How to prove that a motion is Simple Harmonic Motion (SHM)? I would like to know how one could show and prove that a given motion is simple harmonic motion.
Once given an answer, I'll apply that technique to an example I am trying to figure out. 
Thank you in advance!
I believe a motion can be proved simple harmonic, if the relation between its is as such:
$$
a_x = - \omega^2\cdot x
$$
And as such the period time is:
$$
T =\frac{2\pi}{\omega}
$$
Question-so-far: How do you prove such for a given force $F = \frac{G\cdot m_e \cdot M}{R_E} \cdot r$ ? Or any force that has non-trivial constants?
 A: If the total force $F$ on a mass $m$ follows Hooke's law, 
$$F~=~-kx,$$ 
then one can use Newton's 2nd law 
$$F=ma,$$ 
to infer that the motion is a simple harmonic motion
$$ a =-\omega^2x, \qquad\qquad \frac{2\pi}{T}~=~\omega~=~ \sqrt{\frac{k}{m}}~,$$
cf. OP's correct belief. Now it only remains to solve the ODE
$$ \frac{d^2x(t)}{dt^2}~=~-\omega^2x(t), $$
which is a pure math exercise.
A: 
In mechanics and physics, simple harmonic motion is a type of periodic
  motion or oscillation motion where the restoring force is directly
  proportional to the displacement and acts in the direction opposite to
  that of displacement .

So proving the restoring force F is directly proportional would be enough. For example, let's say there is mass m mounted between 2 spring with spring constant $k_1$, $k_2$  which are attached to wall and have its original length when system is at equilibrium. When the mass is disturbed slightly by displacement $x$, the restoring force will be $F = -(k_2 + k_1)x$ which proves the motion will be SHM.
A: $$F = ma$$
$$F = -mg \sin \theta$$
$$ma = -\frac{mgx}{L}$$
$$a = -\frac{gx}{L}$$
Since $-g/L$ is a constant, $a \propto -x$. Hence proved.
