Is there a name for this fact in physics? is it a theorem?

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Discussion Overview

The discussion revolves around the relationship between force and potential energy in physics, specifically the expression F = -∂V/∂x. Participants explore whether this relationship has a specific name or is classified as a theorem, and they touch on its implications in both classical and quantum mechanics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant inquires about the origin and nomenclature of the relationship between force and potential energy.
  • Another participant identifies the force as a "conservative force" and notes that not all forces fit this description, citing kinetic friction as an example.
  • A different participant mentions that while the relationship is not a theorem in physics, it relates to a mathematical concept known as "the lemma of Poincare," which connects conservative vector fields to potential functions.
  • Another participant references the Ehrenfest Theorem in quantum mechanics as an equivalent concept to the classical relationship discussed.

Areas of Agreement / Disagreement

Participants express differing views on whether the relationship is a theorem, with some asserting it is not a theorem in physics while others reference mathematical principles. The discussion remains unresolved regarding a definitive classification.

Contextual Notes

There are limitations in the discussion regarding the definitions of terms like "conservative force" and the applicability of the lemma of Poincare, as well as the context in which the Ehrenfest Theorem is relevant.

Dixanadu
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Hi guys,

So just wondering - the fact that the force is always the negative derivative of potential with respect to distance:
[itex]F=-\dfrac{\partial V}{\partial x}[/itex]

Where does this come from and does it have a name or something? like a theorem perhaps?

Thanks!
 
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Such a force is called a "conservative force", a "force derivable from a potential energy function".
However, not all forces are like this... for example, kinetic friction force is not of this type.
 
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It is not a theorem in physics, but one in mathematics called "the lemma of Poincare", which, when applied to R^3, states that any conservative vector field (thus defined to have a 0 circulation along a closed path, equivalently having 0 curl) is the gradient of a scalar field we call potential.
 
Dixanadu said:
Hi guys,

So just wondering - the fact that the force is always the negative derivative of potential with respect to distance:
[itex]F=-\dfrac{\partial V}{\partial x}[/itex]

Where does this come from and does it have a name or something? like a theorem perhaps?

Thanks!

The equivalent in Quantum Mechanics is called the Ehrenfest Theorem. There's lots about that online.
 

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