A The force from the energy gradient

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The discussion centers on the interpretation of gradient energy in classical field theory, specifically identifying the term E as elastic potential energy. It questions whether the associated force F can be expressed as -∂xφ, derived from the energy term. The conversation clarifies that changing the coefficient from 1/2 to 2 affects the force calculation, as the force is not scale invariant. It emphasizes that the factor in the energy term cannot be ignored, as it directly impacts the force's definition. Understanding the relationship between energy and force is crucial for accurate physical interpretations.
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From this post-gradient energy in classical field theory, one identifies the term ##E\equiv\frac{1}{2}\left(\partial_x\phi\right)^2## as the gradient energy which can be interpreted as elastic potential energy.

Can one then say that $$F\equiv -\frac{\partial E}{\partial\left(\partial_x\phi\right)}=-\partial_x\phi$$
is the associated force?

In addition, if one has the factor as ##2## instead of ##\frac{1}{2}##, can one just ignore the factor of ##4## and claim that the associated force is ##-\partial_x\phi## since the factor is just a scaling?
 
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No and no. The second no is obvious; the force is not scale invariant, so the scale matters. To understand the first no, ask yourself: the force on what?
 

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