A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum
The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close rel...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Mahmut Akyigit
2021-04-01
|
| Series: | Journal of Mathematical Sciences and Modelling |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/tr/download/article-file/1538801 |
| _version_ | 1797350798655488000 |
|---|---|
| author | Kahraman Esen Özen Murat Tosun |
| author_facet | Kahraman Esen Özen Murat Tosun |
| author_sort | Kahraman Esen Özen |
| collection | DOAJ |
| description | The theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented by \"Ozen et al. (JTAM 57(2)(2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future. |
| first_indexed | 2024-03-08T12:51:22Z |
| format | Article |
| id | doaj.art-4f83a9c4465f40ddbc790f5dc0059aa9 |
| institution | Directory Open Access Journal |
| issn | 2636-8692 |
| language | English |
| last_indexed | 2024-03-08T12:51:22Z |
| publishDate | 2021-04-01 |
| publisher | Mahmut Akyigit |
| record_format | Article |
| series | Journal of Mathematical Sciences and Modelling |
| spelling | doaj.art-4f83a9c4465f40ddbc790f5dc0059aa92024-01-20T09:04:12ZengMahmut AkyigitJournal of Mathematical Sciences and Modelling2636-86922021-04-014171810.33187/jmsm.8696981408A New Moving Frame for Trajectories with Non-Vanishing Angular MomentumKahraman Esen Özen0Murat Tosun1SAKARYA ÜNİVERSİTESİ, FEN-EDEBİYAT FAKÜLTESİ, MATEMATİK BÖLÜMÜSAKARYA UNIVERSITYThe theory of curves has a very long history. Moving frames defined on curves are important parts of this theory. They have never lost their importance. A point particle of constant mass moving along a trajectory in space may be seen as a point of the trajectory. Therefore, there is a very close relationship between the differential geometry of the trajectory and the kinematics of the particle moving on it. One of the most important elements of the particle kinematics is the jerk vector of the moving particle. Recently, a new resolution of the jerk vector, along the tangential direction and two special radial directions, has been presented by \"Ozen et al. (JTAM 57(2)(2019)). By means of these two special radial directions, we introduce a new moving frame for the trajectory of a moving particle with non vanishing angular momentum in this study. Then, according to this frame, some characterizations for the trajectory to be a rectifying curve, an osculating curve, a normal curve, a planar curve and a general helix are given. Also, slant helical trajectories are defined with respect to this frame. Afterwards, the necessary and sufficient conditions for the trajectory to be a slant helical trajectory (according to this frame) are obtained and some special cases of these trajectories are investigated. Moreover, we provide an illustrative numerical example to explain how this frame is constructed. This frame is a new contribution to the field and it may be useful in some specific applications of differential geometry, kinematics and robotics in the future.https://dergipark.org.tr/tr/download/article-file/1538801kinematics of a particleplane and space curvesmoving frameangular momentumslant helices |
| spellingShingle | Kahraman Esen Özen Murat Tosun A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum Journal of Mathematical Sciences and Modelling kinematics of a particle plane and space curves moving frame angular momentum slant helices |
| title | A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum |
| title_full | A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum |
| title_fullStr | A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum |
| title_full_unstemmed | A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum |
| title_short | A New Moving Frame for Trajectories with Non-Vanishing Angular Momentum |
| title_sort | new moving frame for trajectories with non vanishing angular momentum |
| topic | kinematics of a particle plane and space curves moving frame angular momentum slant helices |
| url | https://dergipark.org.tr/tr/download/article-file/1538801 |
| work_keys_str_mv | AT kahramanesenozen anewmovingframefortrajectorieswithnonvanishingangularmomentum AT murattosun anewmovingframefortrajectorieswithnonvanishingangularmomentum AT kahramanesenozen newmovingframefortrajectorieswithnonvanishingangularmomentum AT murattosun newmovingframefortrajectorieswithnonvanishingangularmomentum |