Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space
UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ by using the Bishop orthonormal vector frame and associated Bishop components.  Then we compute the energy of the ligh...
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| author | Körpinar, T. Demirkol, R. C. Körpinar, T. Demirkol, R. C. |
| author_facet | Körpinar, T. Demirkol, R. C. Körpinar, T. Demirkol, R. C. |
| author_sort | Körpinar, T. |
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| description | UDC 515.1
We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ by using the Bishop orthonormal vector frame and associated Bishop components.  Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ and investigate its relationship with the energy of the same curve in Bishop vector fields in $\mathbb{E}_{1}^{4}$.  Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space $\mathbb{E}_{1}^{4}$. |
| doi_str_mv | 10.37863/umzh.v72i8.847 |
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DOI: 10.37863/umzh.v72i8.847
UDC 515.1
T. Körpinar, R. C. Demirkol (Muş Alparslan Univ., Turkey)
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA
AND NONELASTICA FOR A LIGHTLIKE CURVE IN THE MINKOWSKI SPACE
ЗАЛЕЖНА ВIД КРИВИНИ ТА КРУЧЕННЯ ЕНЕРГIЯ ПРУЖНОСТI
ТА НЕПРУЖНОСТI ДЛЯ СВIТЛОПОДIБНИХ КРИВИХ
У ПРОСТОРI МIНКОВСЬКОГО
We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space
\BbbE 4
1 by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the
lightlike elastic and nonelastic Cartan curve in the Minkowski space \BbbE 4
1 and investigate its relationship with the energy of
the same curve in Bishop vector fields in \BbbE 4
1 . Here, energy functionals are computed in terms of Bishop curvatures of the
lightlike Cartan curve lying in the Minkowski space \BbbE 4
1 .
Спочатку описано умови пружностi та непружностi свiтлоподiбних пружних кривих Картана у просторi Мiнков-
ського \BbbE 4
1 за допомогою ортонормального векторного репера та вiдповiдних компонент Бiшопа. Потiм обчислено
енергiю пружних та непружних свiтлоподiбних кривих Картана у просторi Мiнковського \BbbE 4
1 та вивчено iхнiй зв’язок
iз енергiєю тiєї ж кривої у векторних полях Бiшопа у \BbbE 4
1 . Тут функцiонали енергiї обчислюються у термiнах кривини
Бiшопа для свiтлоподiбних кривих Картана, що належать простору Мiнковського \BbbE 4
1 .
1. Introduction. Minkowski space-time is an important structure to define many well-known phy-
sical and geometrical concepts such as black holes, gravitational dilation of time, cosmology, length
string theory, contraction, etc. In this Minkowski space, mass-energy equivalence states the rela-
tionship between mass and energy and special relativity estimates this equivalence by the formula
E = mc2, where c is the light’s speed in a vacuum [1, 2].
Some traditional geometric topics such as local and global features of different types of curves are
used to many physical subjects. For instance, Altin [3] calculated the energy of Frenet orthonormal
vector fields by using nonlightlike curves. Körpınar [4] considered a timelike biharmonic particle
and computed its energy functional in Heisenberg spacetime. Lightlike curves are a thoroughly
complicated field to study since their tangent vectors cannot be normalized in an ordinary manner
in contrast to nonlightlike curves. Körpinar characterized the energy of different types of lightlike
curves in Minkowski space \BbbE 4
1 [5].
The fundamental ingredient for the study of the geodesic lightlike congruences, gravitational ra-
diation, Killing horizon are determined via tetrad formalism of Newman – Penrose, which is deduced
by a lightlike curve. Furthermore, it is known that relativistic string can be defined as a surface in
Minkowski space such that it is a Lorentzian analogue of the equations of minimal surface. Wave
equations can also be simplified by string equations and solving 2-dimensional wave equations leads
that strings and pair of lightlike curves are equivalents [6 – 10].
Materials having the feature of deformable property such as flexible metals, paper, cloth, rubber
are the main instances and study fields of the elasticity theory. However, elastic theory enlightens a
broad range of other physical and mathematical studies such as the study of variational problems, the
solution of the elliptic integral, mechanical equilibrium, equilibrium of moments, which constitutes
c\bigcirc T. KÖRPINAR, R. C. DEMIRKOL, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8 1095
1096 T. KÖRPINAR, R. C. DEMIRKOL
the elementary principle of statics. Further, it is seen that elastica gives a natural solution for the
variational problem, which deals with the minimizing of bending energy of the elastic curve. Later,
the equivalence between the motion of the simple pendulum and fundamental differential equation
of elastica was investigated. Recently, numerical computation implemented on the elastica is used to
develop mathematical spline theory [11].
