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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: U (r) = U 0 {\displaystyle U(r)=U_{0}\left}

The equilibrium points of the function are then found: d U d r = 0 = U 0 ⇒ r ≈ r 0 {\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left\\\Rightarrow r&\approx r_{0}\end{aligned}}}

The second derivative is then found, and used to be the effective potential constant: γ eff = d 2 U d r 2 | r = r 0 = U 0 = 114 U 0 r 2 {\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left\\&={\frac {114U_{0}}{r^{2}}}\end{aligned}}}

The system will undergo oscillations near the equilibrium point. The force that creates these oscillations is derived from the effective potential constant above: F = − γ eff (r − r 0) = m eff r ¨ {\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}}

This differential equation can be re-written in the form of a simple harmonic oscillator: r ¨ + γ eff m eff (r − r 0) = 0 {\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0}

Thus, the frequency of small oscillations is: ω 0 = γ eff m eff = 114 U 0 r 2 m eff {\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}}

Or, in general form ω 0 = d 2 U d r 2 | r = r 0 {\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}}

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. d U d t = − F (r) {\displaystyle {\frac {dU}{dt}}=-F(r)}

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 r min {\displaystyle r_{\text{min}}} and r max {\displaystyle r_{\text{max}}}. This approximation is also useful for thinking of Kepler orbits.