If I consider equations of motion derived from the principle of least action for an explicitly time dependent Lagrangian
$$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$
under what circumstances (i.e. which explicit functional $t$-dependence) is the force conservative?
By force I understand here the term on the right hand side of the equation, if I shove everything to the right except the expression $mq''(t)$.
As a side note, besides the technical answer I'd be interested here in some words about the physical motivations involved. I'm somewhat unhappy with a formal $\text{curl}[F]=0$ condition, since it seems to be to easy to fulfill (namely we have to consider closed circles only at single points in time, respectively). The physical motivation behind conservative forces is the conservation of energy on closed paths, where any parametrization $q(s)$ of curves can be considered. But practically, only loops tracked in finite time are physically realizable, i.e. we would move in a circle while t changes.
I guess as soon as one computes the r.h.s. for the equations of motion, one would also be able to define a more physical alternative to the above stated idea of conservative forces in this case. I.e. a ask-if-the-forces-integrate-to-zero-on-a-closed-loop functional for a rout between two points in time $t_1$ and $t_2$. This would be an integral where the momentarily force along the point in the path I'm taking would be taken into account. It wouldn't be path independent of course. (We could then even construct another optimization problem on its own, by asking for path with the smallest energy difference, which really would be a sensible question if friction is involved.)
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