In this study, we firstly determine differential equations satisfied by non-rigid deformable light-
like Cartan curves in order to model the behavior of lightlike elastic Cartan curves in 4-dimensional
Minkowski space \BbbE 4
1 . We compute energy on the lightlike elastic Cartan curves by using the vari-
ational method. Finally, we define lightlike nonelastic Cartan curves and compute their energy to
characterize their structure and investigate the relationship between the elastic and nonelastic cases.
2. Lightlike curves in Minkowski space \BbbE 4
1 . Minkowski space \BbbE 4
1 is the 4-dimensional
standard real vector space equipped with the usual indefinite metric (\cdot , \cdot ) described by
(x, y) = x1y1 + x2y2 + x3y3 - x4y4,
where x, y \in \BbbE 4
1. Owing to the intrinsic features of the indefinite metric (\cdot , \cdot ), any vector x \in \BbbE 4
1
possesses three causal characters, which is determined by the norm of the given vector \| x\| =
=
\sqrt{}
| (x, x)| . To be more specific, any vector x \in \BbbE 4
1 can be characterized as follows:
x is spacelike, if (x,x) is positive,
x is timelike, if (x,x) is negative,
x is lightlike, if (x,x) is zero.
One can give a similar characterization for any space curve defined in \BbbE 4
1. Namely, an arbitrary space
curve \alpha : I \rightarrow \BbbE 4
1 is called a spacelike, timelike or lightlike curve provided that all tangent vectors
of \alpha are spacelike, timelike or lightlike along with the curve. In this study, we mainly focus on a
special class of lightlike curves which is known as a Cartan lightlike curve. A curve \alpha is called
a lightlike Cartan curve supposed that its parametrization is determined by the pseudo-arc function
defined by
s(t) =
t\int
0
\sqrt{}
\| \alpha \prime \prime (u)\| du.
As we recall, some of the geometric and physical features of a space curve are defined with the
help of the geometric quantities known as the torsion and curvature. To handle with the intrinsic
characterization for a lightlike Cartan curve in \BbbE 4
1, these quantities are also defined for lightlike
Cartan curves with the name of the first, second, and third Cartan curvatures and denoted respectively
by k1, k2, k3. A unique orthonormal vectors \{ \bfT ,\bfN ,\bfB 1,\bfB 2\} satisfy the following set of equation
systems along the nongeodesic lightlike Cartan curve \alpha and it is called by the Frenet – Serret frame
of lightlike Cartan curve [12]:
\nabla T\bfT = k1\bfN ,
\nabla T\bfN = - k2\bfT + k1\bfB 1, (1)
\nabla T\bfB 1 = - k2\bfN +k3\bfB 2,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA AND NONELASTICA . . . 1097
\nabla T\bfB 2 = k3\bfT ,
where k1 = 1, k2, and k3 are arbitrary functions. Here, we also have
0 = (\bfT ,\bfT ) = (\bfB 1,\bfB 1) = (\bfN ,\bfB 1) = (\bfB 1,\bfB 2) = (\bfT ,\bfN ) = (\bfT ,\bfB 2),
1 = (\bfN ,\bfN ) = (\bfB 2,\bfB 2), (\bfT ,\bfB 1) = - 1.
Bishop frame is known as the relatively parallel frame and it is obtained by conducting some modifi-
cation on the Frenet – Serret frame. Using the Bishop frame provides some advantages and effective-
ness compared to the classical Frenet – Serret frame. For this reason, we will state Bishop’s frame of
the lightlike Cartan curve and consider it as the main frame for the rest of the paper.
Case 1: Let \alpha be a lightlike Cartan curve in \BbbE 4
1 and k1 = 1, k3 = 0, and k2 be an arbitrary
function, then Bishop frame equations for orthonormal vectors \{ \bfT ,\bfN 1,\bfN 2,\bfN 3\} are stated as the
following [12]:
\nabla T\bfT = \pi 2\bfT + \pi 1\bfN 1,
\nabla T\bfN 1 = \pi 1\bfN 2, (2)
\nabla T\bfN 2 = - \pi 2\bfN 2,
\nabla T\bfN 3 = 0,
where the first, second and third Bishop curvature satisfies respectively that \pi 1 = 1, \pi \prime
2 = - 1
2
\pi 2
2 - k2,
\pi 3 = 0. Here, we also have
0 = (\bfT ,\bfT ) = (\bfN 2,\bfN 2) = (\bfT ,\bfN 1) = (\bfN 1,\bfN 2) = (\bfT ,\bfN 3) = (\bfN 2,\bfN 3),
(\bfN 1,\bfN 1) = (\bfN 3,\bfN 3) = 1, (\bfT ,\bfN 2) = - 1.
