I agree, ut I think your comment needs more details. I'll try:
The idea is that the Newton's Laws an be rewritten and get equivalent equations that look very different. One of the rewrites is "Lagrangian Mechanics". Instead of a differential equation, you have a calculus of variations. In this new formulation it's more clear that the solution is not unique, and you get the particle that rest on top of the dome, the one that falls down, and even some solutions where the particle waits a little and then "decides" to fall down without an eternal trigger. For this formulation, two of the man
concepts are virtual work and virtual displacement. More details in: https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alember...
