# General Scientific Meeting 2016 of the Belgian Physical Society

18 May 2016
UGent - Het Pand
Europe/Brussels timezone
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# Contribution Oral contribution

UGent - Het Pand - Priorzaal
Biological, Medical, Statistical and Mathematical Physics

# Indeterminism in Newtonian physics: a quest for probabilities

## Speakers

• Prof. Sylvia WENMACKERS

## Content

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': $$r(t)= \begin{array}{cl} 0 & (\textrm{if }\ t \leq T); \ \frac{1}{144}(t-T)^{4} & (\textrm{if }\ t \geq T),\end{array}$$ 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.