Case 2: Let \alpha be a lightlike Cartan curve in \BbbE 4
1 and k1 = 1, k2, k3 be arbitrary functions, then
Bishop frame equations for orthonormal vectors \{ \bfT ,\bfN 1,\bfN 2,\bfN 3\} are stated as the following [12]:
\nabla T\bfT = \pi 2\bfT + \pi 1\bfN 1 - \pi 3\bfN 3,
\nabla T\bfN 1 = \pi 1\bfN 2, (3)
\nabla T\bfN 2 = - \pi 2\bfN 2,
\nabla T\bfN 3 = - \pi 3\bfN 2,
where the first, second and third Bishop curvature satisfies respectively that \pi 1 = \mathrm{s}\mathrm{i}\mathrm{n} \theta , \pi 2 =
= - k3 - \theta \prime \prime
\theta \prime
, \pi 3 = \mathrm{c}\mathrm{o}\mathrm{s} \theta (\theta \prime \not = 0) and
2\theta \prime (\theta \prime \prime \prime - k\prime 3) + 2\theta \prime \prime (k3 - \theta \prime \prime ) + \theta \prime 4 - (k3 - \theta \prime \prime )2 - 2k2\theta
\prime 2 = 0.
Here, we also have
0 = (\bfT ,\bfT ) = (\bfN 2,\bfN 2) = (\bfT ,\bfN 1) = (\bfN 1,\bfN 2) = (\bfT ,\bfN 3) = (\bfN 2,\bfN 3),
(\bfN 1,\bfN 1) = (\bfN 3,\bfN 3) = 1, (\bfT ,\bfN 2) = - 1.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1098 T. KÖRPINAR, R. C. DEMIRKOL
3. Energy on the classical Bernoulli – Euler elastica. 3.1. Energy on the Frenet vector field.
We first give fundamental definitions and propositions, which are used to compute the energy of the
vector field.
Definition 3.1. Let (M,\rho ) and (N,h) betwo Riemannian manifolds, then the energy of a dif-
ferentiable map f : (M,\rho ) \rightarrow (N,h) can be defined as
\varepsilon nergy(f) =
1
2
\int
M
n\sum
a=1
h
\bigl(
df(ea), df(ea)
\bigr)
v, (4)
where \{ ea\} is a local basis of the tangent space and v is the canonical volume form in M [3, 13].
Proposition 3.1. Let Q : T
\bigl(
T 1M
\bigr)
\rightarrow T 1M be the connection map. Then the following two
conditions hold:
i) \omega \circ Q = \omega \circ d\omega and \omega \circ Q = \omega \circ \~\omega , where \~\omega : T (T 1M) \rightarrow T 1M is the tangent bundle
projection;
ii) for \varrho \in TxM and a section \xi : M \rightarrow T 1M, we have
Q
\bigl(
d\xi (\varrho )
\bigr)
= \nabla \varrho \xi , (5)
where \nabla is the Levi – Civita covariant derivative [3].
Definition 3.2. Let \varsigma 1, \varsigma 2 \in T\xi (T
1M), then we define
\rho S(\varsigma 1, \varsigma 2) =
\bigl(
d\omega (\varsigma 1), d\omega (\varsigma 2)
\bigr)
+
\bigl(
Q(\varsigma 1), Q(\varsigma 2)
\bigr)
. (6)
This yields a Riemannian metric on TM . As known \rho S is called the Sasaki metric that also makes
the projection \omega : T 1M \rightarrow M a Riemannian submersion.
3.2. Energy on the lightlike elastic Cartan curves in \BbbE 4
1. The research on the curvature-
based energies for space curves began with Bernoulli and Euler. They focus on elastic thin beams
and rods. This type of energy is both essential in the mechanical context and it is also signifi-
cant in computer vision, image processing and computer vision besides mathematical and physical
importance [14 – 17].
Let \alpha \in \BbbE 4
1 be a regular curve defined on any fixed interval [y1, y2] so that
\alpha : [y1, y2] \rightarrow \BbbE 4
1
is parameterized by the pseudo arc-length \rho
\bigl(
\alpha \prime \prime (s), \alpha \prime \prime (s)
\bigr)
= 1.
Elastica of bending energy is defined for the curve \alpha in \BbbE 4
1 over each point on a fixed interval
[y1, y2] as
\scrG =
1
2
y2\int
y1
\| \alpha \prime \prime \| 2dt
with some boundary conditions [18, 19].
