newtonian mechanics - When is "Inertia" Negligible?

I have started to read Strogatz's Nonlinear Dynamics and Chaos and I have come across an interesting bit. He states certain damped oscillators may be modeled as having no inertia term, I.E.

$$m \ddot{x} + b \dot{x} = F(x)$$

But if $m \ddot{x}\approx 0$ (he calls this the "Inertia Term")Then it is equally valid to write

$$b \dot{x} = F(x)$$

To me, this implies that the velocity is allowed to vary depending on the spatial coordinate. However I just said a second ago that either the acceleration or the mass is negligible. So shouldn't velocity be constant?

I think what is actually going on is while a temporal change in velocity is not allowed, a spatial one still may exist. Anyways, this is still a little strange to me so does anyone know of good resources on such subjects or times when this approximation is valid?

The equation appears on page 29, concerning the impossibility of oscillations for first order ordinary differential equations.

UPDATE:

After doing a bit of reflection and dimensional analysis, I have something else to add. $m$ has dimensions of [mass], while $b$ has dimensions of $\frac{[mass]}{[time]}$. Does this mean that a viable situation when this approximation holds is for some small time scales (how small and corresponding to what, I do not know). For if some time corresponding to $b$ is small, then $b$ is large, and maybe $m \ddot{x}$ can be neglected? Can someone verify this?

Answer

As you have noticed we cannot just solve the equation by setting $m$ to zero, this is because the term with small parameter is the only one containing the highest order derivative. In particular, Cauchy problem for the system still contains initial conditions both for, say, $x(0)$ and $\dot{x}(0)$. From the mathematical point of view such system is an example of a singular perturbation.

There are quite a lot of methods for analysis of such systems such as method of matched asymptotic expansions. Wikipedia page for it contains an example quite similar to the problem from the question.

Following this example, we apply this method here and build two approximate solutions (using notation from the question):

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  outer solution, which is valid for late times on the order of $t=O(b \cdot l / F)$ (where $l$ is the typical variation of $x$ and $F$ is the typical value of for $F(x)$) for which the governing approximate equation is $$b\, \dot{x} = F(x)$$. This is 1st order ODE, so the solution will have one integration constant.

  
  


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  inner solution, valid for small times $t=O(m/b)$. For this we rewrite the problem using rescaled time $\tau = b \cdot t/m $. The approximate equation will be $$\partial_\tau^2 x(\tau) + \partial_\tau x(\tau) =0$$. This is second order ODE, and much simpler than original system.

  
  

Then we match solutions in the overlap region, where both approximations are valid. Formally this region corresponds to dual limit: $$\lim_{\tau \to \infty} x(\tau) = \lim _{t \to 0} x (t).$$ From this equation we express one of the constant through the other two and obtain composite solution for the whole domain.