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1. Introduction
Magnetic curves have many applications in physics and differential geometry and play an important role in these areas. When a charged particle enters a magnetic field, the Frenet vectors of this particle are affected by this magnetic field and with this effect a force which is called the Lorenz force occurs. Thus, the particle starts to follow a trajectory in this magnetic field thanks to Lorenz force. This trajectory is called a magnetic curve. The motion of a particle entering the magnetic field with the effect of the Lorenz force is explained as; if the tangent vector field T is parallel to the magnetic field, the Lorentz force will be zero, so the particle moves parallel to the magnetic field. If the tangent vector field T is perpendicular to the magnetic field, the Lorentz force is maximum and the particle moves in a circle in the magnetic field. If the tangent vector field T is at a constant angle with the magnetic field, the particle follows a helical trajectory under the influence of the Lorentz force,1. These curves have attracted the attention of many authors in different disciplines. For this reason, many studies have been carried out by considering these curves in different ways2-9.
On the other hand, fractional analysis means derivative and integral accounts that are not integers. The phrase fractional derivative first appears in a letter sent by Leibniz to L’Hospital in 1695,10. In this letter, Leibniz is asked L’Hospital a question, “Can integer order derivatives be extended to fractional order derivatives?” Afterwards, this subject, which attracted the attention of many mathematicians, took part in many studies11-16. Today, the subject of fractional analysis become very popular and study by many researchers in different fields17-20. Since it is believed to be the better modelling the physical systems with fractional order derivative, they have many studies on this subject. Because, the classical derivative is beneficial to model the physical systems locally but the fractional order derivative is beneficial to model physical systems globally. Fractional analysis have many applications in many branches of science in recent years. The study of this subject by many mathematicians is led to the emergence of many different definitions of fractional derivatives and integrals. Riemann-Liouville, Caputo, Cauchy, and conformable fractional derivatives and integrals are just a few of these definitions. Different fractional derivative and integral definitions naturally brought with them different properties. For example, the derivative of zero is not constant for many types of fractional derivatives, except for the conformable fractional derivative and Caputo fractional derivative. Moreover, except for the conformable fractional derivative, other fractional derivatives do not have features such as the derivative of the product, the derivative of the quotient, or the chain rule, as in the classical sense21. In addition, the conformable fractional derivative is the local fractional derivative, unlike the Riemann-Liouville and Caputo fractional derivative. Conformable fractional derivative has many critical aspects, as it is equivalent to a simple change of variables for differentiable function22. However, the effect of conformable fractional derivatives and integrals on some physical phenomena is worth investigating. It will be interesting that fractional derivatives do not have a geometric interpretation as in the classical sense. However, there are many mathematicians investigating the effect of fractional calculus on differential geometry23-25.
In this study, the effects of conformable fractional derivatives and integrals on magnetic curves are investigated. In addition, a geometric inference is tried to be obtained with the help of examples. Moreover, we are visualize their images for different fractional values using the Mathematica program. The difference of this study from others is the use of conformable fractional derivatives and integrals in calculations. Fractional derivatives and integrals are more precision than ordinary derivatives and integrals because they give more accurate results. So, fractional calculus has applications in many different fields such as physics, engineering, mathematical biology, etc. This article is a complicated study that includes differential geometry, physics and fractional analysis fields. So, we hope that it will contribute to those working in these fields.
2. Preliminaries
2.1. Basic definitions and theorems of differential geometry
In section definitions and theorems, the curves in ℝ3 will be introduced in a nutshell.
Definition 1. Let the curve x(s) be given in n-dimensional Euclidean space with (I; α) coordinate neighborhood. The arc length of the curve x from a to b, is calculated as
which is the length between the points x(a) and x(b) of the curve. The parameter s is said to be arc-length.
Theorem 1. Let x = x(s) be a regular unit speed curve in the Euclidean 3-space where s measures its arc length. Also, let T = x’ be its unit tangent vector, N = T’/||T0’|| be its principal normal vector and B = T(N be its binormal vector. The triple {T;N;B} be the Frenet frame of the curve x: Then the Frenet formula of the curve is given by
where ((s) = ||d2x=ds2|| and ((s) =(dN/ds, B〉 are curvature and torsion of x respectively26.
