Motion along a Space Curve with a Quasi-Frame in Euclidean 3-Space: Acceleration and Jerk

The geometry of motion in a 3-dimensional space has constantly been crucial in terms of understanding physical phenomena. Specifically, the mathematical description of a particle’s motion as a consequence of geometrical analysis is a subject that is required in a wide variety of applications, such as water wave theory, relativity, non-linear optics, sigma models, fluid dynamics, and so on. In Newtonian physics, the force acting on a particle, as defined by the equation $F=m\mathbb{A}$, is proportional to its acceleration $\mathbb{A}$. In certain cases, it is simpler to deal with the acceleration vector when decomposing it into normal and tangential components. However, it is more practical to represent the acceleration vector as the sum of its tangential and radial components when the angular momentum of the particle is constant.

In 1879, for the osculating plane to the curve, Siacci [1] described the acceleration vector as the sum of two particular oblique components. Subsequently in 1944, Whittaker [2] dealt with Siacci’s theorem for the plane and gave a geometrical proof of it. Although Siacci’s formulas are very remarkable, his formulation of the theorem is inaccurate and his proof is burdensome. Therefore, in 2011, Casey [3] used the Serret–Frenet frame to prove Siacci’s theorem in space. Subsequently, in the Finsler manifold ${\mathbf{F}}^3$, in 2012, Küçükarslan et al. [4] investigated Siacci’s theorem for curves. In 2017, Özen et al. [5] investigated Siacci’s theorem for the curves on regular surfaces in $E^3$ according to the Darboux frame. Recently, in 2020, Özen et al. [6] studied Siacci’s theorem for the curves in $E^3$ according to the modified orthogonal frame. In the same year, Özen [7] studied Siacci’s theorem for the curves in Minkowski 3-space by using the Serret–Frenet frame.

In contrast, the jerk vector $\mathbb{J}$ is the time derivative of the acceleration vector. Thus, for a particle with a constant mass, the equality $\mathbb{J}=\left(1/m\right)\left(\mathrm{d}F/\mathrm{d}t\right)$ is satisfied. In the literature, there has been considerable interest in the resolution of the jerk vector for the curves in $E^3$, and several methods and frames for studying the resolution of the jerk vector have been proposed, for example, the Serret–Frenet frame [8,9], the modified orthogonal frame [6], and the Bishop frame [10]. Acceleration varies abruptly when a machinist operates a high-speed train, a stock-car racer races on a track, or a gymnast does gymnastic exercises. Estimating the lower threshold of merely an observable shock and the highest values of the jerk that humans can tolerate without undue discomfort is critical in these situations (see [11]). In addition, in 2017, Tsirlin [12], using the jerk vector formula, gave the maximum permissible speed on a space curve at all trajectory points.

The Serret–Frenet frame is inadequate for studying the space curves in which the curvatures have discrete zero points as, in this case, the principal normal and binormal vectors are discontinuous at points of inflections or along the straight sections of the curve. Therefore, to solve this problem, Dede et al. [13] introduced a new adapted frame along a space curve as an alternative frame to the Serret–Frenet frame and denoted this as the quasi-frame. Numerous studies on the quasi-frame have been discussed; see for example, optical Hasimoto map [14], Berry phase of the linearly polarized light wave along an optical fiber and its electromagnetic curves [15], magnetic flux flows with Heisenberg ferromagnetic spin [16], and evolution of the ruled surfaces [17].

Motivated by these papers, we consider a particle moving on a space curve according to a quasi-frame in the Euclidean 3-space under the influence of arbitrary forces.

The paper is organized as follows: In Section 2, we present some basic definitions concerning the Serret–Frenet frame and quasi-frame in the Euclidean 3-spac $E^3$ and the relation between them. In Section 3, we resolve the acceleration vector $\mathbb{A}$ and the jerk vector $\mathbb{J}$ of a particle moving on a space curve according to the quasi basis. Moreover, we give alternative resolutions of acceleration and jerk vectors. In Section 4, we provide informative examples to demonstrate how our results work.

We give some preliminaries in this part that will be used in our later discussion.