For any two points p1, p2 \in \BbbR 4 and any two non-zero vectors p\prime 1, p
\prime
2 space of smooth curves is
defined as
\varphi =
\bigl\{
\alpha : \alpha (yi) = pi, \alpha
\prime (yi) = p\prime i
\bigr\}
.
It is also defined as the smooth curves of unit speed as a subspace of \varphi in the form
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA AND NONELASTICA . . . 1099
\varphi a =
\bigl\{
\alpha \in \varphi :
\bigm\| \bigm\| \alpha \prime \bigm\| \bigm\| = 1
\bigr\}
.
Then \scrG \pi : \varphi \rightarrow \BbbR can be defined by
\scrG \pi (\alpha ) =
1
2
\int
\alpha
\| \alpha \prime \prime \| 2 + \Gamma (t)
\bigl(
\| \alpha \prime \| 2 - 1
\bigr)
dt, (7)
where \Gamma (t) is a pointwise multiplier. A stationary point of \scrG \pi is the minimum of \scrG on \varphi a for some
\Gamma (t) according to the multiplier principle of Lagrange.
Let \alpha be an extremum of \scrG \pi and V be a vector field along \alpha , which is a curve’s infinitesimal
variation, then we get [20]
\partial \scrG \pi (V ) =
\partial
\partial \Upsilon
\scrG \pi (\alpha +\Upsilon V )
\bigm| \bigm|
\Upsilon =0
= 0. (8)
We obtain significant differences in conditions that have to be satisfied by lightlike elastic Cartan
curves by using the Bishop equations formulae given by Eqs. (2), (3). Further, we also compute the
energy of lightlike elastic Cartan curves by using the Lorentzian and Sasaki metrics.
Case 1: Let \alpha \in \BbbE 4
1 be a lightlike elastic Cartan curve defined on any fixed interval [y1, y2] so
that
\alpha : [y1, y2] \rightarrow \BbbE 4
1
and k1 = 1, k3 = 0, and k2 be an arbitrary function.
By considering the orthonormal frame given by (2), the energy of Bishop vectors (\bfT ,\bfN 1,\bfN 2,\bfN 3)
for lightlike Cartan curve \alpha \in \BbbE 4
1 can be computed if one follows similar steps as in following re-
search completed by K. Demirkol [4]. This study is helpful to see a relation between the energy of
Bishop vectors and bending energy functional which is defined as
\scrG =
1
2
\int
\alpha
\| \nabla T\bfT \| 2 ds.
Let V be a vector field along \alpha such that it is a curve’s infinitesimal variation. By using
equations (7) and (8), we get
0 =
1
2
\partial
\partial \Upsilon
y2\int
y1
\bigm\| \bigm\| (\alpha +\Upsilon V )\prime \prime
\bigm\| \bigm\| 2 + \Gamma
\Bigl( \bigm\| \bigm\| (\alpha +\Upsilon V )\prime
\bigm\| \bigm\| 2 - 1
\Bigr)
dt =
y2\int
y1
(\alpha \prime \prime , V \prime \prime )dt.
Applying integration by parts, we obtain
0 = (\alpha \prime \prime , V \prime )
\bigm| \bigm| \bigm| y2
y1
- (V, \alpha \prime \prime \prime )
\bigm| \bigm| \bigm| y2
y1
+
y2\int
y1
(V, \scrE (\alpha ))dt,
where \scrE (\alpha ) = \alpha \prime \prime \prime \prime . Being elastica implies that
\scrE (\alpha ) = \alpha \prime \prime \prime \prime \equiv 0. (9)
Thanks to Noether’s theorem we know that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1100 T. KÖRPINAR, R. C. DEMIRKOL
\scrJ = \alpha \prime \prime \prime
is a constant vector field. For a parameterized curve \alpha with the arc-length s, we have from Eq. (2)
\alpha \prime = \bfT , \alpha \prime \prime = \bfT \prime = \pi 2\bfT + \pi 1\bfN 1,
\alpha \prime \prime \prime =
\bigl(
\pi 2
2 + \pi \prime
2
\bigr)
\bfT + (\pi 1\pi 2 + \pi \prime
1)\bfN 1 + \pi 2
1\bfN 2.
Thus, we get
\scrJ = (\pi 2
2 + \pi \prime
2)\bfT + (\pi 1\pi 2 + \pi \prime
1)\bfN 1 + \pi 2
1\bfN 2.
