Why is the Harmonic Oscillator so common in physics?

[barnflakes]

I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation?

I only ask because I just solved a problem in my research by assuming that the resistive force to something is F=-kx which of course leads to harmonic oscillation - and this was completely unexpected - nobody knew that the resistive force should be, it just happens that it fits the data bang on.

 

[AlephZero]

I've heard before that it's because when you expand around a minimum point in the potential energy you get a quadratic function, but I can't recall where I read this. Can anyone point me in the right direction, or give their own explanation?

When you expand any "smooth" function as a Taylor series about $x = a$ you have
$f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2$ plus higher powers of $(x-a)$

At a minimum (or maximum) the first derivative $f'(a) = 0$ so $f(x)$ is approximately a quadratic.

Any book on optimisation theory or multi-variable calculus should give the corresponding results for functions of more than one variable.

 

[haruspex]

To expand a little on what AlephZero wrote...
If the potential approximates f(a) + f''(a)x² then the restorative force, the derivative of the potential, is linear. At an energy minimum, the coefficient is negative, producing SHM.

 

[barnflakes]

That's really helpful guys, cheers.