The Euclidean space $E^3=\left({\mathbb{R}}^3,〈·,·〉\right)$ is a metric space with the standard inner product $〈·,·〉$, which given by for any two vectors $G=\left(g_1,g_2,g_3\right)$ and $H=\left(h_1,h_2,h_3\right)$ in $E^3$. Based on this metric, the norm of a vector $G\in E^3$ is given by $∥G∥=\sqrt{〈G,G〉}$. A curve $\zeta =\zeta \left(\ell \right):I\subseteq \mathbb{R}\to E^3$ is a unit speed curve if $∥{\zeta }^{\prime }\left(\ell \right)∥=1$ for all $\ell \in I$. In this case, ℓ is called arc-length parameter of the curve $\zeta \left(\ell \right)$.

Let $\zeta \left(\ell \right)$ be a space curve in $E^3$, parameterized by arc-length ℓ. Denote by $\top \left(\ell \right),\aleph \left(\ell \right)$, $\mathfrak{B}\left(\ell \right)$ the moving Serret–Frenet frame along the unit speed curve $\zeta \left(\ell \right)$, where $\top \left(\ell \right)$, $\aleph \left(\ell \right)$, and $\mathfrak{B}\left(\ell \right)$ are the unit tangent, principal normal, and binormal vectors defined as respectively. In contrast, the Serret–Frenet formulas are defined by where $\kappa (\ell )$ is the curvature function and $\tau \left(\ell \right)$ is the torsion function defined as follows: $\kappa =\kappa (\ell )=∥{\top }^{\prime }\left(\ell \right)∥$, $\tau =\tau \left(\ell \right)=-〈{\mathfrak{B}}^{\prime }\left(\ell \right),\aleph \left(\ell \right)〉$, Ref. [18].

Now, as an alternative to the Serret–Frenet frame, which is donated by { $\top \left(\ell \right)$, ${\aleph }_q\left(\ell \right)$, ${\mathfrak{B}}_q\left(\ell \right)$, $\varrho$}, the quasi-frame (or simply q -frame) along a space curve $\zeta \left(\ell \right)$, where $\top \left(\ell \right)$, ${\aleph }_q\left(\ell \right)$, ${\mathfrak{B}}_q\left(\ell \right)$, and $\varrho$ are the unit tangent, quasi-normal, quasi-binormal, and projection vectors, respectively, and they are defined as follows: where $\varrho$ is the projection vector and can be chosen as $\varrho =\left(1,0,0\right)$ or $\varrho =\left(0,1,0\right)$ or $\varrho =\left(0,0,1\right)$. For simplicity, we choose the projection vector $\varrho =\left(1,0,0\right)$ in this paper. However, the q-frame is singular in all cases where ⊤ and $\varrho$ are parallel. Thus, in those cases where ⊤ and $\varrho$ are parallel, the projection vector $\varrho$ can be chosen as $\varrho =\left(0,1,0\right)$ or $\varrho =\left(0,0,1\right)$. We can define the Euclidean angle $\theta$ between the principal normal ℵ and quasi-normal ${\aleph }_q$ vectors. Then, the relation between the q-frame and the classical Serret–Frenet frame is given as follows:

Thus, we have

By taking the derivative of (4) with respect to ℓ, then substituting (2) and (5) into the results, we obtain the variation equations of the q-frame in the following form: where

The triple $\left({\kappa }_1,{\kappa }_2,{\kappa }_3\right)$ is called the quasi-curvature functions of $\zeta \left(\ell \right)$, Ref. [13].

Therefore, to solve this problem and prevent the occurrence of two reverse oriented principal normal and binormal vectors, we use the q-frame as an alternative frame to the Serret–Frenet frame (Figure 1). If we consider (3) and choose the projection vector $\varrho =\left(0,0,1\right)$, we get the following q-frame:

In this section, we obtain a new resolution of the acceleration and jerk vectors for a particle via the q-frame. Thereafter, in the osculating plane, we give alternative resolutions of the acceleration vector along the radial direction and tangential direction. In addition, in the osculating and rectifying planes, the jerk vector is resolved along the tangential direction and two special radial directions.

With the help of Theorem 2 and substituting (25) into (17), we can state the following theorem.

In this section, we present applications of the results derived to calculate the components of acceleration and jerk vectors with respect to a q-frame by applying Theorems 1–3 and Corollaries 1 and 2.