What is the definition of simple harmonic motion? The motion of a particle where its displacement is given by a sinusoidal function, that is: $$x=A\sin(ωt+φ)$$ or the total force that is acting on the particle is: $$ΣF=-kx$$
Also why we need to show that for a particle undergoing simple harmonic motion the force is in the form of the second equation? At some instant some force (other than the initial forces) may act on the system so the 2nd equation isn't satisfied. Would it better to say that instantaneously the particle undergoes simple harmonic motion? In other words can we say that for a net force that is time dependent (more general case) $ΣF(t)=ma$ at some instant $ΣF(t_{shm})=-kx$ the particle will undergo simple harmonic motion?