Definition 2. Let x: I ⊂ ℝ → E3 be a unit speed curve in Euclidean 3-space E3. If any U fixed direction with the unit tangent vector of the curve x makes a fixed angle, the curve x is called the general helix27. The most well-known characterization of the helix curve is (/( = constant (Lancret theorem)26.
Definition 3. Let x: I ⊂ ℝ → E3 be a unit speed curve in Euclidean 3-space E3. If any U fixed direction with the principal unit normal vector of the curve x makes a fixed angle, the curve x is called the slant helix. Izumiya and Takeuchi obtain a necessary and sufficient condition for a curve to be slant helix: a curve is an oblique propeller if its geodetic curvature and the principal normal satisfy the expression
is constant finction28.
2.2. Basic definitions and theorems of conformable fractional calculus
In this part, some basic definitions and theorems of conformable fractionally derivative and integral are given.
Definition 4. Let us give a function ƒ : [0, ∞) → R. Then the conformable fractional derivative for f of order α is defined by
for all s > 0, 0 < α < 1. If f is α-differentiable in some (0, a), a > 0 and lim s→0 + f(α) (s) exist, then define f(α) (0) = lims→0 + f(α) (s),29.
Theorem 2. Let f : [0, ∞) → R be a function. If a function ƒ is α-differentiable at s0> 0, 0 < α <1, then ƒ is continuous at s0,29.
Accordingly, it is easily visible that the conformable fractional derivative provides all the properties given in the theorem below.
Theorem 3. Let f, g : [0, ∞) → R be α-differentiable at each s > 0, 0 < α < 1. Then
(1) D((aƒ + bg)(s) = aD((f)(s)+bD((g)(s); for all a; b ∈ R.
(2) D(λsp ) = psp-( ; for all p ∈ R.
(3) D(λ) = 0; for all constant functions ƒ(s) = λ.
(4) D((ƒg)(s) = ƒ(s)D((g)(s) + g(s)D((ƒ)(s).
(5)
(6) If ƒ is a differentiable function, then
Theorem 4. Let f, g : [0, ∞) → R be α−differentiable at s0> 0, 0 < α < 1. If (f ◦ g) is α−differentiable and for all s with s ≠ 0 and f (s) ≠ 0, the equation
D((ƒ ◦ g)(s) = ƒ(s)(α-1 D(ƒ(s)D((g)(ƒ( s));
is provided,30.
Definition 5. Let f : [0, ∞) → R be a function. The expression
is called a conformable fractional integral, where α > 0,30.
Theorem 5. Let ƒ : [a; ∞) → R be a function. Then for all s > 0 the following equation exists,30
2.3. Basic definitions and theorems of magnetic field and curves
In this subsection, some basic definitions and theorems of magnetic field and magnetic curve are introduced.
Let M be a (n ≥ 2)-dimensional oriented Riemannian manifold. The Lorenz force of a magnetic field F on M is defined to be a skew symmetric operator 𝜙 given by
for all X, Y ∈ (M), where Χ(M) is the space of vector fields. The magnetic trajectories of F are curves x on M which satisfy the Lorenz equation
The mixed product of the vector fields X, Y, Z ∈ Χ(M), is defined by
Let V be a Killing vector field on M and Fv = ιv dvg be the corresponding Killing magnetic field, where ι is denoted the inner product. Then, the Lorentz force of the Fv is
Consequently, the Lorentz force equation may be written as
A unit speed curve x is a magnetic trajectory of a magnetic field V if and only if V can be written along x as
where the function w(s) associated with each magnetic curve will be called its quasislope measured with respect to the magnetic field V,31.
