Another Unexpectedly Simple Failure of Determinism

Here’s a simple Newtonian system that fails to determine a unique Cauchy evolution, in the spirit of Norton’s dome. In the system I will describe, one begins with a mass sitting at rest. There are then two possible evolutions: the ball might just sit there for all of time, or it might start bouncing for no apparent reason.

Take a spring with a mass in the middle that’s sitting at rest. Let’s set the center of the mass to be the center of our coordinate system x = 0. The differential equation describing the motion of this system (in a neighborhood of the origin) is that of a harmonic oscillator,

x” = -k2x,


where k is a constant describing the tension of the spring. Now, imagine that we wind up the spring very tightly, and then start letting it loose again, at a rate determined by the equation,

-k2 = 6/t2.


The equation of motion for this system is then given by,

x” = 6x/t2.


One obvious solution to this differential equation is x(t) = 0, at all times t. This represents an evolution in which the ball just sits there at rest, for all of time. However, it is easily shown that there is also a second solution:

x(t) = 0 for t ≤ 0
= t3 for t ≥ 0


So, determinism fails for this spring system: there is more than one Cauchy evolution for the initial conditions x = 0, t ≤ 0. It might be that the ball just sits there; or, it might be that the ball starts bouncing for times t > 0.


What’s the pathology? (Left as a puzzle…)


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