Classical mechanics - Conservative system

  1. Could you please address the following?

    (1) For a conservative system, total energy (E = KE + PE) is a constant. It does not depend on time. Is it because of this time-independent nature that E is a constant of motion?

    (2) For a conservative system, the PE is a function of position coordinates alone. It does not depend on time 'explicitly'. If so, is PE a constant of motion?

    (3) Is the time-independent nature of PE a characteristic feature of a conservative system?

    (4) Is PE a function of position coordinates and time for a non-conservative system?

  2. jcsd
  3. (1) I'm not sure if I understood your question, but if a quantity doesn't change over time, then by definition it's a constant of motion.

    (2) The total potential energy of a conservative system can change, so then it is not a constant of motion. If the potential energy function depends on position, then the potential energy of an object will necessarily depend on time, because the object's motion will cause it to change position over time.

    (3) and (4) By "time-independent", I take it you mean that the potential energy of an object U is a function like U(r) and not U(r, t). I wouldn't call this time-independent, because the position r can still change with time. All this means is that if the object(s) don't change their position (or relative position if we're considering the potential energy of two objects), then the potential energy doesn't change.

    At any rate, your question is an interesting one. If the potential energy isn't "time-independent" (using your definition of time-independent), then it's worth noting that by definition, it isn't potential energy proper. Potential energy MUST be of the form U(r), without time mentioned explicitly. To see why U(r, t) wouldn't be conservative, just imagined that you took an object and moved it around in a closed loop. For a conservative force, there cannot be a change in potential energy. But if the potential energy was "time-dependent", then there would almost surely be a change, because there's no guarantee that U(r, t_1) = U(r, t_2).

    Alternatively, suppose two interacting objects initially at rest had a pole attached between them, so that their distance from one another couldn't change. By the conservation of momentum, neither of them can start moving (there must be a zero net momentum, and they can't start moving towards one another because of the pole), so the total kinetic energy will remain zero. However, the potential energy, if it is "time-dependent", will change, so that the total energy is not constant.

    On the other hand, if a potential energy function is "time-independent", that does not mean it is conservative. Friction is a good example.
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