Time dependent force.
What happens if you have a time dependent force that still manages to satisfy
- [math]\vec \nabla \times \vec {F}(\vec r, t) = 0[/math]?
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
- [math]\Delta T = W = \int \vec F \cdot d \vec r[/math]
or
- [math]d T = \frac{dT}{dt} dt = (m \vec \dot v \cdot v) dt = \vec F \cdot d \vec r[/math]
If a potential U for the force exists such that
- [math]\vec F = - \vec \nabla U(r,t) \cdot d \vec r[/math]
or
- [math]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 [/math]
- [math]= - \vec F \cdot d \vec r + \frac{\partial U}{\partial t} dt [/math]
- [math]= - dT + \frac{\partial U}{\partial t} dt [/math]
or
- [math]dT + dU = \frac{\partial U}{\partial t} dt [/math]
Forest_UCM_Energy#Time_Dependent_PE