# Non-deterministic particle system

This question is in the spirit of Norton's dome, an example of an apparently non-deterministic system in Newtonian mechanics. Under certain restrictions, the Picard–Lindelöf theorem guarantees the existence and uniqueness of solutions to differential equations with initial conditions. However, Norton's dome does not satisfy these restrictions. The ball may sit on top of the dome for an arbitrary period of time before sliding down in a particular trajectory, without violating Newton's laws of motion. Is there a plausible example of a system of interacting point-particles such that the equations of a point-particle are continuous but not Lipschitz continuous, and therefore fail to have a unique solution?

• More on Norton's dome: physics.stackexchange.com/q/39632 – Kyle Kanos Oct 13 '14 at 16:13
• Norton's dome is easily falsified. The appearance of multiple solutions is more because of inadequately applied constraints (and the square root makes the others easy to intuit) but they're non-Newtonian and energy is not conserved in any of them. Time stitching them at arbitrary T might allowed mathematically but has no physical justification whatsover and introduces non-determinism. Full explanation here: blog.gruffdavies.com/2017/12/24/… – Gruff Dec 27 '17 at 8:51

I do not know if it is "plausible" (I do not think so), however a trivial model can be constructed for the one-dimensional case with continuous forces depending on velocities, for $c>0$ constant: $$F_{12}(v_1,v_2) = c\sqrt{|v_1-v_2|} \quad\mbox{and}\quad F_{21}(v_1,v_2) = -c\sqrt{|v_1-v_2|}$$ The system of these two particles does not admit a unique solution if initial conditions of the form $x_1(0)=y_1$, $x_2(0)= y_2$ and $v_1(0)=v_2(0) = V$ are given.