Let's say I'm given a potential $U(x)=\frac{\epsilon}{2}\left ((\frac{x}{a})^2 +(\frac{a}{x})^2\right )$ and that I need to find the frequency associated with it. I'm given that the particle of mass $m$ moves in one dimension. How would I do this?
My attempt:
Since this is just a simple oscillating motion problem, I took a look at Taylor's Classical Mechanics and reviewed its section on such motion. I saw that he did
$$F=ma=-kx$$ $$a=-\frac{k}{m}x=-w^2x \quad ; \quad w=\sqrt{k/m}$$
He called $w$ the "angular frequency" with which the cart (in his example) moves. Is this the frequency I'm looking for? If it is, how do I find it, given the potential above?
The "ordinary" frequency is $\omega/(2\pi)$. That is, the frequency of oscillation of a harmonic oscillator with mass $m$ and stiffness constant $k$ is $\frac{k^{1/2}}{2 \pi m^{1/2}}$.
To understand the frequency of small-amplitude oscillations about the minimum of your given potential, first find its minimum. Take the second derivative of the potential at this minimum. This will correspond to $k$ in the harmonic oscillator. Nonlinear effects of course mean that the real frequency is not exactly the one given by this linearization; but these effects become less and less significant as you send the amplitude of the oscillation to zero.
It will turn out that this frequency is (up to a multiplicative dependence on a dimensionless function of the amplitude) the only way to combine $\epsilon,a,m$ to get the units of frequency. Hence how you could also do this using dimensional analysis.
But that just boils down to figuring out the units of $\epsilon,a,m$, which I guess are energy, length, and mass, and choosing powers of them that give you units of frequency. That's routine: energy is $M L^2 T^{-2}$, length is of course $L$, mass is of course $M$. So you want $\epsilon^{k_1} a^{k_2} m^{k_3}$ where $k_1+k_3=0,2k_1+k_2=0,-2k_1=-1$.