# Conservative system <-> time independent potential?

1. Mar 21, 2015

### Coffee_

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. Mar 21, 2015

### AlephNumbers

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.

3. Mar 21, 2015

### Coffee_

What? I'm not sure I follow. The potential energy in that case is only a function of the position, not time explicitly.

4. Mar 21, 2015

### assed

5. Mar 21, 2015

### Coffee_

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

6. Mar 21, 2015

### assed

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.