Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$
We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space $\mathbb{E}_1^3$ and give some applications.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507891954352128 |
|---|---|
| author | Gorgula, V. I. Gürbüz, N. Горгула, В. И. Гурбуз, Н. |
| author_facet | Gorgula, V. I. Gürbüz, N. Горгула, В. И. Гурбуз, Н. |
| author_sort | Gorgula, V. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:48:08Z |
| description | We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space $\mathbb{E}_1^3$ and give some applications. |
| first_indexed | 2026-03-24T02:16:31Z |
| format | Article |
| fulltext |
UDC 515.16
N. Gürbüz, A. Görgülü (Osmangazi Univ., Turkey)
GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE
BY AN EXTERNAL FIELD IN THE 3-DIMENSIONAL
SEMI-EUCLIDEAN SPACE E3
1
УЗАГАЛЬНЕНА ПРУЖНА ЛIНIЯ, ДЕФОРМОВАНА НА НЕНУЛЬОВIЙ
ПОВЕРХНI ЗОВНIШНIМ ПОЛЕМ У ТРИВИМIРНОМУ
НАПIВЕВКIДОВОМУ ПРОСТОРI E3
1
We deduce intrinsic equations for a generalized elastic line deformed on the non-null surface by an external field in
semi-Euclidean space E3
1 and give some applications.
Виведено природнi рiвняння для узагальненої пружної лiнiї, деформованої на ненульовiй поверхнi зовнiшнiм полем
у тривимiрному напiвевклiдовому просторi E3
1, та наведено деякi застосування.
1. Introduction. In this section we give some definitions and theorems.
En with the metric, 〈v, w〉 = −
∑ν
i=1
viwi +
∑n
j=ν+1
vjwj , v, w ∈ En, 0 ≤ ν ≤ n, is called
semi-Euclidean space and is denoted by Enν , where is called the index of the metric. For n = 3, E3
1
is 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. Spacelike
and 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 curve
γ. At the point γ(s) of γ, let E1(s) = γ′(s) denote the unit tangent vector to γ, N denote the unit
normal of non-null surface and E2(s) = ε2N ∧ E1. Then {E1, E2, N} gives an orthonormal basis
for all vectors at γ(s) [2].
The analogue of the Frenet – Serret formulas is given
E′1
E′2
N ′
=
0 ε2cg ε3cn
−ε1cg 0 −ε3τg
−ε1cn ε2τg 0
E1
E2
N
, (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 each
other passing through point γ(s) of any curve γ. χ is angle between u = constant curve with tangent
vector of timelike surface, (cg)1, (cg)2 are curvatures u = constant and v = constant curves. The
geodesic curvature is [2]
cg = (cg)1 coshχ− (cg)2 sinhχ− dχ
ds
. (2)
Here (cg)1 = −1
2
Ev
|E| |G|1/2
, (cg)2 =
1
2
Gu
|E|1/2 |G|
and the normal curvature is
cn = c1 cosh2 χ− c2 sinh2 χ, (3)
c© N. GÜRBÜZ, A. GÖRGÜLÜ, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3 333
334 N. GÜRBÜZ, A. GÖRGÜLÜ
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 elastic
curve, if it is a critical point of total squared curvature energy functional∫
γ
κ2ds. (5)
The study of elastic curves have long research history. In the 1730, Bernoulli and Euler studied
the bending energy functional (5) for in R2. In last years, elastic problem has been reconsidered
many geometers [3 – 8]. Problem play important role in the connection between the motion of curves
and integrable systems [9]: the equation describing the evolution of the torsion with respect to certain
length preserving vector fields coincides with the nonlinear Schrödinger equation. Vortex filaments
and patches in fluids [10 – 13], classical magnetic spin chains [14, 15], interface dynamic contexts
have such curve motions [16].
Generalization of the (5) is given by
H =
∫
γ
h(κ, τ)ds. (6)
Here, κ, τ denote curvature and torsion of non-null curve γ. The arc γ is called a generalized elastic
line if it is extremal for variational problem of minimizing the value of (6) within the family of all
arcs of length l on non-null surface having the same initial point and initial direction as γ in the
semi-Euclidean 3-space E3
1 [17, 18].
In this paper, we study a special case of (6)
I1 =
l∫
0
τ2κds. (7)
In [19], Manning study the intrinsic equations for elastic line deformed an external field in
Euclidean 3-space. In this paper, we study the equilibrium conditions for generalized elastic line
deformed on a non-null surface in semi-Euclidean space E3
1.
