What is the physical meaning of curvature?

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SUMMARY

The discussion centers on the mathematical definition of curvature, specifically its application to a parabola represented by the equation y = x², yielding a curvature formula of κ = 2/(1 + 4x²)^(3/2). Participants explore the relationship between the curvature of potential energy functions and their corresponding distribution functions, particularly in the context of harmonic oscillators in parabolic potentials. The consensus indicates that while curvature is related to distribution functions, the nature of this relationship is complex and warrants further investigation, particularly through the concept of parallel transport.

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  • Understanding of curvature in differential geometry
  • Familiarity with potential energy functions in physics
  • Knowledge of harmonic oscillators and their properties
  • Basic concepts of parallel transport in differential geometry
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  • Study the mathematical definition and applications of curvature in differential geometry
  • Explore the relationship between potential energy functions and their distribution functions
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Students and professionals in physics and mathematics, particularly those interested in the geometric interpretation of physical concepts and the relationship between curvature and energy distributions.

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