Conservative overdamped harmonic oscillator?

This isn't homework. I'm reviewing calculus and basic physics after many years of neglect.

I want to show that a damped harmonic oscillator in one dimension is nonconservative. Given F = -kx - v, if F were conservative then there would exist P(x) such that . I want to show that no such function, P(x), exists.

The easy way would be to find a closed curve around which the integral of Fdx would be zero, but since Fdx is a 1-dimensional 1-form, this doesn't seem to be a meaningful way to do it.

So I think brute force has to prevail. It should be true that:

So let

For underdamped

Therefore x(t) is not 1-1 is multivalued implies W is not a function implies p(x) doesn't exist (since W=-P) implies F is not conservative. Similarly for .

But in the overdamped case, >1, x(t) is a non-oscillating decaying exponential which never crosses equilibrium, implying x(t) is 1-1, implying W is a function, implying F is conservative. But how can this be? How can a frictional damping force, which dissipates energy as heat, ever be conservative?