Optical manipulation provides new insight into a wide range of physical phenomena and has engendered advanced applications in various fields. By utilizing near-field methods that overcome the diffraction limit, the ability to manipulate nanoparticles became feasible. In this paper, we analyze the displacement trajectories of a gold nanoparticle in the field of an ultrashort pulsed beam in symmetric (photonic nanojet) and symmetry-breaking (photonic hook) systems. We generate a symmetric optical force by illuminating a dielectric, micro-cylinder, and break the symmetry by adding a dielectric plate. We developed an efficient numerical method for calculating nanoparticle displacement under pulsed illumination which uses a stroboscopic map. Within this method, we revealed the influence of the plate position on the likeliness of different types of nanoparticle motion (i.e. stable, negative, or positive motion) in symmetric and symmetry-breaking configurations. Our work stimulates the development of experimental methods for optomechanical manipulation and opens a venue for future fundamental investigations for a range of practical applications, where accurate control over the mechanical motion of nanoscale objects is required.