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Spelling, grammar, and general cleanup, typos fixed: Therefore → Therefore,, a equilibrium → an equilibrium
===Derivation===
===Derivation===
{{Math-h}}
{{Math-h}}
Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as:
Suppose there is a potential, <math>V(x)</math>, that has an equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as:
<math>
<math>
</math>
</math>
where <math>V'(x_0)</math>, <math>V''(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net [[force]] is zero). Therefore the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately:
where <math>V'(x_0)</math>, <math>V''(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net [[force]] is zero). Therefore, the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately:
<math>
<math>
</math>
</math>
==See Also==
==See also==
*[[Newtonian mechanics]]
*[[Newtonian mechanics]]
*[[Potential energy]]
*[[Potential energy]]
==References==
==References==
{{Reflist}}
[[Category:Mechanics]]
[[Category:Mechanics]]