In this paper the problem of a particle constrained to move on an axisymmetric surface embedded in
three-dimensional Euclidean space, under the influence of a gravitational field, is addressed from a
geometrical point of view. Using a covariant geometrical approach at undergraduate level, a
variational framework is implemented in order to obtain the particle’s motion equation, which is
solved in an analytical way for the case where the particle moves on a cylinder and numerically when
the particle moves on a catenoid. The trajectories obtained are expressed as functions of conserved
quantities such as total energy, E , and the z component of the particle’s angular momentum, L 3 .
Moreover, expressions for normal and geodesic curvatures as well as the speed, tangential
acceleration and normal force constraining the particle on the surface along the trajectories are
obtained.