Why can't conservative forces depend on velocity? In my mechanics lecture notes, it is written that, for a force $F$,
To be conservative, $F$ must be a function of position only: forces that depend on velocity, time, etc. cannot be conservative.
How do I go about proving, mathematically, such statement?
In the case of a drag force (like air resistance), I know that we get energy dissipation and therefore such force cannot be conservative. How should I argue in a more general scenario?
 A: By definition, the work done against a conservative force to move an object from position $P$ to position $Q$ must be independent of exactly how the object moves from $P$ to $Q$. If the force depended on velocity, or changed with time etc. then the work done would not be independent of how the object moved from $P$ to $Q$.
A: *

*One of the ideas behind a conservative force is that it should be a state function on the configuration space $M$ of possible positions. Since the velocity is not a coordinate in the configuration space $M$, the force is forbidden to depend on velocity.

*Now the above conventional definition excludes e.g. the Lorentz force and the Coriolis force. If we want to modify the conventional definition of a conservative force to allow velocity-dependence, we should first of all replace the configuration space $M$ with its tangent bundle $TM$, which is the space of possible kinematic states of the system, described by a position and a velocity. Such a relaxed notion of conservative force is discussed in my Phys.SE answer here. 
