Caption: "An animation demonstrating the simple harmonic motion of a mass on a spring (device) in both real space and phase space (which has dimensions position and velocity in this context). Note that the phase space axes are switched from the standard convention in order to align the two diagrams." (Slightly edited.)

Features:

1. A harmonic oscillator is a system in which there is a restoring force that is linearly proportional to the displacement from mechanical equilibrium.

   The force law is called Hooke's law. The formula is

       F = - kx ,

   where k is a constant which depends on the system, x is the displacement from mechanical equilibrium at x = 0, and the minus sign makes the force point opposite to the displacement.

   The minus sign makes the force a restoring force.

2. The potential energy (energy of position) of the Hooke's law force is given by

       V = (1/2)kx**2 .

3. An ideal spring is an exact harmonic oscillator. Real springs approximate exact harmonic oscillators.

4. If a harmonic oscillator is subject only to the Hooke's law force, it undergoes simple harmonic motion which is illustrated in the animation. Note that the absolute value of position is a maximum when velocity is zero and vice versa.

5. The differential equation for simple harmonic motion follows from Newton's 2nd law of motion (AKA F=ma) with Hooke's law providing the force:

       m(d²x/dt²) = -kx

   which has general solution

       x=A*sin(ωt)+B*cos(ωt) ,

   where A and B are constants determined by the initial conditions and the angular frequency ω = sqrt(k/m).

6. Hooke's law is super-important because almost all sufficiently small displacements from stable equilibrium are subject to a Hooke's law restoring force. So Hooke's law is everywhere.

7. Hooke's law was discovered by Robert Hooke (1635--1703) (who is the eponym) in 1660, but NOT effectively published until 1678.