Effective potentials in geodesic curves on surfaces

10th Workshop of the Gravitation and Mathematical Physics Division, Mexican Physical Society

物理学;数学

Santiago, J.A.^1 ; Chacón-Acosta, G.^1 ; González-Gaxiola, O.^1

Applied Mathematics and Systems Deptartment, Universidad Autónoma Metropolitana-Cuajimalpa, Vasco de Quiroga 4871, Cuajimalpa, México D. F.

C.P. 05348, Mexico^1

关键词: Axially symmetric;    Effective potentials;    Euclidean spaces;    First integral;    Geodesic curvatures;    Geodesic curves;    Geodesic equations;    Normal curvature;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/545/1/012014/pdf
DOI  :  10.1088/1742-6596/545/1/012014

来源: IOP

【 摘 要 】

In this work, the equations of geodesic curves on surfaces embedded in euclidean space are obtained. By introducing a vector Lagrange multiplier, we show that the geodesic curvature of the curves are zero and the normal curvature of them can be identified with the force transmitted to the surface. We then obtain the corresponding formulas in the case of axially symmetric surfaces, where a first integral of the geodesic equations can be interpreted as a particle moving in an effective potential (being zero the total energy), and the angular momenta is conserved. The methodology developed is illustrated with some examples: the catenoid and the pseudosphere.