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Homework Help: Conservative system <-> time independent potential?

  1. Mar 21, 2015 #1
    Is it possible that the potential energy depends on the time and the system is conservative?

    Let me elaborate. Consider a function ##U(\vec{r},t)## and consider the case where the forces in space at any moment are given by ##\vec{F}=-\nabla{U}##. So in this case, according to the definition of conservative, the force field is conservative.

    However the energy is time dependent ( since the Lagrangian would be time dependent the energy is not conserved ).

    QUESTION: Often I have seen conservative and time-independent potential energy used interchangably. As you see, it seems to be not the case. So what's the real connection between conservative and time independent?
  2. jcsd
  3. Mar 21, 2015 #2
    A mass attached to a spring that is allowed to oscillate up and down (in the absence of air resistance) seems to fit your parameters.
  4. Mar 21, 2015 #3
    What? I'm not sure I follow. The potential energy in that case is only a function of the position, not time explicitly.
  5. Mar 21, 2015 #4
  6. Mar 21, 2015 #5
    Yes I understand that energy is not conserved, but what about the definitions of a conservative force as :

    "A force is conservative if there exists a function of which the force is a gradient" : See : http://en.wikipedia.org/wiki/Conservative_force
  7. Mar 21, 2015 #6
    Oh...so all you are worried about is nomenclature. Call it what you want, I call a conservative force a force that results in conservation of energy and one necessary condition is that it can be written a the gradient of a time-independent scalar function.
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