Variational equations and Killing magnetic trajectories on timelike surfaces in semi-Riemannian manifolds

Abstract

In this article, Darboux frame variations for timelike surfaces in semi-Riemannian manifolds are discussed. In addition, the Killing equations are examined by using the Darboux frame curvature variations. Then, magnetic trajectories are generated by means of the variational vector fields. Furthermore, parametric representations of all magnetic trajectories on the de Sitter space $\mathbb{S}_{1}^{2}$ are presented. Moreover, various examples of magnetic trajectories are given in order to illustrate the theoretical results.

Keywords

• vector fields
• magnetic flows
• ordinary differential equations
• special curves
• motion of a rigid body with a fixed point
• variational methods

References

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