Registering seconds with a conic clock

The feasibility of registering seconds using a nonlinear integrable oscillator, namely, the frictionless motion of a point-like particle that slides under gravity on an inverted conical surface, is studied. Depending on the integer part of the relation between the angular and radial frequencies of the particle trajectory, only an angular interval for the cone is available for this purpose. For each one of these possible angles, there exists a unique trajectory that has the capability of registering seconds. The method to obtain the geometrical properties of these trajectories and the necessary initial conditions to reach them are then established.