A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, X-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X-$$\end{document}equilibrium curves, X-moment conservative curves, X-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X-$$\end{document}gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where X∈Ts,Ns,Bs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$\end{document} and we examine these curves and we give their properties.


Introduction
From the physical point of view, the concepts of linear velocity and angular velocity and hence the concepts of angular moment and torque are very important in the dynamics of the motion of a body in a certain orbit. In terms of physics, unless an external force is applied to an object, the object is stationary and will not have a velocity and an orbit, therefore its moment vector and torque vector will not occur. When a force is applied to the object, the object will have a linear velocity and hence linear momentum, angular velocity and angular moment relative to a reference point taken in space. In addition, an orbit and angular moment of the object relative to the reference point and a torque vector relative to the position of the object will occur. The linear moment of a body of mass m with its linear velocity v is P = mv , and the moment vector (or angular moment vector) of the object relative to the point taken, with its position r relative to the reference point taken, is defined by L = r × P . Torque vector is defined as the change of the angular moment of the object with respect to time during the motion of the object as Γ = dL dt such that where dr dt = v, dP dt = F and since P and v are linearly dependent then we have Γ = r × F [1]. Γ is the torque vector caused by the force F acting on the object with respect to the reference point and its magnitude is the size of the torque vector. If the torque is zero, angular momentum is conserved during the movement, if the torque vector and angular moment vector are linearly dependent, the body rotates only in the plane perpendicular to the torque vector with respect to the reference point, if the torque vector and angular moment vector are perpendicular, the body has a gyroscopic effect [2]. There are three types of equilibrium in physics which are called stable, unstable and neutral systems. If the direction of the displacement vector of an object and the directions of the torque vector are opposite, the body is in stable equilibrium in its orbit. If the direction of the displacement vector of an object and the directions of the torque vector are the same, it means that the object is in unstable equilibrium in its orbit. If the balance of the object is independent of the displacement vector from its original position, it is said that the object is in a neutral equilibrium state in its orbit [3]. If we look at the subject in terms of differential geometry, we assume that every regular (s) curve is an orbit of a moving body with unit mass and with unit velocity which is the same of the drawing velocity of the curve. The body has a helix motion at every point of the curve. This motion is called Frenet motion [4]. This motion is the movement of linear independent orthonormal T(s) , N(s) and B(s) vectors (which are called tangent, principal normal and binormal) at each point of the curve. During this movement, it has a curvature (s) and torsion (s).
In this study, we will define and examine new curves by using the physical vector quantities such as torque, equilibrium moment conservative and gyroscope mentioned above in the theory of curves in differential geometry.
Let (s) be a regular curve with arclenght parameter and with the Frenet vectors T(s) , N(s) B(s) and with the curvature (s) and torsion (s) . The variation of the Frenet frame according to time is defined as where (s) = (s) (s) and the functions. f (s) , g(s) and h(s) are at least C 0 −functions and so the position vector of the curve can be given as Here, the functions f (s) , g(s) and h(s) satisfy the relations  of the curve  are called normal, rectifying and osculatory planes, respectively. An (s) curve is called a normal, rectifying, and osculatory curve if the position vector lies in the normal, rectifying, and osculatory plane, respectively [6,7]. Helices are known as famous Lancret's curves and occupy a very special place in the theory of curves. For a curve to be a generalized helix, the necessary and sufficient condition is (s) = constant. Venant first proved this characterization [8][9][10]. In particular, if curvature and torsion are constant, it is a curved circular helix. If a curve is a linear combination of constant multiples of Frenet vectors of another curve, the curve is called a Smarandache curve [11]. Tuncer, defined and examined the moment vectors ( T− dual, N−dual and B−dual curve) of the curve with respect to the origin of the vector by using T(s) , N(s) , B(s) vector and by using the position vector of the curve [5]. Şenyurt et al. examined the vectorial moment of the unit Darboux vector [12]. Şenyurt and Çalışkan expressed and examined the vectorial moments in terms of alternative frame and they applied these to ruled surfaces [13]. If T−dual, N−dual and B−dual curves of a curve are denoted by L T (s) , L N (s) and L B (s) according to the origin, then these curves are defined as respectively [5].
On the other hand, while a body with the unit mass and with the unit speed moving along the curve (s) , it has displacements on the T−direction, N−direction and B− direction at every moment of the motion due to the curvature and torsion of the (s) which are geometrically deflected from the tangent direction (or rotation around the center of curvature) and deflected from the osculatory plane (or rotation around the torsion center).
Throughout this study, external forces such as gravity and friction were neglected.

