Potential Energy: Rigorous Understanding & E=mc²

[Mohammed Ayaz Quadri]

I used to believe that potential energy of a particular particle is of no meaning. It is always linked with a system, and Potential Energy of a system means negative of work done by INTERNAL conservative forces from an initial stage of assumed zero potential energy. And energy cannot be said to be stored in one part of a system. Its just IN the system SOMEWHERE. Am I correct.

When I go through advanced courses online or in books like Kleppner and Kolenkow they do not mention anything about constraint of internal forces only.

Please someone give me rigorous understanding if Potential Energy and its inclusion in E=mc²

 

[vanhees71]

What do you mean by "internal forces"? What is in your opinion missing in the standard textbooks?

 

[PeroK]

Why not believe Kleppner & Kolenkow?

If you consider the Earth and a satellite as a two-body system, then the PE comes from an internal force. But, if you consider the motion of the satellite only, then the PE of the satellite comes from a gravitational field, which represents an external force.

 

[vanhees71]

What are "internal and external forces"? Isn't this overcomplicating the simple issue of a Newtonian two-body problem?

The potential of the forces in this case simply is (just simplifying it to pointlike Earth and sattelite)
$$V(\vec{x}_1,\vec{x}_2)=-\frac{G}{|\vec{x}_1-\vec{x}_2|}$$
with the force excerted on the Earth being
$$\vec{F}_1=-\vec{\nabla}_1 V=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_1-\vec{x}_2),$$
and the force excerted on the satellite being
$$\vec{F}_2=-\vec{\nabla}_1 V=-\vec{F}_1=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_2-\vec{x}_1).$$

 

[Mohammed Ayaz Quadri]

What do you mean by "internal forces"? What is in your opinion missing in the standard textbooks?

I mean by Internal Forces, forces operating within different parts of system, whose dynamic effect cancels out (forming action reaction pair).

 

[Mohammed Ayaz Quadri]

What are "internal and external forces"? Isn't this overcomplicating the simple issue of a Newtonian two-body problem?

The potential of the forces in this case simply is (just simplifying it to pointlike Earth and sattelite)
$$V(\vec{x}_1,\vec{x}_2)=-\frac{G}{|\vec{x}_1-\vec{x}_2|}$$
with the force excerted on the Earth being
$$\vec{F}_1=-\vec{\nabla}_1 V=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_1-\vec{x}_2),$$
and the force excerted on the satellite being
$$\vec{F}_2=-\vec{\nabla}_1 V=-\vec{F}_1=-\frac{G}{|\vec{x}_1-\vec{x}_2|^3} (\vec{x}_2-\vec{x}_1).$$

This seems interesting. I believe with potential you are talking about scalar potential of the helmholtz decomposition. But idk much about that. Instead of pointing issues in the view I mentioned, can you please give me your detailed view of PE?

Thank You.

 

[vanhees71]

In classical mechanics a potential is a scalar function, from which the forces on the particles are given as the gradients of this scalar function. In the example above you have a potential for a closed two-body system, which (as a closed system must) obeys all the symmetry principles of Galilei-Newton spacetime. As you see, as a consequence of spatial translation invariance, the potential only depends on the difference $\vec{x}_1-\vec{x}_2$ of the position vectors of the particles and thus the center of mass is unaccelerated and thus you have $\vec{F}_1=-\vec{F}_2$, which is also known as Newton's Third Law of actio=reactio. Further $V$ is a scalar function, i.e., a function of $|\vec{x}_1-\vec{x}_2|$ only, and this must be so, because of the invariance under spatial rotations (isotropy of space). Finally, the potential is time-independent, which is due to the temporal translation invariance (homogeneity of time). All these symmetries together, which build a symmetry group of transformations of the space-time coordinates lead to the known conservation laws:
$$\vec{P}=m_1 \dot{\vec{x}}_1 + m_2 \dot{\vec{x}}_2=\text{const}., \quad \vec{J}=m_1 \vec{x}_1 \times \vec{p}_1 + m_2 \vec{x}_2 \times \vec{p}_2=\text{const}, \quad E=\vec{p}_1^2/(2m_1) + \vec{p}_2^2/(2m_2) +V(|\vec{x}_1-\vec{x}_2|)=\text{const}.,$$
and the center of mass
$$\vec{R}=\frac{m_1 \vec{x}_1 + m_2 \vec{x}_2}{m_1+m_2}$$
moves with constant velocity.

In this context of point-particle mechanics it doesn't make sense to ask about, "where the energy sits". It's a property of the system as a whole. Only in continuum mechanics and field theory you have the notion of an energy density, and to some extent it can make sense to ask, how "energy is distributed" over the system, but that's not such a trivial issue as it might seem!

 

[Dale]

What are "internal and external forces"? Isn't this overcomplicating the simple issue of a Newtonian two-body problem?

I don’t think this is an over complication. It is a very important exercise to define “the system” since many laws describe the exchange of conserved quantities across the system boundaries. So it is important to be able to identify what is in the system and what is external. It is not always obvious for a new student.

 

[vanhees71]

You mean the description as a closed system (which I had in mind) vs. the approximation to treat the heavy body as a source of an external field, within which the light body moves without taking into account the back reaction to the heavy body? That's of course an important point, and it's not a simple one too. Of course then you neglect to motion of the heavy body and you loose some of the symmetries (particularly translational invariance in this case) and you have only a restricted set of conserved quantities left (in this case energy from temporal translation invariance and angular momentum from the rotational invariance around the then fixed center of force). Then "internal" and "external" makes sense of course.