∂ρ∧ ∂2Ψ∂ρ2,∂3Ψ∂ρ3〉2∂∂t(∣∣∣∣〈∂Ψ∂ρ ∧ ∂2Ψ∂ρ2 , ∂Ψ∂ρ ∧ ∂2Ψ∂ρ2〉∣∣∣∣−3/2 ∣∣∣∣〈∂Ψ∂ρ , ∂Ψ∂ρ〉∣∣∣∣−1)dρ. (19)〈∂2Ψ∂ρ2,∂Ψ∂ρ〉vanishes at t = 0, since 〈E′1, E1〉 = 0. After complicated computations, with differen-tiating of (10) at t = 0 [8],I ′2(0) =l∫0[(∂φ∂t)∣∣∣∣t=0− ε1δcg(φ− φ(l)]ds. (20)As function of coordinates along Ψφ = φ[u(ρ) + tδ(ρ)f(ρ), v(ρ) + tδ(ρ)g(ρ)]and (∂φ∂t)∣∣∣∣t=0= δ[f(∂φ∂u)+ g(∂φ∂v)].From (19) for t = 0 and (20), J ′(0) is obtained as following:J ′(0) =l∫0δ[(UΥ)′′ − 2ε2(cnU)′′ + (UΩ)′ + ε1U(l) −− ζ(f(∂φ∂u)+ g(∂φ∂v)− ε1cg(φ− φ(l)))]ds −− 2ε2δ′′(l)cn(l)U(l) + δ′(l)[(−2ε2(cnU)′(l) + U(l)Υ(l)] ++ δ(l)[(2ε2(cnU)′′(l)− (UΥ)′(l) + U(l)Ω(l)], (21)whereU =(cgcn)′ + ε3(cnτg)− ε2(cgτg)∣∣ε2c2g + ε3c2n∣∣3/2 , V =∣∣ε2c2g + ε3c2n∣∣−1 ((cgcn)′ + τg(ε3cn − ε2cg)),Ω = (2c3n + 2(3− 4ε1)cnc2g + 4ε1cgτ2g + 4ε3c′nτg − 6ε3cnτ2g − 6cgτ ′g − 3(ε2cnτ ′g + cgτ2g )V ), (22)Υ = (2ε1ε3c′n − 4cgτg + 3ε1cgV ).3. Intrinsic equations for generalized elastic line deformed on timelike surface with timelikearc. For E1 is timelike, E2 and N are spacelike, respectively,〈E1, E1〉 = ε1 = −1, 〈E2, E2〉 = ε2 = 1 and 〈N,N〉 = ε3 = 1. (23)We consider the case c2g > c2n.ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3338 N. GU¨RBU¨Z, A. GO¨RGU¨LU¨Using (21), (22) and (23) for all choices of the function δ(s) with arbitrary values of δ(l),δ′(l), δ′′(l), and J ′(0) = 0, the path of timelike arc γ(s) must satisfy as following boundary conditionsand differential equationcn(l)U(l) = 0, (24)2(cnU)′(l) = −U(l)Υ(l), (25)2(cnU)′′(l) = (UΥ)′(l)− U(l)Ω(l), (26)(UΩ)′ + (UΥ)′′ − 2(cnU)′′ − U(l)− ζ(f(∂φ∂u)+ g(∂φ∂v)+ cg(φ− φ(l))) = 0, (27)whereU =(cgcn)′ + τg(cn − cg)(c2g + c2n)3/2, V = (c2g + c2n)−1((cgcn)′ + τg(cn − cg)),Ω = (2c3n + 14cnc2g − 4cgτ2g + 4c′nτg − 6cnτ2g − 6cgτ ′g − 3(cnτ ′g + 3cgτ2g )V ),Υ = −(2c′n + 4cgτg + 3cgV ).4. Intrinsic equations for generalized elastic line deformed on spacelike surface. If E, E2are spacelike and N is timelike,〈E1, E1〉 = ε1 = 1, 〈E2, E2〉 = ε2 = 1 and 〈N,N〉 = ε3 = −1. (28)We consider the case c2g < c2n. With aid (21), (22) and (28), for all choices of the function δ(s) witharbitrary values of δ(l), δ′(l), δ′′(l), and J ′(0) = 0, the path of spacelike arc γ(s) must satisfy asfollowing conditions and differential equationcn(l)U(l) = 0,−2(cnU)′(l) + U(l)Υ(l) = 0,(29)2(cnU)′′(l)− (UΥ)′(l) + U(l)Ω(l) = 0,−(UΥ)′′ − 2(cnU)′′ − (UΩ)′ + U(l)− ζ(f(∂φ∂u)+ g(∂φ∂v)− cg(φ− φ(l))= 0,whereU =(cgcn)′ − τg(cn − cg)(c2n − c2g)3/2, V = (c2n − c2g)−1((cgcn)′ − τg(cn − cg)),Υ = (−2c′n − 4cgτg + 3cgV ),Ω = (2c3n − 2cnc2g + 4cgτ2g − 4c′nτg + 6cnτ2g − 6cgτ ′g − 3cn(−cnτ ′g + cgτ2g )V ).ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE BY AN EXTERNAL FIELD . . . 