Siacci's Theorem for Frenet Curves in Minkowski 3-Space

Abstract

For motion of a material point along a space curve, due to Siacci [1], a resolution of the acceleration vector is well known. In this resolution, the acceleration vector is stated as the sum of two special oblique components in the osculating plane to the curve. In this paper, we have studied the Siacci's theorem for non-relativistic particles moving along the Frenet curves at very low speeds relative to the speed of light in Minkowski 3-space. Moreover, an illustrative example is given to show how the aforesaid theorem works. This theorem is a new contribution to the field and it may be useful for some specific applications in theoretical, mathematical and computational physics.

Resolutions of the Jerk and Snap Vectors for a Quasi Curve in Euclidean 3-Space

“…respectively. From Definition 2, the jerk and snap vectors of the point particle P according to the quasi frame are expressed as in (10) and (11), respectively. The proof is complete.…”

“…Özen et al [9] studied Siacci's theorem for Darboux curves on regular surfaces in Euclidean 3-space E 3 . Özen [10] studied Siacci's theorem for Frenet curves in Minkowski 3-space E 3 1 . Résal [11] obtained a resolution of the jerk vector for Frenet curves in Euclidean 3-space E 3 .…”

A new method for resolving the jerk and jounce vectors in Euclidean 3‐space

“…In 2011, the radial component is located on the trajectory's instantaneous osculating plane, while the tangent component is located along the trajectory's tangent line vector 2 . In 2020, Siacci theorem is studied in Minkowski 3‐space 3 . In 2021, a new visualization of fundamental mechanical elements and concepts according to differential geometry is given in Elsayied et al 4…”

“…2 In 2020, Siacci theorem is studied in Minkowski 3-space. 3 In 2021, a new visualization of fundamental mechanical elements and concepts according to differential geometry is given in Elsayied et al 4 The time derivative of an acceleration vector is the jerk vector of a particle. Jerk is essential when evaluating the destructive effect of motion on a mechanism and the discomfort caused to passengers in vehicles.…”

On the Resolution of the Acceleration Vector According to Bishop Frame

“…Afterwards, Ozen et al [9] discussed the Siacci's theorem in the space endowed with the modified orthogonal frame. Finally, Ozen expressed and proved the Siacci's theorem for Frenet curves in 3-dimensional Minkowski space [10]. In the theory of curves, Serret-Frenet frame is a moving frame which is very useful and has an important place.…”