- Norton's dome & Malament's mounds
1.1. Introduction
1.2. Assigning probabilities
1.3. Phase space vector field
In 2003, the philosopher of science John D. Norton presented a thought experiment which exhibits non-deterministic behaviour within the context of Newtonian mechanics. The setup is quite simple. Imagine a dome (hill or mound) with a shape such that a point mass located on the apex (where it has a zero velocity) may slide downward following the hill's curvature exactly due to the downward gravitational pull. The shape of the hill is given by the equation:
The dynamical problem can also be expressed in terms of the arc length r(t)
This appears to be an innocent example of an unstable equilibrium situation. However, the dome's shape is chosen such that
the initial value problem is not
Lipschitz-continuous
. As a result,
there is a continuum of solutions: the mass may stay at the apex forever, or it may start sliding off after any possible
delay time T.
Although the latter solutions could be considered as the result of small perturbations, just as in the case of deterministic
systems with an unstable equilibrium, no perturbation is required in this thought experiment. Here, the movement is initiated
spontaneously, without any perturbation, giving rise to non-deterministic behaviour in a system that looks like it could come straight
from a textbook on Newtonian physics.
Norton's dome problem can be generalized to an entire family of surfaces of the shape:
Cross-section of Malament's mounds for various values of the power a. The two limiting cases are indicated by the dashed lines.
Initial value problems that are indeterministic due to non-Lipschitz continuity do not come with a probability distribution.
We set out to find a natural procedure for assigning probabilities to their solutions[1].
Since Norton's dome received much attention by philosophers of science, as a simple example of indeterminism in Newtonian mechanics, we have used this toy example and Malament's generalization to develop a method for assigning probabilities to the different solutions.
Our approach starts from a discretization of time, which results in difference equations instead of differential equations. This approach is familiar to most physicists, but we have formalized the idea of infinitesimal time steps and perturbations with a recent mathematical formalism: Alpha-theory[4]. Before considering the Alpha-limit, we had to study the difference equations with non-infinitesimal time steps and perturbations. Since there is no known solution for the discrete version of Norton's dome and Malament's mounds, an important component of our study consisted of numerical experiments. This prompted us to develop this program. Some results can be seen in [1].
Our assignment of probabilities to solutions of the differential equation is based on a measure of the phase space of solutions
to the difference equations in the Alpha-limit. (This is possible because the map from the latter to the former is many-to-one.)
Dynamics of the discretized version of Norton's dome system on the phase space of initial conditions.
We found that the relation of the delay time T as function of the (infinitesimal) initial conditions is highly non-linear, with the positive
(red) delay times being very localised.
Dependence of T on R1 at a constant value of R0.
As a result of this, we found that regular solutions with any observable delay time have zero probability.
The regular solution with T=0, which describes that the mass slides off without any observable delay, has unit probability.
(The trivial solution of the mass remaining at the apex forever has zero probability as well.)
