Harmonic motion is a frequent phenomenon in physics, and not surprisingly the study of harmonic motion in its different aspects constitutes one of the first encounters with theoretical physics, and it continues to be a subject at higher levels. Perhaps the simplest mechanical system that exhibits harmonic motion is the pendulum, consisting of a mass connected through a string with some hook to support it. As soon as the mass is brought out of its equilibrium position it is allowed to freely swing back and forth under the influence of gravity. In the highly idealised case of small deflections, i.e. small angles, from equilibrium and frictionless motion this system indeed exhibits harmonic motion. The main features of this motion can found in many other systems, mechanical, electromagnetic, and so on, and this renders the simple mathematical pendulum the paragon example of this kind of motion. Typically then, some other features are added to the discussion, namely a driving force and some damping.
However, most undergraduate textbooks stop at this stage. In this and the next lecture we will set out for a numerical treatment of real oscillatory systems. Starting with the highly idealised case of a frictionless mathematical pendulum we will discuss how to treat simple harmonic motion. Having sharpened our tools, we will then include the effect of friction and a driving force. In the next lecture we will add non-linearities to the system and observe how it is driven into chaos. While chaos has some intuitive meaning for all of us (typically inherited from mums describing our rooms), it is hard to define it in a quantitative, physical manner. By struggling with this definition we will recover some key issues for the description of chaotic systems.
The subject of this lecture is also discussed in chapter 3, sections 1 and 2 of Giordano & Nakanishi.
Assume a pendulum with mass m, connected with a fixed support through a rigid massless rod of length l. The deflection angle θ is taken with respect to a vertical line, such that the equilibrium position of the mass is at θ=0. The conjugate angular velocity will be denoted by ω. Gravity is directed vertically, but due to the rod its effect amounts to a force perpendicular to the rod, driving the mass into its equilibrium position, Newton's law then implies the following equation of motion for the mass in terms of the deflection angle: |
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There's a variety of equivalent ways to write the solution of this equation, the one chosen here reads