By the fact that \scrJ is a constant vector field, we find \scrJ s = 0. From this, we have the system
3\pi 2\pi
\prime
2 + \pi \prime \prime
2 + \pi 3
2 = 0,
2\pi 1\pi
\prime
2 + \pi 1\pi
2
2 + \pi \prime
1\pi 2 + \pi \prime \prime
1 = 0,
\pi 1\pi
\prime
1 = 0.
Thus, we have following sample solution family of the nonlinear ordinary differential equation system
for certain values of \pi 1, \pi 2, and \pi 3.
Theorem 3.1. Energy of lightlike elastic Cartan curves having the Bishop characterization in
the Eq. (2) in constant vector field \scrJ in \BbbE 4
1 is stated by using the Sasaki metric as follows:
\varepsilon nergy1(\scrJ ) = 0.
Proof. From (4) and (5) we know
\varepsilon nergy1 (\scrJ ) =
1
2
s\int
0
\rho S (d\scrJ (\bfT ), d\scrJ (\bfT )) ds.
If one considers the Eq. (6) then it is obtained that
\rho S (d\scrJ (\bfT ), d\scrJ (\bfT )) = \rho (d\omega (\scrJ (\bfT )), d\omega (\scrJ (\bfT )))+\rho (Q(\scrJ (\bfT )), Q(\scrJ (\bfT ))).
Since \scrJ is a section, we get
d(\omega ) \circ d(\scrJ ) = d(\omega \circ \scrJ ) =d(idC) = idTC .
We also know
Q(\scrJ (\bfT )) = \bigtriangledown T\scrJ = 0.
Thus, we find from the former statements that
\rho S
\bigl(
d\scrJ (\bfT ), d\scrJ (\bfT )
\bigr)
= (\bfT ,\bfT ) + (\nabla T\scrJ ,\nabla T\scrJ ) = 0.
So, we can easily obtain that \varepsilon nergy1(\scrJ ) = 0.
Theorem 3.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA AND NONELASTICA . . . 1101
Theorem 3.2. Energy of lightlike Cartan curves of Bishop orthonormal vectors of Eq. (2) are
defined by using the Sasaki metric and Bishop curvatures given by the Eq. (2) in the following
manner:
\varepsilon nergy1(\bfT ) =
1
2
s\int
0
\pi 2
1ds,
\varepsilon nergy1(\bfN 1) = 0,
\varepsilon nergy1(\bfN 2) = 0,
\varepsilon nergy1(\bfN 3) = 0.
Proof. It is obvious if one considers Eqs. (4) – (6).
Corollary 3.1. \BbbE 4
1 be a lightlike elastic Cartan curve having the Bishop characterization given
by the Eq. (2), then we have the following relation:
\varepsilon nergy1(\bfT ) - \varepsilon nergy1(\scrJ ) =
s
2
,
\varepsilon nergy1 (\scrJ ) = \varepsilon nergy1 (\bfN 1) = \varepsilon nergy1(\bfN 2) = \varepsilon nergy1(\bfN 3).
Proof. If one uses Theorems 3.1 and 3.2 it gives the result immediately.
Case 2: Let \alpha \in \BbbE 4
1 be a lightlike elastic Cartan curve defined on any fixed interval [y1, y2] so
that \alpha : [y1, y2] \rightarrow \BbbE 4
1 and k1 = 1, k2, k3 be arbitrary functions.
Theorem 3.3. Let \alpha be a lightlike elastic Cartan curve with the Bishop characterization given
by the Eq. (3). If V is a vector field, which is an infinitesimal variation of the curve \alpha , then we have
constant vector field J and some restrictions as the following:
\scrJ =(\pi \prime
2 + \pi 2
2)\bfT + (\pi \prime
1 + \pi 1\pi 2)\bfN 1+( - \pi \prime
3 - \pi 2\pi 3)\bfN 3,
0 = \pi \prime
2 + \pi 2
2,
0 = \pi \prime
1 + \pi 1\pi 2,
0 = - \pi \prime
3 - \pi 2\pi 3.
Proof. If we follow a similar procedure as in Case 1 and use the characterization given in the
Eq. (3), it is obvious.
Thus, we have following sample solution family of the nonlinear ordinary differential equation
system for certain values of \pi 1, \pi 2, and \pi 3.
Theorem 3.4. Energy of lightlike Cartan curves of Bishop orthonormal vectors of Eq. (3) are
defined by using the Sasaki metric and Bishop curvatures given by the Eq. (3) in the following
manner:
\varepsilon nergy1(\bfT ) = 0,
\varepsilon nergy1(\bfN 1) = 0,
\varepsilon nergy1(\bfN 2) = 0,
\varepsilon nergy1(\bfN 3) = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1102 T. KÖRPINAR, R. C. DEMIRKOL
Proof. It is obvious if one considers Eqs. (4) – (6).
