- #1

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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?

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?