Discontinuous systems? And why do we need uniqueness anyway?

[askmathquestions]

Much of the theory of ordinary differential equations is based around continuous derivatives. A lot of nice theories came together with semi-group theory of linear systems and the Banach contraction theorem, but these are limited to continuous functions. Then you get into partial differential equations and uniqueness is generally applied to PDE with analytic coefficients.

Can we introduce functions with a finite number of step discontinuities and still achieve uniqueness? Such functions are Riemann-integrable correct? So, does uniqueness hold for some classes of $L^1$ functions?

If not, can we identify the points where the trajectories are not unique, and decide "which" trajectory is chosen with some book-keeping of the inertia or "direction" of a trajectory?

If you think about this physically, a car or bike on a 2D surface might go in a loop-de-loop, it's trajectory will overlap with itself at multiple points. You'll notice this doesn't magically stop the car from attaining a smooth, continuous trajectory, even though mathematically, multiple trajectories may intersect with themselves or each other and and are therefore not unique.

Analogously, I don't believe uniqueness should truly be a problem for non-unique solutions to differential equations, so long as we can keep track of the "direction" or "inertia". But, how do we do that? What constraint or assumptions do we introduce to differential equations to attain continuous and practical trajectories that may model physical phenomena where trajectories are not necessarily unique?

 

[pbuk]

Can we introduce functions with a finite number of step discontinuities and still achieve uniqueness? Such functions are Riemann-integrable correct? So, does uniqueness hold for some classes of $L^1$ functions?

What have you learned about this in your studies so far?

If not, can we identify the points where the trajectories are not unique, and decide "which" trajectory is chosen with some book-keeping of the inertia or "direction" of a trajectory?

Are you asking "if we have a problem which has multiple solutions, can we introduce additional information that makes the solution unique"? What do you think the answer is?

If you think about this physically, a car or bike on a 2D surface might go in a loop-de-loop, it's trajectory will overlap with itself at multiple points. You'll notice this doesn't magically stop the car from attaining a smooth, continuous trajectory, even though mathematically, multiple trajectories may intersect with themselves or each other and and are therefore not unique.

Yes, so $\dfrac {dy}{dx} = f(x, y)$ is not a good model for a car's motion. Can you think of a better one?

Analogously, I don't believe uniqueness should truly be a problem for non-unique solutions to differential equations, so long as we can keep track of the "direction" or "inertia". But, how do we do that? What constraint or assumptions do we introduce to differential equations to attain continuous and practical trajectories that may model physical phenomena where trajectories are not necessarily unique?

Given your answers to my questions above, do you think there is a unique way of determining these additional constraints?

 

[PeterDonis]

a car or bike on a 2D surface might go in a loop-de-loop, it's trajectory will overlap with itself at multiple points

Its trajectory in ordinary space, but ordinary space is not the correct space to be looking at if you want uniqueness. For a deterministic system, the trajectory in phase space is what must be unique (no overlapping with itself--although it is possible for an idealized system to have a phase space trajectory that is a closed curve).