My title - Departamento de Matemática da Universidade do Minho

7 OSCILLATIONS AND CYCLES 51
7 Oscillations and cycles
7.1 Harmonic oscillator
Harmonic oscillator.
equation
The harmonic oscillator is the (phenomenon modeled by the) Newton
¨q = −ω 2 q .
This is a quite universal equation, since it describes small oscillations around a “generic” stable
equilibrium of any one-dimensional Newtonian system 16 (indeed, take a Newton equation mẍ =
−dU ′ (x) of a particle in a potential field U. An equilibrium position is a zero of the force, i.e. a
point x 0 where U ′ (x 0 ) = 0. It is “stable” if x 0 is a local minimum of the potential, so that the Taylor
expansion of the potential around x 0 in powers of q = x − x 0 starts with U(x) = α + 1 2 βq2 + . . . ,
for some positive second derivative U ′′ (x 0 ) = β. If we are only interested in small displacements
of x around x 0 , we can safely disregard high order terms and approximate the Newton equation
as m¨q ≃ −βq, which is an harmonic oscillator with resonant frequency ω = √ β/m).
Call p = ˙q the momentum. The Newton equation ¨q = −ω 2 q is equivalent to Hamilton’s first
order equations
˙q = p
ṗ = −ω 2 q .
If we define the complex variable z = ωq + i ˙q, Newton equation then takes the form of a first order
linear equation in the complex line, namely ż = −iωz, whose solution is z (t) = e −iωt z (0).
In terms of the original (physical) variables, the solutions read
periodicity of
exp
q (t) = q 0 cos (ωt) + v 0
ω
sin (ωt) = A sin(ωt + φ)
where the amplitude A and the initial phase φ depend on the initial conditions q(0) = q 0 and
˙q(0) = v 0 . So, all trajectories are periodic with common period 2π/ω, and orbits are ellipses in
the q- ˙q plane, determined by the conserved energy
E = 1 2
(
˙q 2 + ω 2 q 2) = ω 2 A 2 .
Harmonic oscillator, phase curves and time series.
16 “The harmonic oscillator, which we are about to study, has close analogs in many other fields; although we start
with a mechanical example of a weight on a spring, or a pendulum with a small swing, or certain other mechanical
devices, we are really studying a certain differential equation. This equation appears again and again in physics and
other sciences, and in fact is a part of so many phenomena that its close study is well worth our while. Some of the
phenomena involving this equation are the oscillations os a mass on a spring; the oscillations of charge flowing back
and forth in an electrical circuit; the vibrations of a tuning fork which is generating sound waves; the analogous
vibrations of the electrons in an atom, which generate light waves; the equations for the operation of a servosystem,
such as a thermostat trying to adjust a temperature; complicated interactions in chemical reactions; the growth of
a colony of bacteria in interaction with the food supply and the poison the bacteria produce; foxes eating rabbits
eating grass, and so on; ...”
Richard P. Feynman [Fe63]