Editing Oscillation (section)

== Small oscillation approximation ==
In physics, a system with a set of conservative forces and an equilibrium point can be approximated as a harmonic oscillator near equilibrium. An example of this is the [[Lennard-Jones potential]], where the potential is given by:
<math display="block">U(r) = U_0 \left[ \left(\frac{r_0} r \right)^{12} - \left(\frac{r_0} r \right)^6 \right]</math>

The equilibrium points of the function are then found:
<math display="block">\begin{align}
\frac{dU}{dr} &= 0 = U_0 \left[-12 r_0^{12} r^{-13} + 6r_0^6r^{-7}\right] \\
\Rightarrow r &\approx r_0
\end{align}</math>

The second derivative is then found, and used to be the effective potential constant:
<math display="block">\begin{align}
\gamma_\text{eff} &= \left.\frac{d^2U}{dr^2} \right|_{r=r_0} = U_0 \left[ 12(13) r_0^{12} r^{-14} - 6 (7) r_0^6 r^{-8} \right] \\[1ex]
&= \frac{114 U_0}{r^2}
\end{align}</math>

The system will undergo oscillations near the equilibrium point. The force that creates these oscillations is derived from the effective potential constant above:
<math display="block">F= - \gamma_\text{eff}(r-r_0) = m_\text{eff} \ddot r</math>

This differential equation can be re-written in the form of a simple harmonic oscillator:
<math display="block">\ddot r + \frac {\gamma_\text{eff}} {m_\text{eff}} (r-r_0) = 0</math>

Thus, the frequency of small oscillations is:
<math display="block">\omega_0 = \sqrt { \frac {\gamma_\text{eff}} {m_\text{eff}}} = \sqrt {\frac {114 U_0} {r^2 m_\text{eff}}}</math>

Or, in general form<ref>{{Cite web |date=2020-07-01 |title=23.7: Small Oscillations |url=https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/23%3A_Simple_Harmonic_Motion/23.07%3A_Small_Oscillations |access-date=2022-04-21 |website=Physics LibreTexts |language=en}}</ref>
<math display="block">\omega_0 = \sqrt{\left.\frac {d^2U} {dr^2} \right\vert_{r=r_0}}</math>

This approximation can be better understood by looking at the potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between potential energy and force.
<math display="block">\frac {dU} {dt} = - F(r)</math>

By thinking of the potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between <math>r_\text{min}</math> and <math>r_\text{max}</math>. This approximation is also useful for thinking of [[Kepler orbit|Kepler orbits]].