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I'm not sure if this belongs here or in the physics section. The mathematical definition of curvature is the derivative of the unit tangent vector normalized to the arc length: [itex]\kappa[/itex] = [itex]\frac{dT}{ds}[/itex]. If we apply this to a parabola with equation y = [itex]x^{2}[/itex] we get [itex]\frac{2}{(1+4x^{2})^{3/2}}[/itex]. This resembles a lorenzian line shape which is the distribution function (amplitude vs frequency) of a harmonic oscillator in a parabolic potential (i.e. a graph of amplitude vs frequency shows a resonance at some frequency.)

My question:

Is the curvature of a potential energy function related to the distribution function of the potential energy function or does it have some other physical relationship that I am missing? Or is the resemblance of the curvature of a parabola and the lorenzian line shape a coincidence?

Sorry if this is kind of vague let me know if you have questions.

My question:

Is the curvature of a potential energy function related to the distribution function of the potential energy function or does it have some other physical relationship that I am missing? Or is the resemblance of the curvature of a parabola and the lorenzian line shape a coincidence?

Sorry if this is kind of vague let me know if you have questions.

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