3395. Applications.Theorem 1. On S11 ×R Lorentz cylinder in E31, there is not generalized elastic line deformed.Proof. Parametric equation for r radius Lorentz cylinderx(u, v) = (r sinhu, r coshu, v).Using shape operator, principal curvatures are obtained as following:c1 = −1rand c2 = 0.With aid (3) and (4), respectively,cn = −1rcosh2 χ,cg =1rcoshχ sinhχ,(30)boundary conditions (24) are just satisfied at χ = 0, using (30). Therefore, on S11 × R Lorentzcylinder, there is not generalized elastic line deformed.Theorem 2. There is not generalized elastic line deformed on H2(r) hyperbolic 2-space in E31.Proof. H2(r) hyperbolic 2-space satisfy−x2 + y2 + z2 = −r2and matrix for shape operator −1r 00 −1r,c1 = −1rand c2 = −1r, cn = c1 cos2 χ+ c2 sin2 χ = −1r6= 0. From (29), proof is clear.Theorem 3. If γ is generalized elastic line deformed for non-null surface which geodesic torsionand normal curvature is zero, γ must satisfy the following differential equation:f(∂φ∂u)+ g(∂φ∂v)= ε1cg(φ− φ(l)).Proof. If cn = 0 and τg = 0, proof is trivial from (21).Theorem 4. Any arc on spacelike plane is generalized elastic line deformed.Proof. For spacelike plane, cn = 0 and τg = 0. Thus, (29) is satisfied.1. O’Neill B. Semi-Riemannian geometry. – New York: Acad. Press, 1983.2. Ugˇurlu H. H., C¸alıs¸kan A. On the geometry of space-like surfaces // J. Inst. Sci. and Technology Gazi Univ. – 1997.3. Capovilla R., Chryssomalakos C., Guven. Hamiltonians for curves // J. Phys. A: Math. and Gen. – 2002. – 35. –P. 6571 – 6587.4. Langer J., Singer D. A. The total squared curvature of closed curves // J. Different. Geom. – 1984. – 20. – P. 1 – 22.5. Arroyo J., Garay O. J., Mencia J. J. Closed generalized elastic curves in S2(1) // J. Math. Phys. – 2003. – 48. –P. 339 – 353.ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3340 N. GU¨RBU¨Z, A. GO¨RGU¨LU¨6. Garay O. J. Extremals of the generalized Euler – Bernoulli energy and applications // J. Geom. Symmetry Phys. –2008. – 12. – P. 27 – 61.7. Barros M., Ferrandez A., Javaloyes M. A., Lucas P. 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Langer J. Recursion in curve geometry // J. Math. – 1999. – 5. – P. 25 – 51.17. Rong-pei Huang, Dong-hu Shang. Generalized elastic curves in Lorentz flat space L4 // Appl. Math. and Mech. –2009. – 30. – P. 1193 – 1200.18. Ekici C., Go¨rgu¨lu¨ A. Intrinsic equations for a generalized elastic line on an oriented surface in the Minkowski spaceE31 // Turk. J. Math. – 2009. – 33. – P. 397 – 407.19. Manning G. Elastic line deformed on a surface by an external field: Intrinsic formulation and preliminary applicationto nucleosome energetics // Phys. Rev. A. – 1988. – 38. – P. 3073 – 3081.Received 04.12.12,after revision — 17.12.14ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3

Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space E₁³

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Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space E₁³

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)