In this work, we examine the impact of magnetic fields on the moving particle trajectories by variational approach to the magnetic flow associated with the Killing magnetic field on 2−dimensional lightlike cone Q 2 ⊂ E 3 1 . We give some characterizations for x−magnetic curve and x−magnetic surface of rotation using the Killing magnetic field of this curve in Q 2 and we give the different types of axes of rotation, then creates three different types of magnetic surfaces of rotation in 2−dimensional lightlike cone Q 2 ⊂ E 3 1 .

In this study, the tube surfaces generated by the curve defined in Galilean 3-space are examined and some certain results of describing the geodesics on the surfaces are also given. Furthermore, the conditions of being geodesic on the tubular surface are obtained with the help of Clairaut’s theorem, which allows us to constitute the specific energy. The physical meaning of the specific energy and the angular momentum is of course related with the physical meaning itself. Our results show that the specific energy and the angular momentum obtained on tubular surfaces can be expressed using arbitrary geodesic curve in Galilean space. In addition, some characterizations are given for these surfaces, with the obtained mean and Gaussian curvatures.

By thinking the magnetic flow connected by the Killing magnetic field, the magnetic field on the setting out particle orbit is investigated in $ ⊂ % &. Clearly, dealing with the Killing magnetic field of-magnetic curve, the rotational surface generated by-magnetic is expressed in $ ⊂ % & , and the variant kinds of axes of rotation in lightlike cone $ ⊂ % & is given. Furthermore, the specific kinetic energy, specific angular momentum and conditions being geodesic on rotational surface generated by-magnetic curve are expressed with the help of Clairaut's theorem.

In this study, we provide a brief description of rotational surfaces in 4-dimensional (4D) Galilean space using a curve and matrices in [Formula: see text]. That is, we provide different types of rotational matrices, which are the subgroups of [Formula: see text] by rotating a selected axis in [Formula: see text]. Hence, we choose two parameter matrices groups of rotations and we give the matrices of rotation corresponding to the appropriate subgroup in Galilean 4-space and we generate rotated surfaces.

In this paper, we examine the notion of the involute-evolute curves for the curves lying the surfaces in Minkowski 3-space E3 1 . We call these new associated curves as involute-evolute and by using the Darboux frame of the curves. We give the representation formulae for spacelike curves in Minkowski 3-space E3 1 and using this formulae we give some characterizations of these curves. Besides, we find the relations between the normal curvatures, the geodesic curvatures and the geodesic torsions of these curves.