If generalized elastic line is deformed an external field, it minimizes the sum of its generalized
elastic energy and its energy of interaction with the field.
The problem is to minimize the energy
J =
l∫
0
(τ2κ− ζφ)ds = I1(t)− ζI2(t), (8)
I1(t) =
l∫
0
τ2κds, (9)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE BY AN EXTERNAL FIELD . . . 335
I2(t) =
l∫
0
φds. (10)
Here ζ is a constant measuring the strength of the external field and φ(u, v) gives its shape. For γ
regular curve in E3
1,
κ =
|〈γ′ ∧ γ′′, γ′ ∧ γ′′〉|1/2
|〈γ′, γ′〉|3/2
and 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
∂u
du
ds
+
∂x
∂v
dv
ds
, 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 define
Ψ(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 length
in general. For fixed t, |t| < ε, let L∗(t) denote the length of the non-null arc Ψ(ρ; t), 0 ≤ ρ ≤ l∗.
Then
L∗(t) =
l∗∫
0
(∣∣∣∣〈∂Ψ
∂ρ
(ρ; t) ,
∂Ψ
∂ρ
(ρ; t)
〉∣∣∣∣)1/2
dρ
with L∗(0) = l∗ > l.We can restrict Ψ(ρ; t), to an non-null arc of length l by restricting the parameter
ρ to an interval 0 ≤ ρ ≤ ω(t) ≤ l∗, ω(0) = l by requiring
ω(t)∫
0
(∣∣∣∣〈∂Ψ
∂ρ
,
∂Ψ
∂ρ
〉∣∣∣∣)1/2
dρ = l (12)
and
dω
dt
∣∣∣∣
t=0
= ε1
l∫
0
δcgds. (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, № 3
336 N. GÜRBÜZ, A. GÖRGÜLÜ
which gives
∂2Ψ
∂ρ2
∣∣∣∣
t=0
= E′1 = ε2cgE2 + ε3cnN, (14)
∂3Ψ
∂ρ3
∣∣∣∣
t=0
= E′1 = −ε1(ε2c2g + ε3c
2
n)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,
dω
dt
∣∣∣∣
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 have
I ′1(t) =
dω
dt
{〈
∂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, № 3
GENERALIZED 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 Ψ
φ = 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∫
0
δ
[
(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)
where
U =
(cgcn)′ + ε3(cnτg)− ε2(cgτg)∣∣ε2c2g + ε3c2n
∣∣3/2 , V =
∣∣ε2c2g + ε3c
2
n
∣∣−1 ((cgcn)′ + τg(ε3cn − ε2cg)),
Ω = (2c3n + 2(3− 4ε1)cnc
2
g + 4ε1cgτ
2
g + 4ε3c
′
nτg − 6ε3cnτ
2
g − 6cgτ
′
g − 3(ε2cnτ
′
g + cgτ
2
g )V ), (22)
Υ = (2ε1ε3c
′
n − 4cgτg + 3ε1cgV ).
3. Intrinsic equations for generalized elastic line deformed on timelike surface with timelike
arc. 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, № 3
338 N. GÜRBÜZ, A. GÖRGÜLÜ
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 conditions
and differential equation
cn(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)
where
U =
(cgcn)′ + τg(cn − cg)
(c2g + c2n)3/2
, V = (c2g + c2n)−1((cgcn)′ + τg(cn − cg)),
Ω = (2c3n + 14cnc
2
g − 4cgτ
2
g + 4c′nτg − 6cnτ
2
g − 6cgτ
′
g − 3(cnτ
′
g + 3cgτ
2
g )V ),
Υ = −(2c′n + 4cgτg + 3cgV ).
4. Intrinsic equations for generalized elastic line deformed on spacelike surface. If E, E2
are 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) with
arbitrary values of δ(l), δ′(l), δ′′(l), and J ′(0) = 0, the path of spacelike arc γ(s) must satisfy as
following conditions and differential equation
cn(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,
where
U =
(cgcn)′ − τg(cn − cg)
(c2n − c2g)3/2
, V = (c2n − c2g)−1((cgcn)′ − τg(cn − cg)),
Υ = (−2c′n − 4cgτg + 3cgV ),
Ω = (2c3n − 2cnc
2
g + 4cgτ
2
g − 4c′nτg + 6cnτ
2
g − 6cgτ
′
g − 3cn(−cnτ ′g + cgτ
2
g )V ).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
GENERALIZED ELASTIC LINE DEFORMED ON NON-NULL SURFACE BY AN EXTERNAL FIELD . . . 339
5. Applications.
Theorem 1. On S1
1 ×R Lorentz cylinder in E3
1, there is not generalized elastic line deformed.