Corollary 3.2. Let \alpha \in \BbbE 4
1 be a lightlike elastic Cartan curve having the Bishop characterization
given by the Eq. (3), then we have the following relation:
\varepsilon nergy1(\bfT ) = \varepsilon nergy1(\scrJ ) = \varepsilon nergy1(\bfN 1) =
= \varepsilon nergy1(\bfN 2) = \varepsilon nergy1(\bfN 3).
Proof is obvious.
4. Energy on nonelastic lightlike Cartan curve in \BbbE 4
1 . In this section, we deal with the concept
of different types of nonelastic lightlike Cartan curves and their energies in \BbbE 4
1.
Case 1: Let \alpha \in \BbbE 4
1 be a lightlike Cartan curve defined on any fixed interval [y1, y2] so that
it has the Bishop characterization same as in the equation (2). For a vector field V, which is an
infinitesimal variation of the curve \alpha , by using Eqs. (7), (8) we get
0 =
\bigl\langle
\alpha \prime \prime , V \prime \bigr\rangle \bigm| \bigm| y2
y1
-
\bigl\langle
V, \alpha \prime \prime \prime \bigr\rangle \bigm| \bigm| y2
y1
+
y2\int
y1
\langle V, \scrE (\alpha )\rangle dt,
where \scrE (\alpha ) = \alpha \prime \prime \prime \prime . As opposed to (9), if we assume that the curve is nonelastic, then we have
\scrE (\alpha ) = (3\pi 2\pi
\prime
2 + \pi \prime \prime
2 + \pi 3
2)\bfT + (2\pi 1\pi
\prime
2 + \pi 1\pi
2
2 + \pi \prime
1\pi 2 + \pi \prime \prime
1)\bfN 1+3\pi 1\pi
\prime
1\bfN 2.
Theorem 4.1. Energy of lightlike nonelastic Cartan curve having the Bishop characterization
same as in equation (2) can be computed by using the Sasaki metric as follows:
\varepsilon nergy1
\bigl(
\scrE (\alpha )
\bigr)
=
1
2
s\int
0
((3\pi \prime
1\pi
\prime
2 + 3\pi 1\pi
\prime \prime
2 + 3\pi 1\pi 2\pi
\prime
2 + \pi \prime
1\pi
2
2+
+2\pi 1\pi
\prime
2 + \pi \prime \prime
1\pi 2 + \pi \prime \prime \prime
1 + \pi 1\pi
3
2)
2 - (3\pi
\prime 2
2 + 4\pi 2\pi
\prime \prime
2+
+6\pi \prime
2\pi
2
2 + \pi \prime \prime \prime
2 + \pi 4
2)(3\pi
\prime 2
1 + 4\pi 1\pi
\prime \prime
1 + 2\pi 2
1\pi
\prime
2 + \pi 2
1\pi
2
2 - 2\pi 1\pi
\prime
1\pi 2))ds.
Proof. It is obvious if we apply the Sasaki metric given by Eqs. (4) – (6) to the vector field
of \scrE (\alpha ).
Corollary 4.1. As stated in the Eq. (2), we have the following case for the energy of lightlike
nonelastic Cartan curve \alpha \in \BbbE 4
1 depending on Bishop and Frenet curvatures:
\varepsilon nergy1
\bigl(
\scrE (\alpha )
\bigr)
=
1
2
s\int
0
\biggl( \biggl(
- 3
\biggl(
1
2
\pi 2
2 + k2
\biggr) \prime
- 3\pi 2
\biggl(
1
2
\pi 2
2 + k2
\biggr)
-
- 2
\biggl(
1
2
\pi 2
2 + k2
\biggr)
+ \pi 3
2
\biggr) 2
-
\biggl(
- 4\pi 2
\biggl(
1
2
\pi 2
2 + k2
\biggr) \prime
- 6
\biggl(
1
2
\pi 2
2 + k2
\biggr)
\pi 2
2 -
-
\biggl(
1
2
\pi 2
2 + k2
\biggr) \prime \prime
+ \pi 4
2
\biggr) \biggl(
- 2
\biggl(
1
2
\pi 2
2 + k2
\biggr)
+ \pi 2
2
\biggr) \biggr)
ds.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA AND NONELASTICA . . . 1103
Case 2: Let \alpha \in \BbbE 4
1 be a lightlike Cartan curve defined on any fixed interval [y1, y2] so that it has
the Bishop characterization same as in equation (3). For a vector field V, which is an infinitesimal
variation of the curve \alpha , by using Eqs. (7), (8), we get
0 =
\bigl\langle
\alpha \prime \prime , V \prime \bigr\rangle + \bigl\langle V,\Gamma \alpha \prime - \alpha \prime \prime \bigr\rangle + y2\int
y1
\langle V, \scrE (\alpha )\rangle dt,
where \scrE (\alpha ) = \alpha \prime \prime \prime \prime . As opposed to (9), if we assume that the curve is not elastica then we have
\scrE (\alpha ) = (\pi \prime
2 + \pi 2
2)\bfT + (\pi \prime
1 + \pi 1\pi 2)\bfN 1+(\pi 2
1+\pi 2
2)\bfN 2 - (\pi \prime
3 + \pi 2\pi 3)\bfN 3.
