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I had this thought in the shower one day. Assume we have a particle of mass $m$ at the top of an inclined plane of length $L$ and angle $\theta$. What if instead of some constant or deterministic friction coefficient $\mu(x)$, the friction coefficient randomly varies at each infinitesimal step $dx$? I assume that this can be interpreted in two ways: we could see the random $\mu$ as some kind of white noise from a distribution $p(x)$, where $\mu(x)$ has no correlation with $\mu(x+dx)$, or we could alternatively assume that while $\mu$ is random, it has some inherent structure like continuity.

In either case, can we find some differential equation to model the motion of this particle?

To me, this seems like a natural extension of the usual freshman year problem.

I know people like seeing attempts at solving the problem by the problem author, but I have no clue on how to even start. I unfortunately don't have any background in stochastic processes/SDEs, which I assume would be useful here.

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    $\begingroup$ You have an object sliding down the ramp. What is the surface area of the bottom of the object? The force of friction would be the integral over that area. This would change as the object slides, but it would be a continuous function. Suppose the object is a rectangle. If the object slide a distance $dx$, an area $ydx$ would be exposed at the back and the same area would be covered in front. The friction averaged over those strips would tell you how the force changes with time. $\endgroup$ Commented Feb 3 at 18:46
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    $\begingroup$ Why not simply write the equations of motion, generate a random $\mu(x)$ and solve the equation numerically? If the coefficient is deterministic there is no need for SDE $\endgroup$ Commented Feb 3 at 19:54

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