Geodesic motions in SO(2,1)
Geodesic motions in SO(2,1)
In this study, we have considered the rotational motions of a particle around the origin of the unit 2-sphere S2 2 with constant angular velocity in semi-Euclidean 3-space with index two E3 2 , namely geodesic motions of SO(2, 1). Then we have obtained the vector and the matrix representations of the spherical rotations around the origin of a particle onS 2 2 . Furthermore, we consider some relations between semi-Riemann spaces SO(2, 1) and T1S2 2 such as diffeomorphismand isometry. We have obtained the system of differential equations giving geodesics of Sasaki semi-Riemann manifold(T1S 2 2 , gS ) . Moreover, we consider the stationary motion of a particle on S2 2 corresponding to one parameter curve of SO(2, 1), which determines a geodesic of SO(2, 1). Finally, we obtain the system of differential equations givinggeodesics of the semi-Riemann space (SO(2, 1), h) and we show that the system of differential equations giving geodesics of Riemann space (SO(2, 1), h) is equal to that of (T1S2 2 , gS ).
___
- [1] Arnold V. On the differential geometry of infinite dimensional Lie groups and its applications to the hydrodynamics
of perfect fluids. Annales de L’Institut Fourier 1966; 16 (1): 319-361.
- [2] Ayhan I. On the sphere bundle with the Sasaki semi Riemann metric of a space form. Global Journal of Advanced
Research on Classical and Modern Geometries 2014; 3 (1): 25-35.
- [3] Ayhan I. On geodesics of SO(1,2). In: 7th International Eurasian Conference on Mathematical Sciences and
Applications; Kyiv, Ukraine; 2018.
- [4] Cheng H, Gupta KC. An historical note on finite rotations. Journal of Applied Mechanics 1989; 56 (1): 139-145.
- [5] Hartley R, Trumpf J, Dai Y, Li H. Rotation averaging. International Journal of Computer Vision 2013; 103 (3):
267-305.
- [6] Jaferi M, Yayli Y. Generalized quaternions and rotation in 3-space E3
αβ . TWMS Journal of Pure and Applied
Mathematics 2015; 2 (6): 224-232.
- [7] Kilingenberg W, Sasaki S. On the tangent sphere bundle of a 2-sphere. Tohuku Mathematics Journal 1975; 27:
49-56.
- [8] Korolko A, Leite FS. Kinematics for rolling a Lorentzian sphere. In: IEEE Conference on Decision and Control;
2011. pp. 6522-6527.
- [9] Novelia A, O’Reilly OM. On geodesics of the rotation group SO(3). Regular and Chaotic Dynamic 2015; 20 (6):
729-738.
- [10] O’Neill B. Semi-Riemann Geometry. New York, NY, USA: Academic Press, 1983.
- [11] Özdemir M, Ergin AA. Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and
Physics 2006; 56 (2): 322-336.
- [12] Park FC, Bobrow JE, Ploen SR. A Lie group formulation of robot dynamics. International Journal of Robotics
Research 1995; 14 (6): 606-618.