Proposition 1. Let x : I ⊂ R → M3 be a curve in a 3D oriented Riemannian Manifold (M3; g) and V be a vector field along the curve x. One can take a variation of x in the direction of V, say, a map Γ : I x(-ε, ε) → M3 which satisfies Γ(s; 0) = x(s); (d(/ds)(s; k) = V(s): In this setting, we have the following funtions:
(1) The speed funtion
(2) The curvature function κ(s, k) of x(s),
3) The torsion function ((s,k) of x(s). The variations of those functions at k = 0 are
where R is the curvature tensor of M3,31.
Proposition 2. Let V (s) be the restriction to x(s) of a Killing vector field, say V of M3, then,31
Definition 6. Let x : I ⊂ R → M3 be a curve in 3D oriented Riemannian space (M3 , g) and F be a magnetic field on M. We call the curve x is a T-magnetic curve if the tangent vector field of the curve satisfy the Lorentz force equation, that is,31
Definition 7. Let x : I ⊂ R → M3 be a curve in 3D oriented Riemannian space (M3 , g) and F be a magnetic field on M. We call the curve x is a N-magnetic curve if the normal vector field of the curve satisfy the Lorentz force equation, that is,32
Definition 8. Let x : I ⊂ R → M3 be a curve in 3D oriented Riemannian space (M3 , g) and F be a magnetic field on M. We call the curve x is a B-magnetic curve if the binormal vector field of the curve satisfy the Lorentz force equation, that is,32
2.4. Basic definitions and theorems of conformable fractional curves
In this part of the preliminaries section, we present brief information about conformable curves using conformable fractional derivative.
Definition 9. Let x = x(s) be a curve. If x : (0,∞) → R3 is (-differentiable curve, then x is called a conformable curve in R 3 ,33.
Definition 10. Let x : (0,∞) ( R3 be a conformable curve in R3 . Velocity vector of x is determined by
for all s ∈ (0,∞),33.
Definition 11. Let x: (0,∞) → R3 be a conformable curve in R3: Then the velocity function v of x is defined by
for all s ∈ (0,∞),33.
Definition 12. Let x: (0,∞) → R3 be a conformable curve in R3 . The arc length function s of x is defined by
for all s ∈ (0,∞). If v(s) = 1 for all s0 ∈ (0,∞), it’s said that x has unit speed,33.
Conclusion 1. Let x: (0,∞) → R3 be a conformable curve in R3. The concepts velocity vector, velocity function and arc length function obtained according to conformable fractional derivative are equivalent to the standard concepts.
Definition 13. Let x be a conformable curve. If D((x)(s) ≠ 0 for all s ∈ (0,∞), x is called a conformable regular curve,33.
Definition 14. Let x = x(s) be a regular unit speed conformable curve in 3D Riemannian manifold where s measures its arc length. Also, let t = D((x)(s)s(-1 be its unit tangent vector, n = D((t)(s) / ||D((t)(s)|| be its principal normal vector and b = t(n be its binormal vector. The triple {t, n, b} be the conformable Frenet frame of the curve x(s). Then the conformable Frenet formula of the curve is given by
where κ((s) = || D((t)(s) || and ((((s) = (D((n)(s), b(s)〉 are curvature and torsion of x, respectively.
Conclusion 2. Let x = x(s) be a regular unit speed conformable curve where s measures its arc length. The following relation exists between the curvature and torsion of x according to Frenet frame and the conformable curvature and torsion of x according to conformable Frenet frame as
Conclusion 3. Let x = x(s) be a regular unit speed conformable curve where s measures its arc length. As can be seen fromEq. (13), when x is a unit speed curve, the conformable derivative has no effect on the Frenet frame, so the Frenet elements do not undergo any change. However, consideringEqs. (14)and(15); the curvature and torsion of the x curve has changed under the conformable fractional derivative.
3. Main results
3.1. Fractional t-magnetic curves
In this subsection, we define the fractional t-magnetic curve with a conformable fractional derivative focus. We are also obtained some characterizations of this curve.
