Conservative and non-conservative fields

[e2m2a]

The gravitational field is defined as a conservative field for small displacements of a body in the field. The potential energy of a body in the field would only be dependent on the position of the body.

Any sufficiently small displacements of the body in a closed path in this field starting at point A, going to point B, and returning to point A would yield a zero net change in energy.

But what happens if the field varies in intensity in time as the body moves in its closed path? Suppose, for example, that the mass of the Earth was constantly decreasing in time. (There are continuous mass ejections at escape velocity from opposite ends of the equator.)

Would the gravitational field cease to be conservative? Would a closed loop displacement yield changes in energy not equal to zero? Could F = -dU/dx be modified to include a time-variant field U(x,t) that makes sense?

 

[mfb]

The gravitational field is defined as a conservative field for small displacements of a body in the field.

A constant gravitational field (in Newtonian gravity) is conservative everywhere, for all displacements.

But what happens if the field varies in intensity in time as the body moves in its closed path?

The gravitational field at each point in time is conservative, but now you can extract energy by moving around over some time period. But: a real closed loop would have to go back in time, which is a bit unphysical.

 

[e2m2a]

But: a real closed loop would have to go back in time, which is a bit unphysical.

No. A physical closed loop. A → B → A. A physcial loop in space, not a "loop" in time. In the hypothetical scenario, a person at point A at time 0 could measure g to be 9.8 m/s sq. At point B at time 0 + delta t could measure g to be 9.5 m/s sq. At point A again at time 0 + 2*delta t, the person could measure g to be 9.0 m/s sq.

Clearly, the field would cease to be conservative.

Imagine a similar situtation. An observer moves a 1 kg mass body 1 meter up in the Earth's gravitational field. The increase in GPE of the body would be about 9.8 joules. This increase in GPE could easily be accounted for by the 9.8 joules of work done on the body to move it to a higher PE.

If the observer allows the body to fall back 1 meter, the KE of the body will exactly equal the increase in PE of the body prior to the fall. Everything adds up, mechanical energy is conserved in this field.

But suppose while the body fell down, the Earth acquired mass by bombardments of meteorites or a hugh influx of electromagnetic energy which was completely absorbed by the Earth with no reflection. Now, the observer will measure the KE to be greater than its initial PE because the intensity of the field has increased as the body falls downward. Mechanical energy will not be conserved.

The observer, (unware of the influx of mass/energy to the earth), would conclude something very strange just happened to the field. There is an increase in KE unaccounted for by the initial work done to give the body its initial GPE.

The field is not a conservative field in this case. So, a time-dependent gravitational field must yield strange results not normally observed with a time-independent gravitational field.

 

[mfb]

A physcial loop in space, not a "loop" in time.

You return to the same point in space. But if the field varies in time, this position is different from the starting position: You are at a different time.

If you take all relevant energy into account, energy is conserved in Newtonian gravity. See this post for details.