Graduate The force from the energy gradient

[user1139]

TL;DR

Please see below

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?

 

[Demystifier]

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?