
#1
Oct1711, 03:16 PM

P: 5

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. 



#2
Oct1811, 09:09 AM

P: 336

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.




#3
Oct1811, 10:52 AM

P: 34

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




#4
Oct1811, 12:26 PM

P: 5

What is the physical meaning of curvature?
Thanks for the responses. I'm not familiar with parallel transport so I'll study up and see if it can answer my question.



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