Particle in a conservative force field + SHM

[m00npirate]

Homework Statement

I read online in several places that any particle in motion in a conservative force field undergoes simple harmonic motion for small amplitudes.
I am attempting to prove this is true out of my own curiosity, but I don't know if I have the tools necessary to prove it. My math background is Calculus through half of multivariate, basic real analysis, and very simple differential equations (separation of variables).
Thanks in advance.

Homework Equations

I know that for a field to be conservative, $F=\nabla f$ for some f.
Also, a particle in simple harmonic motion must be expressible in the form:
$$E = \frac{\alpha}{2} q^{2} + \frac{\beta}{2} \dot{q}^{2}$$
where $\omega = \sqrt{\frac{\alpha}{\beta}}$

or
$$\ddot{x} + \omega^{2}x = 0$$I have no idea how to proceed other than to solve some partial differential equations with unknown functions. Is there any easier way to prove this for the general case?

 

[Asteroid1]

This might not be the answer you may have hoped for. I wonder what sites told you that all conservative fields can produce an harmonic motion.

I read online in several places that any particle in motion in a conservative force field undergoes simple harmonic motion for small amplitudes.

As a counter example:
We have the potential function $$f=1x$$
then the vector field will become $$F=\nabla f=1$$

Good luck getting an harmonic oscillator out of such a vector field ;)

 

[m00npirate]

Perhaps they meant an attractive force? I'll try to find where I saw that again.

Edit: here's one
"The problem of the simple harmonic oscillator occurs frequently in physics because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, will behave as a simple harmonic oscillator."
http://en.wikipedia.org/wiki/Harmonic_oscillator#Applications
I saw it in a textbook too on google books. I'll try to find it again.

 

[m00npirate]

All that it requires is for the force to be linear in x for sufficiently small x. Can anyone think of a conservative (attractive) force for which this would be untrue? I would try on my own but we haven't really covered exactly how to construct these in calc yet ><