Some Characteristics of the Classical One-Dimensional Harmonic Oscillator

We can use this force expression to determine an equation of motion for the mass, that is, an equation relating its location in space, x, to its location in time, t  

According to this equation, x(t) is a function that, when differentiated twice, is regenerated with the multiplier —k/m. A general solution is 

the equation of motion [Eq. (3-3)] leads to a function, x t), that describes the trajectory of the oscillator. This function has the trigonometric time dependence characteristic of harmonic motion. From this expression, we see that x(t) repeats itself whenever the argument of the cosine increases by In. This will require a certain time interval t. Thus, the pendulum makes one complete back and forth motion in a time t given by 

Let us calculate and compare the time-averaged potential and kinetic energies for the classical harmonic oscillator. When the particle is at some instantaneous displacement x its potential energy is 

The cumulative value of the potential energy over one complete oscillation, Vc, is given by the integral 

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