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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?

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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 $Q$.

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  • $\begingroup$ How do I prove that mathematically? For example, for a force $F=-k v$, where $v$ is the velocity? $\endgroup$
    – sam wolfe
    Commented Feb 5, 2020 at 16:08
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  1. 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.

  2. 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.

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