Definition 15. Let x : I ⊂ R → M3 be a conformable curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M: If the vector area of the tangent curve of x with respect to the conformable frame satisfies the Lorenz force equation, the curve x is called fractional t-magnetic curve, that is
Proposition 3. Let x : I ⊂ R → M3 be a unit speed fractional t-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, ((, ((}. Then, we have the Lorenz force according to conformable frame as
where Ω1 is a certain function.
Proof. Let x : I ⊂ R → M3 be a unit speed fractional t-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, ((, ((}. Since 𝜙(t) ∈ Sp{ t, n, b}, we get
and thus
From the above equations, we can write
Similarly, we can easily calculate that
This completes the proof. ■
Proposition 4. Let x be a unit speed fractional t-magnetic trajectory of a magnetic field V if and only if V can be written along the curve x as
This completes the proof. ■
Theorem 6. Let x be a unit speed fractional t-magnetic trajectory and V be a Killing vector field on a simply connected space form (M3, g). Then the following equations exist
and
where C is the curvature of the Riemanian space M3
Proof. Let V be a magnetic field in a Riemanian 3D manifold. If the α-th conformable fractional derivative of Eq. (17) is taken with respect to s and conformable frame formulas are applied, we have
It can be easily seen that if V(v) = 0 of Proposition 1, the case is g(D(,V,t) = 0. So, if this equation is used in the above equation,
is obtained. If the conformable derivative of the above equation with respect to s is taken once again from the (-th order and conformable frame formulas are applied, we have
If the above equation is adjusted, we get
Then, if V(v) = 0 in Proposition 1 and Eqs. (9), (10) and (20) are considered in Eq. (18), following equation is obtained
where it is clear that s1-α ≠0. So, as can be clearly seen
Thus, the first part of the theorem is proved. Then (18) and (20) are considered with V(κ)=0 in Proposition 1, we obtain
In particular, if M3 has constant curvature C, then
and so,
and
If the above equation is arranged, we have
Thus, the second part of the theorem is proved. Similarly (18) and (20) are considered with V(τ)=0 in Proposition 1, we obtain
Hence, if M3 has constant curvature C, then
and
So, the last part of the theorem is proved and the proof is completed. ■
Corollary 1. Let x be a unit speed fractional t-magnetic curve in 3D oriended Riemanian manifold (M 3, g). If the function Ω1 is a zero and kα is non-zero constant function, then the curve x is a helix or circle. Moreover, the axis of the helix is the vector field V.
Proof. We assume that x be a fractional t-magnetic curve in 3D Riemann space with Ω1 is a zero and kα is non-zero constant function, then from Eq. (23), we get
and
If necessary algebric operations are done, we obtain
and
Finally, if the above equation is arranged, we get
Since kα is non-zero constant function, we get
■
Remark 1. The conformable derivative for differentiable functions is equivalent to a simple change of variable. Precisely, u = xα/α. It should be noted that a criticism of the conformable derivative is that, although conformable at the limit α→1, it is not conformable α→1. From the point of view of the assertion about the equality of the conformable derivative to a change of variables, one can say that the conformable derivative is not conformable as at the other limit α→0 because tα/α is undefined at α = 0,22.
3.2. Fractional n-magnetic curves
In this section, we redefine the n-magnetic curve with a conformable fractional derivative focus. We are also obtained some characterizations of this curve.
Definition 16. Let x : I ⊂ R → M3 be a conformable curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M. If the vector area of the tangent curve with respect to the conformable frame satisfies the Lorenz force equation, the x curve is called fractional n-magnetic curve, that is
Proposition 5. Let x : I ⊂ R → M3 be a unit speed fractional n-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, kα, kα}. Lorenz force eqations in the conformable frame are written as
where Ω2 is a certain function.
