Given a classical harmonic oscillator with particle mass

, much can be calculated using the principle of energy conservation. We have

where

is the oscillating amplitude and

is the displacement. We need to find the result of integration for the period

,

.

The result of classical harmonic oscillator mechanical behavior analyzed by the probability density function shows that the characteristic of this density function

with six kinds of spring constants, where

varies from

to

.

Snapshots 1, 2, 3, and 4 describe the behavior of the density function for six different spring constants moving simultaneously from its equilibrium point with the given total energy. The density function is plotted against its change of particle position. Graphs show that the lowest spring constant has the furthest displacement. The sharp vertical lines on the graph indicate the furthest displacement from each particle position. These lines are a vertical asymptote at the furthest displacement limit of each particle position showing an infinite value of this function. For the change of particle position

, it increases the total energy

indicating the decrease of density function

towards zero. In fact, dynamically, the difference of the spring constant factor causes the change of the oscillation amplitudes

. As a result of this function, a group of the classical harmonic oscillators doing the vibration motion from the equilibrium position

to the largest displacement simultaneously may result in harmonic and no harmonic motions. The smallest

shows that the value of this function remains constant longer than the larger

with increasing the total energy. This can be studied from the results of demonstration for the four harmonic graphs with determination of the certain position

from the equilibrium point

.