Harmonic Oscillator



After downloading harmonic.jar, please execute it by double-clicking, or typing "java -jar harmonic.jar".

If the above application does not start, please install OpenJDK from adoptium.net.


In this page we treat the harmonic oscillator, governed by a differential equation,

dθ2/dt2 = - θ

where θ denotes the angle from the vertical direction. The analytic solution of this equation is written as

θ = sin t.

That is, this oscillator moves periodically forever according to this solution. So even if you watch this page for a long time, nothing surprising would appear.
We can predict the future precisely in this model. Such a world of view is proposed by a French mathematician Pierre-Simon Laplace (1749-1827). Let us quote from "A philosophical essay on probabilities" written by Pierre-Simon Laplace:

Therefore, we must think that the present state of the universe is a result of the previous state, and the present state is a cause of the future state. Let us consider an intelligence who knows the every force which moves the nature, knows every state of everything, and has an enough ability to analyze them.
(…)
For this intelligence, there would be nothing uncertain, and, for his eye, the future would exist in front of him like the past.
(translated from Japanese by T.K.)

This intelligence is called Laplace's demon.
However, the chaos theory (and the theory of quantum mechanics) proved that such demon cannot exist in this world filled with nonlinearity.
Periodically Forced Pendulum >>

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Introduction to Chaos and Nonlinear Dynamics