Proof. Let x : I ⊂ R → M3 be a unit speed fractional n-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, kα, kα}. Since 𝜙t ∈ Sp{t, n, b} we get
and thus
From the above equations, we can write
Similarly, we can easily calculate that
This completes the proof. ■
Proposition 6. Let x be a unit speed fractional n-magnetic trajectory of a magnetic field V if and only if V can be written along the curve x as
Proof. Let x be a unit speed fractional n-magnetic trajectory of a magnetic field V if Using Proposition 3 and Eq. (6), we can easily see that
This completes the proof. ■
Theorem 7. Let x be a unit speed fractional n-magnetic trajectory and V be a Killing vector field on a simply connected space form (M3, g). Then the following equations exist
where C is the curvature of the Riemanian space M3.
Proof. Let V be a magnetic field in a Riemanian 3D manifold. If the α-th conformable fractional derivative of Eq. (26) is taken with respect to s and conformable frame formulas are applied, we have
It can be easily seen that if V(v) = 0 of Proposition 1, the case is g(DαV, t) = 0. So, if this equation is used in the above equation, we get
is obtained. If the conformable derivative of the above equation with respect to s is taken once again from the α-th order and conformable frame formulas are applied, we have
If the equation is arranged, we obtain
Then, if V(v) = 0 in Proposition 1 and Eqs. (9), (10) and (11) are considered in Eq. (27), we have
Thus, the first part of the theorem is proved. Then Eqs. (27) and (28) are considered with V(k)=0 in Proposition 1, we obtain
In particular, if M3 has constant curvature C, then
and so,
Thus, the second part of the theorem is proved. Similarly Eqs. (27) and (28) are considered with V(k)=0 in Proposition 1, we obtain
Hence, if M3 has constant curvature C, then g(R(V, t)t, b) = Cg(V, b) = Ckα and g(R(V, t)n, b) = 0. So we obtain following
Thus, the last part of the theorem is proved and the proof is completed. ■
Corollary 2. Considering Ω2 is a non-zero constant function, we easily see that the fractional n-magnetic curve is a curve in the Euclidean 3-space.
Corollary 3. Let x be a unit speed fractional n-magnetic curve in 3D oriented Riemann manifold (M 3 , g). If the function Ω2 is non-zero constant, then the curve x is a slant helix. Moreover, the axis of the slant helix is the vector field V
Proof. We assume that x is a fractional n-magnetic curve in Euclidean 3-space with non-zero constant function Ω2, then from (29), (30) and (31), we have
If the above equation is arranged, we get
If necessary arregements are made, we obtain
and
These complete the proof. ■
3.3. Fractional b-Magnetic Curves
In this section, we define the b-magnetic curve with a conformable fractional derivative focus. We are also obtained some characterizations of this curve.
Definition 17. Let x : I ⊂ R → M3 be a conformable curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M. If the vector area of the tangent curve with respect to the conformable frame satisfies the Lorenz force equation, the x curve is called fractional b-magnetic curve, that is
Proposition 7. Let x : I ⊂ R → M3 be a unit speed fractional b-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, kα, kα}. So, Lorenz force according to the conformable frame is written as
where Ω3 is a certain function.
