Confusion about potential energy, field energy, kinetic energy

Question

I have some confusion about potential energy in Newtonian mechanics and field energy in classical Field mechanics. I have many questions but they are all strongly related.

In Newtonian mechanics, we analyze a system of two particles A & B attracted to each other by a conservative force, and say that there is kinetic energy and potential energy. Kinetic energy depends on their masses and velocities, and potential energy depends on their distance.

• Is it correct to say that particle A has kinetic energy? I think so, since it depends solely on properties of that particle.

• If so, is it correct to say that such kinetic energy is located within particle A?

• Is it correct to say that particle A has potential energy? Maybe not, since it depends of the distance between both particles. On the other hand, we can assign a part of the potential energy to particle A and the rest to B. And by taking those derivatives we obtain the force that acts on each one, separately. So I'm not sure if it's a property of the system as a whole, or of each separate particle.

• If so, is it correct to say that such potential energy is located within particle A?

On the other hand, there is Field mechanics for that same system. We say that there is an attractive field. And that this field has energy.

• If we introduce fields and field energy, then we must discard potential energy, right? They are alternate expressions of the same thing?

• Does each point in space has some field energy located within it? Or is that just for calculating purposes, and the field energy is considered to be just a property of the whole system?

• Finally, is it any more correct to talk about field energies than to talk about potential energies? Or are they completely equivalent models?

Answer 1

In Newtonian mechanics, we analyze a system of two particles A & B attracted to each other by a conservative force, and say that there is kinetic energy and potential energy. Kinetic energy depends on their masses and velocities, and potential energy depends on their distance.

You say the two particles are attracted by a conservative force. So yes there is potential energy. You say the particles have kinetic energy, so I am assuming they are moving. But you did not provide a reference frame with respect to which their velocities (and thus kinetic energy) is being measured. This is important since kinetic energy is reference frame dependent.

Is it correct to say that particle A has kinetic energy? I think so, since it depends solely on properties of that particle.

The kinetic energy does not depend solely on the properties of the particle. It depends on the frame of reference of the observer, since velocity (and therefore kinetic energy) is reference frame dependent.

So it is correct to say that particle A has kinetic energy of 1/2 mv$^2$ but only with respect to a frame of reference where its velocity is $v$.

If so, is it correct to say that such kinetic energy is located within particle A?

It is correct to say that particle A possesses kinetic energy with respect to the reference frame of the observer.

Is it correct to say that particle A has potential energy? Maybe not, since it depends of the distance between both particles. On the other hand, we can assign a part of the potential energy to particle A and the rest to B. And by taking those derivatives we obtain the force that acts on each one, separately. So I'm not sure if it's a property of the system as a whole, or of each separate particle.

Potential energy is a system property, not a property of a particle or object alone. That's because potential energy is a property of position and position only has meaning with respect to something else.

In this case, the potential energy is that of the particle AB system.

If so, is it correct to say that such potential energy is located within particle A?

Not wishing to sound repetitive, but the potential energy is located in the particle AB system, not within particle A.

On the other hand, there is Field mechanics for that same system. We say that there is an attractive field. And that this field has energy.

I'm not familiar with the term "Field mechanics". But in classical mechanics the potential energy does not belong to the field alone any more than potential energy belongs to the particle alone. Potential energy belongs to the particle-field system.

Hope this helps.

Answer 2

Yes each point of the space have some field energy located there. The potential energy is another expression of it in a simpler form. This equality can be seen by integration by part. $$\mathcal{E}_{field}=\iiint (\nabla \phi)^2d^3x=-\iiint \phi\Delta\phi d^3x = \iiint\phi \rho d^3x = \mathcal{E}_{potential}$$ where $\phi$ is the potential and $\rho$ the density of charge and we used the Poisson equation $\Delta \phi = -\rho$.