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What is the physical meaning of curvature?

by cnelson
Tags: curvature, meaning, physical
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Oct17-11, 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.
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Oct18-11, 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.
Oct18-11, 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.

Oct18-11, 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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