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m00npirate
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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, [itex]F=\nabla f[/itex] for some f.
Also, a particle in simple harmonic motion must be expressible in the form:
[tex] E = \frac{\alpha}{2} q^{2} + \frac{\beta}{2} \dot{q}^{2} [/tex]
where [itex]\omega = \sqrt{\frac{\alpha}{\beta}}[/itex]
or
[tex]\ddot{x} + \omega^{2}x = 0 [/tex]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?