I Gradient Energy: Definition & Classical Mechanics

[Haorong Wu]

TL;DR Summary

Gradient energy is given by $\frac 1 2 (\nabla \phi)^2$. What does it represent?

In page 40 of Spacetime and geometry by Sean M. Carroll, when consider the classical mechanics of a single real scalar field, it reads that the field will have an energy density including various contributions:
kinetic energy:$\frac 1 2 \dot \phi^2$
gradient energy:$\frac 1 2 (\nabla \phi)^2$
potential energy:$V(\phi)$

I am not familiar with gradient energy. I googled it, but it returns with energy gradient, which I do not think is the same thing.

Also, is this gradient energy introduced because it and kinetic energy can combine into a covariant form?

Thanks!

 

[haushofer]

It's the energy cost due to spatial variation of the field phi. It's just nomenclature.

 

[Haorong Wu]

It's the energy cost due to spatial variation of the field phi. It's just nomenclature.

Thanks. Is there any simple example that could help me memorize it?

I am thiking about electric field. Could I say the field vary from point to point, so the energy associated with one point is different from another one. The difference between two very close points will be something like gradient energy?

 

[haushofer]

Thanks. Is there any simple example that could help me memorize it?

I am thiking about electric field. Could I say the field vary from point to point, so the energy associated with one point is different from another one. The difference between two very close points will be something like gradient energy?

Yes, as long as you don't confuse the electric field for a scalar field ;)

 

[Haorong Wu]

Hi, @haushofer . I am a little confused now. Could a scalar field not represent an electric field? I thought this was valid because in some papers, I read that the scalar field is used to study the transpotation of light. For example, in https://arxiv.org/abs/2009.04217 , the paragraph before Eq. (2).

 

[haushofer]

Maybe in some effective treatment I'm not familiar with, but a general time dependent electric field cannot be written with just a scalar field.