Theorem 4.2. Energy of lightlike nonelastic Cartan curve having the Bishop characterization
same as in equation (3) can be computed by using the Sasaki metric as follows:
\varepsilon nergy2
\bigl(
\scrE (\alpha )
\bigr)
=
1
2
s\int
0
\Bigl(
(\pi \prime \prime
1 + \pi \prime
1\pi 2 + \pi 1\pi
\prime
2 + \pi 1\pi
2
2)
2+
+(\pi \prime \prime
3 + 2\pi \prime
2\pi 3 + \pi 2\pi
\prime
3 + \pi 2
2\pi 3)
2 -
- (\pi \prime \prime
2 + 3\pi 2\pi
\prime
2 + \pi 3
2)(3\pi 1\pi
\prime
1 + 2\pi 2\pi
\prime
2 - \pi 3
2 + \pi 3\pi
\prime
3 - \pi 2\pi
2
3)
\Bigr)
ds.
Proof. It is obvious, if we apply the Sasaki metric given by Eqs. (4) – (6) to the vector field
of \scrE (\alpha ).
Corollary 4.2. As stated in the Eq. (3), we have the following case for the energy of lightlike
nonelastic Cartan curve \alpha \in \BbbE 4
1 depending on Bishop and Frenet curvatures:
\varepsilon nergy2
\bigl(
\scrE (\alpha )
\bigr)
=
1
2
s\int
0
\Biggl( \Biggl(
- \mathrm{s}\mathrm{i}\mathrm{n} \theta + \mathrm{c}\mathrm{o}\mathrm{s} \theta
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr)
- \mathrm{s}\mathrm{i}\mathrm{n} \theta
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) \prime
+
+\mathrm{s}\mathrm{i}\mathrm{n} \theta
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) 2
\Biggr) 2
+
\Biggl(
- \mathrm{c}\mathrm{o}\mathrm{s} \theta - 2
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) \prime
\mathrm{c}\mathrm{o}\mathrm{s} \theta +
+\mathrm{s}\mathrm{i}\mathrm{n} \theta
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr)
+ \mathrm{c}\mathrm{o}\mathrm{s} \theta
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) 2
\Biggr) 2
-
-
\Biggl( \biggl(
- k3 - \theta \prime \prime
\theta \prime
\biggr) \prime \prime
+ 3
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) \biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) \prime
-
-
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) 3
\Biggr) \Biggl(
3 \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s} \theta + 2
\Biggl(
k3 + \theta \prime \prime
\theta \prime
\Biggr) \biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) \prime
-
-
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr) 3
- \mathrm{s}\mathrm{i}\mathrm{n} \theta \mathrm{c}\mathrm{o}\mathrm{s} \theta +
\biggl(
k3 + \theta \prime \prime
\theta \prime
\biggr)
\mathrm{c}\mathrm{o}\mathrm{s}2 \theta
\Biggr) \Biggr)
ds,
where 2\theta \prime (\theta \prime \prime \prime - k\prime 3) + 2\theta \prime \prime (k3 - \theta \prime \prime ) + \theta
\prime 4 - (k3 - \theta \prime \prime )2 - 2k2\theta
\prime 2 = 0 and \theta \prime \not = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1104 T. KÖRPINAR, R. C. DEMIRKOL
5. Application of the argument. In this section, we examine some special cases of lightlike
Cartan curves in the case of being elastic or nonelastic.
