 # The Harmonic Oscillator as a Control System

The classical harmonic oscillator is an example of a primitive control system where there is no representation of the controlled system S as distinct from S itself. Information flows directly from S to the controller C.

A hydrogen molecule can be treated as a harmonic oscillator. Here the controlled system S consists of two protons at a distance x from each other. The agent which is embedded in S is the relative momentum of the protons in the center-of-mass coordinate system; it causes protons to move with respect to each other.

The controller C is the electron shell which holds the protons together at a certain equilibrium distance x0 by exerting on them a force which acts against the coulomb repulsion. (This force is quantum-mechanical in its origins, but this does not prevent us from considering oscillations classically). The distance x is a representation of the two-proton controlled system for the controller, the shell, but there is no part of the controller that would keep x: it is a feature of the proton pair, i.e. the controlled system. Also, there is no separate subsystem to keep the goal of the controller, but it functions as if the goal was to keep the distance x close to x0.

We shall now see that the control processes in a classical oscillator result from the interplay of the three fundamental factors in classical mechanics: coordinate, momentum, and force. The three arrows in our control scheme: action, perception and information will be described by the three familiar equations of the classical oscillator.

Nothing happens as long as the protons are at the equilibrium distance x0 and at rest with respect to each other. Suppose some uncontrollable disturbance pushes one of the protons passing to the controlled system some momentum p. The coordinate x starts changing according to the equation:

dx/dt = p/m (perception)

In terms of the control scheme this is perception, because the agent p of the controlled system determines the representation.

The change in the representation informs the agent F of the controller, the shell, which is assumed to be proportional to the change of x and opposite in direction:

F = -k(x-x0) (information)

This force acts on the proton pair, causing the change of the momentum according to the equation:

dp/dt = F (action)

This will start stopping the movement of the protons and finally will reverse it. In this manner, unstopping oscillations will follow, which will keep the distance x close to x0 (assuming that the initial push was in certain limits).

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