Geodesics on the Momentum Phase Space with metric $^{{C}}{g}$

Year 2018, , 61 - 67, 11.03.2018

İsmet Ayhan

https://doi.org/10.32323/ujma.396109

Abstract

In this paper, a system of the differential equations giving geodesics on the momentum phase space with pseudo Riemann metric $^{C}g$ of a Hamilton space is found by using the Euler Lagrange equations. Then, space like geodesics on pseudo hyperbolic 2-space $H_{1}^{2}$ are obtained. Finally, a system of the differential equations giving geodesics on the cotangent bundle with pseudo Riemann metric $^{C}g$ of $H_{1}^{2}$  is get.

Keywords

The Hamilton mechanic system, geodesics on the cotangent bundle, spacelike surface

References

• [1] R. Abraham and J. E. Marsden, Foundations of mechanics, W. A. Benjamin Inc.,New York, 1967.
• [2] I. Ayhan, On the tangent sphere bundle of the pseudo hyperbolic two space, Global Journal of Advanced Research on Classical and Modern Geometries, vol. 3, no. 2, pp 76-90, 2014.
• [3] I. Ayhan, On the sphere bundle with the Sasaki semi Riemann metric of a space form, Global Journal of Advanced Research on Classical and Modern Geometries, vol. 3, no. 1, pp 25-35, 2014.
• [4] I. Ayhan, Geodesics on the tangent sphere bundle of 3-Sphere, International Electronic Journal of Geometry, vol. 6, no. 2, pp 100-109, 2013.
• [5] A. C. Coken, I Ayhan, On the geometry of the movements of the particles in a Hamilton space, Abstract and Applied Analysis, DOI:10.1155/2013/830147, 2013.
• [6] P. Free, Introduction to general relativity, Lecture Notes, Virgo site, 2003.
• [7] R. Miron, H. Hrimiuc, H. Shimada, and V. S. Sabau, The geometry of Hamilton and Lagrange spaces, Kluwer Academic, New York, USA, 2001.
• [8] A. Polnarev, Motion of a test particle in a gravitational field and Hamilton Jacobi equations, relativity and gravitation, Lectures Notes 5, 2011.
• [9] S. Waner, G. C. Levine, Introduction to differential geometry and general relativity, Lectures Notes, 2005.
• [10] C. K. Wonk, Classical physics in Galilean and Minkowski space-times, Lecture Notes 3, 2009.
• [11] K. Yano and S. Ishihara, Tangent and cotangent bundles, Marcel Decker Inc., New York, 1973.