Is there a measure of internal energy flow?

Question

A system might have internal energy and/or kinetic energy. Kinetic energy in classical mechanics is a form of energy the object has, only because of its relative movement to other objects.

If you have a system like an oscillator, then you have a system where the amount of internal energy (energy of the spring, say) and the kinetic energy oscillate periodically. The amount of energy that is flowing from "internal energy" to "kinetic energy" and back therefore also meassures how this system is different from a system with might have the same amount of energy in total, but all its energy only stored in form of internal energy.

Is there a reasonable meassure of this difference, that is the ''energy-form flow''?

Maybe such a meassure might also be related to the entropy, because if there is kinetic energy around, I expect the entropy to be higher than otherwise.

Then I was thinking mabye that something like $\frac{dE_{kin}}{dt}$ might therefore be relevant. But the only quantity which directly puts energy and time together is the action. However, I'm not sure if the action has an illustative meaning, which goes beyond its definition, anyway.

Answer 1

It's not used particularly often, but there is a measure of energy flow called the "mechanical Poynting vector." Just as the Poynting vector for electromagnetic radiation in a vacuum gives the energy flux carried by the EM wave, the mechanical Poynting vector gives the energy flux carried by waves in a mechanical system. Wikipedia gives the definition as

$$\mathbf{S}_m(\mathbf{r},t) = \frac{1}{2}\sum_i m_i \dot{r}_i^2(t)\dot{\mathbf{r}}_i(t)\delta\bigl(\mathbf{r} - \mathbf{r}_i(t)\bigr)$$

Or more generally, if you have a continuous system (a fluid), it would be

$$\mathbf{S}_m(\mathbf{r},t) = \frac{1}{2}\rho(\mathbf{r},t)v^2(\mathbf{r},t)\mathbf{v}(\mathbf{r},t)$$

Keep in mind that this is a vector field, which makes it a local measure: the value of $\mathbf{S}_m$ at each point only measures energy flow at that point.

Answer 2

The physical notion of energy flow (and momentum flow) is contained in the stress-energy tensor. The momentum is the flow of energy, the stress is the flow of momentum. The reason this is not apparent in potential energy problems is becuase the potential energy is stored in the mutual gravitational field of the object and the Earth, and this field reconfigures itself at the speed of light in response to the motion of the object. This means that the flow of potential energy is to and fro from far away places, essentially instantly.

The stress tensor is reviewed in this answer: Where does the reaction to action come from?

Answer 3

I'm not sure if this is what you were thinking about, but the change in potential energy is (minus) the work done by the conservative force related to the potential energy in question.

For example, in an ideal spring, $W_s = -F_s\,\mathrm{d}x = \mathrm{d}E_e$, with $W_s$ the work done by the spring, $F_s$ its force, and $\mathrm{d}E_e$ the differential of the elastic energy. So, if you divide by the differential of time, you get the instantaneous power: $P_s = -F_s\,v = \frac{\mathrm{d}E_e}{\mathrm{d}t}$. This tells you how fast the elastic potential energy is changing. If this is the only force, it can only go to kinetic energy.

Furthermore, the rate of change in kinetic energy is the instantaneous power of the net force acting on the system: $\frac{\mathrm{d}E_\mathrm{kin}}{\mathrm{d}t} = P_\mathrm{net\ force}$.