T-torque curve, T-equilibrium and T-gyroscobic curve of a space curve
In displacement motion on T−direction along (s) , from equations (3) ve (7), T−dual curve according to origin is and as s changes, angular moment vector changes, T−torque vector is The position vector Γ T s * always lies in rectifying plane of the curve (s) and it is perpendicular to the principal normal vector. Let the parameter s * be arc lenght parameter of T−torque curve then there is the relation between the arclenght parameters of both the T−torque and the curve (s) . As the parameter s changes in an interval I ⊂ IR , the condition for the vector Γ T s * to construct a regular curve is that   (4) and (6), we get c 0 = h(s) = 0 then, this requires the curve (s) to be a planar curve. For this reason, we can say that there are no any curves, whose T−torque vectors are constant. For the same reason, Γ T s * is not a Smarandache curve of (s).
When the T−torque and T−moment vectors are perpendicular, then from (4) and (11), for (s) ≠ 0 , we getg(s)f (s) = 0 , hence we can say all the rectifying and normal curves are T−gyroscobic curves.
If the T−torque and T−dual(angular moment) vectors are linearly dependent then from (4) and (11), for (s) ≠ 0 , we get h(s) = 0 , −c 0 g(s) = f (s) (s) and from (5) (5) and (6) we obtain also, we get the solution is In this case, the position vectors of (s), L T (s) and Γ T s * are (11), then T−torque vector doesn't have any components on T(s) , in this case (s) is a neutral T−equilibrium curve. We can give the following theorem.
Theorem 1 For a regular space curve (s) , the followings are true. The curve (s) is stable or unstable T−equilibrium curve (for = −1 or = 1 respectively) if and only if its position vector is vi. Osculating curves are neutral T−equilibrium curves.
In the case that (s) is a generalised helix, then (s) = = constant and from (5) and we get the solution also from (4) and (6), we get where q(s) = e (1+ 2 ) ∫ r(s)ds and r(s) = (s) 3 − (s)+ � (s) . In the special case of (s) is a circular helix then Thus, we give the following remark. ii. If (s) is a circular helix then its T−dual curve is

N-torque curve, N-equilibrium and N-gyroscobic curve of a space curve
In displacement motion on N-direction along (s) , from (3) and (8)

s) (s)g(s)T(s) − (s)Q(s)N(s) +(1 + (s)g(s))B(s)
Definition 6 Let (s) be a regular space curve, the curve (s) is called N−gyroscobic curve, if the vectors Γ N s * and L N (s) are perpendicular at each point of the curve (s).
If there is no change of N−dual(angular moment) vector, then Γ N s * = 0 , in this case, from (14), (s) = 0 , g(s) = − 1 (s) and h(s) = 0 , and from (4), f (s) = c 0 so the curve (s) is a planar curve, from (5), we obtain the differential equation The solution is (s) =  (4) and (6), also we get and Therefore, if we rewrite (g(s) (s)) � (s) (s) = (s)Q(s) in (17), then we obtain , we get and also Thus, from (19) and (20) In this case, (s) is a generalised helix and then, we give the following theorem.
Theorem 2 Let (s) be a regular space curve in E 3 , then the followings are true.

i. The curve (s) is N−moment conservative if and only if its position vector is
ii.
There is nospace curve, whose N−torque vector is a constant vector, except lines in E 3 , iii. If the N−torque curve of (s) is a Smarandach curve then (s) is a generalised helix and its position vector is If position vectors of N−torque and N−dual curves are linearly dependent then at first, from (13) and (14), later by using (4), (5) and (6), we obtain f (s) = c 0 e s ,   (13) and (14), we have by using (4) and (6), we get Here, we can take f (s) = c 0 cos (s) and h(s) = c 0 sin (s) , where (s) is the angle between the position vector of (s) and T(s) . Therefore, from (4) and (6) and for � (s) ≠ 0 , from (5) we obtain so the solution is Let the vectors Γ N s * , N(s) be linearly dependent such as Γ N s * = N(s) for = ±1 . We have (s)g(s) = 1 + (s)g(s) = 0 and then we obtain (s) = 0 , g(s) = −1 (s) , from (4) and (5), we obtain f (s) = c 0 and The solution is g(s) = ± √ c 1 + 2c 0 s . Also from (6)   On the other hand, it is easy to see that since Γ B s * , B(s) cannot be inearly dependent such that Γ B s * = ±B(s) , then there are no stable or unstable B−equilibrium curves. From (25), if f (s) = 0 then binormal vector is perpendicular to Γ B s * , so we can say all the normal curves are neutral B−equilibrium curves. Thus, we give the following theorem.
Theorem 4 Let (s) be a space curve then, the followings are true.

i.
There are no any B−moment conservative space curves in E 3 , ii.
There are no any space curves whose B−torque vectors are constant vectors, except lines in E 3 , iii. if Γ B s * is a Smarandach curve, then position vector of (s) is

v.
There is no any space curve which are stable or unstable B−equilibrium in E 3 , vi. All the normal curves are neutral B−equilibrium curves.

Conclusions
The physical investigation of the movement of the Frenet frame along the curve in the theory of curves has led to the emergence of new curves, both in terms of pairs of curves and by incorporating physical concepts into the calculations. Especially with the inclusion of physical concepts in the calculations of Frenet motion, X-moment curves, X-torque curves, X-gyroscopic curves, X-equilibrium curves and X-moment conservative curves entered the active field of study as new curves, where X ∈ {T(s), N(s), B(s)} . All the curves presented in this article can be described and analyzed by the position vector of a curve. All rectifying curves and normal curves are T-gyroscopic curves, osculating curves are neutral T-equilibrium curves, if the curve is N-torque curve then it is a Smarandache curve, and in the case the curve is generalized helix, normal curves are also neutral B-equilibrium curves, which are