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I'm trying to grasp basic concepts of energy and I have a question regarding conservation of energy.

According to Feynman's book and Wikipedia, the law of conservation of energy states that isolated physical systems (physical systems with no resulting force being exerted on them, only internal forces) have their energy conserved.

But then I picture some object at some point above the surface of earth (disregarding air resistance). It is clearly not an isolated physical system (gravity is being exerted on it) but somehow, while it's freefalling, its energy is being conserved (its potential energy, based on its position relative to earth, is being transfered to its kinetic energy).

I know the earth-object system can be thought of as an isolated system, but what I'm curious is that for some non-isolated systems (the mass above the earth), energy is being conserved.
So, it seems like the conservation law of energy sometimes works only for isolated systems, but sometimes for non-isolated systems.

So, to what kinds of system does the law of conservation of energy really apply? Or equivalently, what are "isolated physical systems", really?

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    $\begingroup$ Potential energy is a property of the Earth-object system. You can't say it's just a property of the object. $\endgroup$ – John Rennie Feb 13 '15 at 10:04
  • $\begingroup$ This is a bit wierd, because in fact what is causing the gravitational motion of the object is the curvature of the space-time, so while it would still make sense to talk about Potential energy of the object whenever in a certain position of the curved space-time, it becomes wierd now to talk about potential energy of the object-earth system. Don't you agree ? $\endgroup$ – nerdy Feb 13 '15 at 17:19
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As you quoted,

the Law of Conservation of Energy states that Isolated Physical systems have their energy conserved.

But, nobody says that a non-isolated system cannot have its energy conserved.

The mass above the Earth is acted upon by a field which is conservative, the gravitational field. In a conservative field the total energy of a body is conserved, while only transforming, reversibly from one form to another. In your case, the gravitational potential energy transforms into kinetic and back. I.e. by falling on the Earth, if the collision of the object with the Earth is completely elastic (no loss of energy), the object would bounce back, then fall again, and so on.

Absolutely isolated systems don't exist, only approximately isolated. The Universe is full of bodies, s.t. any object is subjected to the attraction of other objects. In our labs we do all sort of experiments in vacuum but we can never realize absolute vacuum.

About the conservation of energy, in the classical physics we can say that it always holds, on condition that we take in consideration all the forms of energy involved in the process under examination. For instance, if when examining your object moving in the gravitational field of the Earth we take in consideration the friction with the air, the conservation of the mechanical energy doesn't hold anymore. But if we add the energy lost by friction, the total energy remains constant.

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Expanding on what Sofia said, the mass above the Earth isn't an isolated system because gravity is acting on it. As she said only approximated isolated systems exit, imagine a lump of iron in a room. There is a magnetic field set up in the room, but because you are inside it you don't know it. The lump begins to move and you measure KE. You would conclude the low is broken, but really you were not considering a closed system. If you knew there was a field, then you would have measure a potential energy before and thus the gain in KE would have made sense

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