Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the differential equation
 |
(1)
|
where 
denotes the second derivative of 
with respect to 
, and 
is the angular frequency of oscillation. This ordinary
differential equation has an irregular singularity
at 
.
The general solution is
 |  |  |
(2)
|
 |  |  |
(3)
|
where the two constants 
and 
(or 
and 
) are determined from the initial conditions.
Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in
the analysis of the flow of current in an electronic CL circuit (which contains a
capacitor and an inductor). If a damping force such as Friction is present, an additional
term 
must be added to the differential equation
and motion dies out over time.
