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It seems rather obvious that kinetic energy can be regarded as a property of a moving mass.. If carry a cup of hot coffee from one room to another, it seems clear that I have transported heat energy. That indicates that energy can also change position.

However, there is the problem of whether mass has a position. An idealized "point mass" can have a position and move. A real mass needs to have a mass density function. Moving a real mass through space moves its density function. So if I want to talk about the position of kinetic or heat energy in real objects, it appears I need an energy density function.

Then there is the problem of potential energy. If a point mass m is at height h above the surface of the Earth we say it "has" potential energy mgh, as if the energy is "in" the mass. But the hope of getting back (m)(g)(h) worth of work if it falls depends on the gravitation field exerting the force (m)(g) throughout the fall. You could say the potential energy depends on the potential of the gravitational field at the specific location of the point mass. However, the fact that a potential function exists depends on the global properties of the gravitational field.

Suppose a mass is at rest in a time-varying graviational field. It seems it ought to have potential energy since a stationary mass in a constant force field does. We could say the mass has a time varying potential energy, but how exactly is its potential energy defined? If its potential energy has a position, where is it? If the potential energy has a density function, what is the extent of it in space?