Simple Harmonic Motion

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.