# Difference between periodic motion vs harmonic motion vs simple harmonic motion

So , I am kind of confused about the difference between these three things:

(i) Periodic motion.
(ii) Harmonic motion.
(iii) Simple harmonic motion

It will be the best if someone showed me the differences by mathematical definitions and function graphs for each of them separately side by side.I struggle with real world examples.

Periodic motion: A system is called periodic if it is invariant under a finite translation in time T.

meaning: If $$\vec{x}(t)$$ is the trajectory of a particle then $$\vec{x}(t+T)=\vec{x}(t)$$ where T is the period of the system.

Simple Harmonic motion: the motion of a particle of mass $$m$$ undergoing Simple Harmonic Motion satisfies the following differential equation (aka Hooke's Law):

$$m \frac{d^2{x}}{dt^2}=-k{x}$$

(in general k could be a matrix, but in 1D it is a constant scalar)

(Complex) Harmonic motion: the motion of a particle that in addition to the linear term in $$x$$ also has other terms in the differential equation. for example a dissipative (frictional) force of the form: $$-\gamma \dot{x}$$ or a driving force of the form $$F(t)$$ (for example $$F(t)=A_dcos(\omega_dt))$$

example of such equation is: $$m \frac{d^2{x}}{dt^2}=-k{x}-\gamma \dot{x}+F(t)$$

Note: periodic motion isn't necessarily harmonic.