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Norton's dome is an example of indeterminism in Newtonian physics, based on a differential equation involving a non-Lipschitz continuous function [1]. It involves a gravitational field, in which a mass is placed with velocity zero at the apex of a dome (x=0,v=0), which has the following shape:

y(x)=−2/3(1−(1−3/2|x|)2/3)3/2.

Besides the trivial, singular solution, r(t)=0 (where r is the arc lenght measured along the dome), there is a one-parameter family of infinitely many solutions to this Cauchy problem, which can be represented geometrically as a `Peano broom':

r(t)=0(if t≤T); 1144(t−T)4(if t≥T),

where parameter T is a positive real number (representing the time of the onset of the movement).

Similar examples have been discussed by Poisson and other 19th century physicists [2]. We analyze and two conflicting intuitions about such cases using discrete models. In particular, we present an alternative model using difference equations and an infinitesimal hidden variable. (We use `infinitesimal' in the sense of non-standard analysis, which is close to Leibniz's formulation of the calculus as well as to physical praxis [3].) Our hyperfinite model for the dome is deterministic. Moreover, it allows us to assign probabilities to the variable in the indeterministic model.

References

[1] J. D. Norton. The dome: An unexpectedly simple failure of determinism. Philosophy of Science (2008) 75:786–798.

[2] M. van Strien. The Norton Dome and the Nineteenth Century Foundations of Determinism. J Gen Philos Sci (2014) 45:167–185.

[3] S. Albeverio, J. E. Fenstad, R. Hoegh-Krøhn, and T. Lindstrøm. Non-Standard Methods in Stochastic Analysis and Mathematical Physics. Pure and Applied Mathematics. Academic Press, Orlando, FL, 1986.