Forest UCM Energy TimeDepPE

Time dependent force.

What happens if you have a time dependent force that still manages to satisfy

$\vec \nabla \times \vec {F}(\vec r, t) = 0$?

Because of the above, and Stoke's Theorem , you would be able to find a close loop where zero work is done at some given time.

If we consider the work energy theorem

$\Delta T = W = \int \vec F \cdot d \vec r$

or

$d T = \frac{dT}{dt} dt = (m \vec \dot v \cdot v) dt = \vec F \cdot d \vec r$

If a potential U for the force exists such that

$\vec F = - \vec \nabla U(r,t) \cdot d \vec r$

or

$dU(r,t) = \frac{\partial U}{\partial x} dx +\frac{\partial U}{\partial y} dy +\frac{\partial U}{\partial z} dz +\frac{\partial U}{\partial t} dt$

$= - \vec F \cdot d \vec r + \frac{\partial U}{\partial t} dt$

$= - dT + \frac{\partial U}{\partial t} dt$

or

$dT + dU = \frac{\partial U}{\partial t} dt$

Forest_UCM_Energy#Time_Dependent_PE