Periodic Motion: Explaining Resolving into Oscillatory Components

[PrincePhoenix]

According to our textbook "..every periodic motion, however complicated it may be, can always be resolved into simple oscillatory components." Can someone explain this? How can the periodic motion of Earth be resolved into oscillatory components?

 

[evanlee]

Hey PrincePhoenix,

I believe what your textbook is referring to is the idea of a Fourier Series. The mathematician and physicist derived a method with which it is possible to decompose any periodic function to a sum of basic periodic functions, namely sines and cosines. By manipulating the amplitudes of several (often infinite) sines and cosines, it is possible to construct the desired function.

To learn more about the actual computation of the series, just search the web for Fourier series and harmonic analysis.

http://mathworld.wolfram.com/FourierSeries.htmlAs far as the periodic motion of the Earth goes, if you can come up with a clean equation for the forces acting on it, it is possible to derive the periodicity of the Earth's motion. Using Fourier, I assume you would be able to decompose this into a form of an infinite sum of sines and cosines.

Hope this helps a bit!

 

[PrincePhoenix]

Thanks.

 

[Bob S]

Simple periodic motion is usually described as being like the motion of a pendulum in a gravitational field, or Hookes Law for a spring
http://en.wikipedia.org/wiki/Pendulum
http://en.wikipedia.org/wiki/Hooke's_law
where the restoring force is (nearly) proportional to the displacement from the center.
http://en.wikipedia.org/wiki/Hooke's_law
But the planetary motion around the Sun is also periodic, and is described by Kepler's laws.
http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

Bob S