Proof. Parametric equation for r radius Lorentz cylinder
x(u, v) = (r sinhu, r coshu, v).
Using shape operator, principal curvatures are obtained as following:
c1 = −1
r
and c2 = 0.
With aid (3) and (4), respectively,
cn = −1
r
cosh2 χ,
cg =
1
r
coshχ sinhχ,
(30)
boundary conditions (24) are just satisfied at χ = 0, using (30). Therefore, on S1
1 × R Lorentz
cylinder, there is not generalized elastic line deformed.
Theorem 2. There is not generalized elastic line deformed on H2(r) hyperbolic 2-space in E3
1.
Proof. H2(r) hyperbolic 2-space satisfy
−x2 + y2 + z2 = −r2
and matrix for shape operator −1
r
0
0 −1
r
,
c1 = −1
r
and c2 = −1
r
, cn = c1 cos2 χ+ c2 sin2 χ = −1
r
6= 0. From (29), proof is clear.
Theorem 3. If γ is generalized elastic line deformed for non-null surface which geodesic torsion
and 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. Uǧurlu H. H., Çalışkan 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, № 3
340 N. GÜRBÜZ, A. GÖRGÜLÜ
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. Relativistic particles with rigidity and torsion in D = 3 spacetimes
// Classical Quantum Gravity. – 2005. – 22. – P. 489 – 513.
8. Gürbüz N. Intrinsic formulation for elastic line deformed on a surface by external field in the Minkowski 3-space //
J. Math. Anal. and Appl. – 2007. – 327. – P. 1086 – 1094.
9. Hasimoto A. Soliton on a vortex filament // J. Fluid Mech. – 1972. – 51.
10. Lamb J. Solitons on moving space curves // J. Math. Phys. – 1977. – 18.
11. Goldstein R. E., Petrich D. M. Solitons, Euler’s equation and vortex patch dynamic // Phys. Rev. Lett. – 1992. – 69. –
P. 555 – 558.
12. Lakshamanan R. Phys. Lett. A. – 1982. – 92.
13. Bryant R., Griffiths P. Reduction for constrained variational problem // Amer. J. Math. – 1986. – 108.
14. Goldstein R. E., Petrich D. M. The Korteweg – de Vries hierarchy as dynamics of closed curves in the plane // Phys.
Rev. Lett. – 1991. – 67. – P. 3203 – 3206.
15. Langer J., Perline R. The planar filament equation // J. Math. Phys. – 1994. – 35.
16. 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., Görgülü A. Intrinsic equations for a generalized elastic line on an oriented surface in the Minkowski space
E3
1 // 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 application
to nucleosome energetics // Phys. Rev. A. – 1988. – 38. – P. 3073 – 3081.
Received 04.12.12,
after revision — 17.12.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
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| id | umjimathkievua-article-1986 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:31Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6f/9ed2a111fc7600612505ccb08883236f.pdf |
| spelling | umjimathkievua-article-19862019-12-05T09:48:08Z Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ Узагальнена пружна лiнiя, деформована на ненульовsq пoвepxнi зовнiшнiм полем у тривимірному напiвевкiдовому просторi $\mathbb{E}_1^3$ Gorgula, V. I. Gürbüz, N. Горгула, В. И. Гурбуз, Н. We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space $\mathbb{E}_1^3$ and give some applications. Виведено природні рівняння для узагальненої пружної лінії, деформованої на ненульовій поверхні зовнішнім полем у тривимірному напівевклідовому просторі $\mathbb{E}_1^3$, та наведено деякі застосування. Institute of Mathematics, NAS of Ukraine 2015-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1986 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 3 (2015); 333–340 Український математичний журнал; Том 67 № 3 (2015); 333–340 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1986/991 https://umj.imath.kiev.ua/index.php/umj/article/view/1986/992 Copyright (c) 2015 Gorgula V. I.; Gürbüz N. |
| spellingShingle | Gorgula, V. I. Gürbüz, N. Горгула, В. И. Гурбуз, Н. Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title | Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title_alt | Узагальнена пружна лiнiя, деформована на ненульовsq пoвepxнi зовнiшнiм полем у тривимірному напiвевкiдовому просторi $\mathbb{E}_1^3$ |
| title_full | Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title_fullStr | Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title_full_unstemmed | Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title_short | Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$ |
| title_sort | generalized elastic line deformed on a nonnull surface by an external field in the 3-dimensional semi-euclidean space $\mathbb{e}_1^3$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1986 |
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