What is the physical meaning of curvature?

[cnelson]

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: $\kappa$ = $\frac{dT}{ds}$. If we apply this to a parabola with equation y = $x^{2}$ we get $\frac{2}{(1+4x^{2})^{3/2}}$. 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.

 

[Eynstone]

The curvature of potential energy is definitely related to distribution function, but the dependence is weird.I can't give any physical meaning to curvature which might suit the dependence.

 

[kdbnlin78]

The curvature of a surface is (or at least can be) defined as moving a vector through parallel transport around a closed loop.

 

[cnelson]

Thanks for the responses. I'm not familiar with parallel transport so I'll study up and see if it can answer my question.