Oscillatory behaviour is ubiquitous in Nature, and in particular the harmonic oscillator is one of the cornerstone models of physics. This material is used at Macquarie University in the unit phys201 Physics IIA where it forms the first half.
The motion for a harmonic oscillator is derived using Newton's second law. Different parametrizations of the solution, the velocity, acceleration and energy are also determined.
The equations of motion and solutions are derived for the simple pendulum and a general pendulum. Dynamical maps are introduced as a way of handling nonlinear oscillators.
The Lotka-Voltera equations are a very simple model of predator-prey dynamics. They lead to oscillatory behaviour in both populations.
The solutions to the harmonic oscillator with a velocity dependant drag force are derived. The relaxation time for the oscillator is defined.
Energy of the damped harmonic oscillator is described. The Q of an oscillator.
The map of phase space of a dynamical system is a really convenient way of summarising the behaviour of a system. In particular it doesn't require a solution to the equations of motion.
The solution to a damped oscillator with a periodic driving force is derived.
When an oscillator is driven at just the right frequency it hits resonance and absorbs energy from the driving. The details of resonance for a driven and damped harmonic oscillator are explored.
Coupling together two or more oscillators introduces a whole new level of complexity. It turns out that there is a particular way of looking at the system that makes it simple to solve the motion.
The loaded string is a classic problem of coupled oscillators where \(N\) small masses are threaded onto a light string. It makes a nice transition for considering the continuum case and waves.
The continuum limit of the loaded string is derived, arriving at the one dimensional wave equation.