x0 and the parameters k and m determine the energy of the oscillations. The force F is a conservative force and so is expressed as the rate of change with x of a potential energy function U (x).
in our example. The total energy E of the oscillating atom at time t is the sum of potential and kinetic energies. Hence
Or, using equation (iii),
Since the sum of the squares of the sines and cosines of any angle equation unity.
We see that, as expected, E is constant for fixed m, k, and x0 and during the motion potential and kinetic energy are interchanged continuously, the former being zero when the speed is at its maximum value, and the latter being zero when the speed is zero at the turning points of the oscillation.
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