Example 5.1. Let \alpha be a lightlike Cartan curve in \BbbE 4
1 and k1 = 1, k2 = k3 = 0. Then modified
Bishop frame equations for orthonormal vectors \{ \bfT ,\bfN 1,\bfN 2,\bfN 3\} having the form in the Eq. (2) can
be stated as the following [12]:
\nabla T\bfT = \pi 2\bfT + \pi 1\bfN 1,
\nabla T\bfN 1 = \pi 1\bfN 2,
\nabla T\bfN 2 = - \pi 2\bfN 2,
\nabla T\bfN 3 = 0,
where \pi 1 = 1, \pi 2 =
2
s
, \pi 3 = 0.
If \alpha is a lightlike elastic Cartan curve, then
\varepsilon nergy1(\scrJ ) = 0.
If \alpha is a lightlike nonelastic Cartan curve, then
\varepsilon nergy1
\bigl(
\scrE (\alpha )
\bigr)
=
1
2
\int \biggl(
12
s3
- 12
s3
- 4
s2
+
8
s3
\biggr) 2
-
\biggl(
32
s4
- 48
s4
- 12
s4
+
16
s4
\biggr) \biggl(
- 4
s4
+
4
s2
\biggr)
ds,
which is equal to
\varepsilon nergy1
\bigl(
\scrE (\alpha )
\bigr)
= 16
\biggl(
3
7s7
- 7
5s5
+
1
s4
- 1
3s3
\biggr)
+ \scrC ,
where \scrC is a constant term and s \not = 0.
Example 5.2. Let \alpha be a lightlike Cartan curve in \BbbE 4
1 and k1 = 1, k2 = k3 = 0. Then modified
Bishop frame equations for orthonormal vectors \{ \bfT ,\bfN 1,\bfN 2,\bfN 3\} having the form in the Eq. (3) can
be stated as the following [12]:
\nabla T\bfT = \pi 1\bfN 1,
\nabla T\bfN 1 = \pi 1\bfN 2,
\nabla T\bfN 2 = 0,
\nabla T\bfN 3 = 0,
where \pi 1 = 1, \pi 2 = 0, \pi 3 = 0.
If \alpha is a lightlike elastic Cartan curve, then
\varepsilon nergy1(\scrJ ) = 0.
If \alpha is a lightlike nonelastic Cartan curve, then
\varepsilon nergy2(\scrE (\alpha )) = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
CURVATURE AND TORSION DEPENDENT ENERGY OF ELASTICA AND NONELASTICA . . . 1105
References
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Received 01.09.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
|
| id | umjimathkievua-article-847 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:52Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/bc/8bda5b57f09ead88ab4a5cabdeef9fbc.pdf |
| spelling | umjimathkievua-article-8472022-03-26T11:01:59Z Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space Körpinar, T. Demirkol, R. C. Körpinar, T. Demirkol, R. C. UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ by using the Bishop orthonormal vector frame and associated Bishop components.  Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space $\mathbb{E}_{1}^{4}$ and investigate its relationship with the energy of the same curve in Bishop vector fields in $\mathbb{E}_{1}^{4}$.  Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space $\mathbb{E}_{1}^{4}$. УДК 515.1 Спочатку описано умови пружності та непружності світлоподібних пружних кривих Картана у просторі Мінковського $\mathbb{E}_{1}^{4}$ за допомогою ортонормального векторного репера та відповідних компонент Бішопа.  Потім обчислено енергію пружних та непружних світлоподібних кривих Картана у просторі Мінковського $\mathbb{E}_{1}^{4}$ та вивчено іхній зв'язок із енергією тієї ж кривої у векторних полях Бішопа у $\mathbb{E}_{1}^{4}$.  Тут функціонали енергії обчислюються у термінах кривини Бішопа для світлоподібних кривих Картана, що належать простору Мінковського $\mathbb{E}_{1}^{4}$. Institute of Mathematics, NAS of Ukraine 2020-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/847 10.37863/umzh.v72i8.847 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 8 (2020); 1095-1105 Український математичний журнал; Том 72 № 8 (2020); 1095-1105 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/847/8741 |
| spellingShingle | Körpinar, T. Demirkol, R. C. Körpinar, T. Demirkol, R. C. Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_alt | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_full | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_fullStr | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_full_unstemmed | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_short | Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space |
| title_sort | curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the minkowski space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/847 |
| work_keys_str_mv | AT korpinart curvatureandtorsiondependentenergyofelasticaandnonelasticaforalightlikecurveintheminkowskispace AT demirkolrc curvatureandtorsiondependentenergyofelasticaandnonelasticaforalightlikecurveintheminkowskispace AT korpinart curvatureandtorsiondependentenergyofelasticaandnonelasticaforalightlikecurveintheminkowskispace AT demirkolrc curvatureandtorsiondependentenergyofelasticaandnonelasticaforalightlikecurveintheminkowskispace |