  In physics, conserved quantities are key to understanding and describing
physical phenomena. These conserved quantities are related to Noether's theorem
and the Lagrangian description both in classical mechanics and in field theory.
In this article we have found the equation of the vortex core trajectory in
terms of two conserved physical quantities, namely the energy, $E$, and a
vector perpendicular to the orbit plane, $\vec{A} = -\vec{L} + \vec{G} \,
|\vec{r}_c|^2/2$ where $\vec{G}$, $\vec{L}$ and $\vec{r}_c$ are the topological
gyrovector, the angular momentum and the position of the vortex core,
respectively. We find that in the absence of a dissipative term, for small
deviations of the vortex core, the trajectory is bounded between two concentric
circles. On the contrary, under the action of a dissipative term proportional
to the damping coefficient, $\vec{A}$ is no longer conservative and the vortex
core moves either towards the center or out of the cylinder, depending on the
circularity of the magnetic vortex and the intensity of the magnetic field
applied in the plane of the cylinder.
