Conservative Overdamped Harmonic Oscillator?

[inkliing]

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 - [math]\small\mu[/math]v, if F were conservative then there would exist P(x) such that [math]\small -\frac{dP}{dx} = F[/math]. 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:

 

[math]\small W=\int_{x_1}^{x_2}Fdx = \int_{x_1}^{x_2}(-kx-\mu v)dx = \frac{1}{2}kx_1^2-\frac{1}{2}kx_2^2-\mu\int_{x_1}^{x_2}\frac{dx}{dt}dx = \frac{1}{2}kx_1^2-\frac{1}{2}kx_2^2-\mu\int_{t_1}^{t_2}\left(\frac{dx}{dt}\right)^2 dt[/math]

So let [math]\small\omega_{\circ}=\sqrt{k/m}\mbox{ , }\zeta=\frac{\mu}{2\sqrt{mk}}\mbox{ , }\omega_1=\left\{\begin{matrix}\omega_{\circ}\sqrt{\zeta^2-1},&\zeta>1\\\omega_{\circ}\sqrt{1-\zeta^2},&\zeta<1\end{matrix}\right.[/math]

 

For underdamped [math]\small\zeta<1\Rightarrow x=e^{-\zeta\omega_{\circ}t}(C_1 cos\omega_1 t + C_2 sin\omega_1 t)[/math]

 

[math]\small\Rightarrow W=\frac{1}{2}kx_1^2-\frac{1}{2}kx_2^2-\mu\int_{t_1}^{t_2}e^{-2\zeta\omega_{\circ}t}[(-\zeta\omega_{\circ}C_1+\omega_1 C_2) cos\omega_1 t + (-\omega_1 C_1-\zeta\omega_{\circ} C_2) sin\omega_1 t]^2 dt[/math]

 

Therefore x(t) is not 1-1 [math]\small\Rightarrow \int_{x_1}^{x_2}vdx[/math] is multivalued implies W is not a function implies p(x) doesn't exist (since W=-[math]\small\Delta[/math]P) implies F is not conservative. Similarly for [math]\small\zeta=1[/math].

 

 

 

But in the overdamped case, [math]\small\zeta[/math]>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?

[Erasmus00]

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.

 

Non-zero. For a non-conservative force, the integral would be non-zero.

 

In general, you are overthinking this. Take a step back- can you find two paths along which a different amount of work is done? What happens if I travel slowly from x = 0, t = 0 to x = x_1, t = t_1. Now, what happens if I travel quickly to arrive at x_1, and then simply wait until t = t_1. Does the force do different amounts of work? Is the work path independent?