The force from the energy gradient

In summary, the term ##E\equiv\frac{1}{2}\left(\partial_x\phi\right)^2## in classical field theory represents the gradient energy, which is equivalent to elastic potential energy. The associated force, $$F\equiv -\frac{\partial E}{\partial\left(\partial_x\phi\right)}=-\partial_x\phi$$, cannot be simplified to ##-\partial_x\phi## and is not scale invariant. The factor of ##2## and ##4## are not interchangeable in this context.
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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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  • #2
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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What is "The force from the energy gradient"?

"The force from the energy gradient" is a concept in physics that describes the force exerted on an object due to a change in energy over a certain distance. It is also known as the force of potential energy or the force of a potential field.

How is "The force from the energy gradient" calculated?

The force from the energy gradient is calculated using the formula F = -∇U, where F is the force, ∇ is the gradient operator, and U is the potential energy. This formula takes into account the change in potential energy over a certain distance.

What is the relationship between "The force from the energy gradient" and potential energy?

"The force from the energy gradient" is directly related to potential energy. In fact, it is the force that is responsible for changes in potential energy. As the potential energy changes, so does the force from the energy gradient.

How does "The force from the energy gradient" affect an object's motion?

"The force from the energy gradient" affects an object's motion by causing it to move in the direction of the energy gradient. This means that the object will move from areas of higher potential energy to areas of lower potential energy.

Can "The force from the energy gradient" be observed in everyday life?

Yes, "The force from the energy gradient" can be observed in everyday life. For example, when a ball is rolled down a hill, the force from the energy gradient is what causes it to accelerate and move towards the bottom of the hill where the potential energy is lower.

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