Proof. Let x : I ⊂ R → M3 be a unit speed fractional b-magnetic curve in 3D oriented Riemannian space (M3, g) and F be a magnetic field on M with the conformable frame elements {t, n, b, kα, τα}. Since 𝜙(t) ∈ Sp{t, n, b} we get
and thus
From the above equations.,we can write
Similarly, we can easily calculate that
This completes the proof. ■
Proposition 8. Let x be a unit speed fractional b-magnetic trajectory of a magnetic field V if and only if V can be written along the curve x as
Proof. Let x be a unit speed fractional b-magnetic trajectory of a magnetic field V. Using Proposition 3 and Eq. (6) we can easily see that
This completes the proof. ■
Theorem 8. Let x be a unit speed fractional b-magnetic trajectory and V be a Killing vector field on a simply connected space form (M3, g). Then the following equations exist
Proof. Let V be a magnetic field in a Riemanian 3D manifold. If the α-th conformable fractional derivative of Eq. (33) is taken with respect to s and conformable frame formulas are applied, we have
It can be easily seen that if V(v) = 0 of Proposition 1, the case is g(DαV, t) = 0 So if this equation is used in the above on,
is obtained. If the conformable derivative of the above equation with respect to s is taken once again from the α-th order and conformable frame formulas are applied, we have
If the equation is arranged, we get
Then, if V(v) = 0 in Proposition 1 and Eqs. (9), (10) and (11) are considered in equation (34), we have
Thus, the first part of the theorem is proved. Then equations (34) and (35) are considered with V(k) = 0 in Proposition 1, we obtain
In particular, if M3 has constant curvature C, then g(R(V, t)t, n) = Cg(V, n) = 0 and so following equation
is obtained. Thus, the second part of the theorem is proved. Similarly if the Eqs. (34) and (35) are considered with V(k) = 0 in Proposition 1, we obtain
Hence, if M3 has constant curvature C, then g(R(V, t)t, b) = Cg(V, b) = Ckα and g(R(V, t)n, b) = 0 So, we have following
Thus, the last part of the theorem is proved and the proof is completed. ■
Corollary 4. Let x be a fractional b-magnetic curve in 3D oriented Riemanian manifold (M3, g). If the function Ω3 is a constant function, then the curve x is a general helix. Moreover, the axis of the general helix is the vector field V.
Proof. We assume that x be a unit speed fractional b-magnetic curve in Euclidean 3-space with Ω3 is a constant function. Then from the equation (37), we get
and
Since Ω3 is a constant function, it can be say that kα is a constant function. In addition, considering the equation (36), the following equation can be easily seen
This completes the proof. ■
Example 1. Let x be a fractional t-magnetic trajectory of a magnetic field V. If the tangent vector field t is perpendicular to the magnetic field, the Lorentz force is maximum and the moves by the particle for different α values are given in Figs. 1 and 2 in the magnetic field.
Figure 1 Fractional t-magnetic curve x(s) for α→1(Black); α = 0.9(Blue) and α = 0.7(Red), respectively.
Figure 2 Fractional t-magnetic curve x(s) for α = 0:5(Orange), α = 0:3(Purple) and α = 0:1(Green), respectively
Example 2. Let x be a fractional t-magnetic trajectory of a magnetic field V. From Corollary 1, we can easily see that Ω1 = 0 and kα is a constant function. The figure of the t-magnetic curve for different α values are given in Figs. 3 and 4 in the magnetic field.
Figure 3 Fractional t-magnetic curve x(s) for α→1(Black), α = 0.9(Blue) and α = 0.7(Red), respectively.
Figure 4 Fractional t-magnetic curve x(s) for α = 0.5(Orange), α = 0.3(Purple) and α = 0.1(Green), respectively.
Example 3. Let x be a fractional n-magnetic trajectory of a magnetic field V: From Corollary 3, we can easily see that Ω2 is non-zero constant. The moves by the particle for different α values are given in Figs. 5 and 6 in the magnetic field.
Figure 5 Fractional n-magnetic curve x(s) for (( 1(Black), ( = 0.9(Blue) and ( = 0.7(Red), respectively
4. Conclusion
In this article, starting from the effect on the curves the effects of conformable fractional derivatives and integrals on magnetic curves are investigated. The Frenet frame has been tried to be formed with the help of conformable derivative of a unit speed conformable curve. However, as can be seen from Eq. (14), the Frenet frame of the unit speed curve is not affected by the conformable derivative, that is, the elements of the Frenet frame have not undergone any change under the conformable derivative. By U. Gözütok et al. are mentioned in article33, the physical properties (velocity, speed, arc-length) of the unit speed conformable curve do not change under the conformable derivative. On the other hand, curvature and torsion concepts are one of the most important factors in determining the characterization of the curve, as those who work on the theorem of curves, which is one of the sub-branches of differential geometry, know very well. Therefore, the difference of this study from the others is that the curvature and torsion of a curve are obtained depending on the fractional derivative. As can be seen from Conclusion 2, the curvatures of the conformable curve have changed under the conformable derivative. In this study, this change in the curvature of the curve is examined and visualized with various examples to better understand the results.
