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Title: | Assigning probabilities to non-Lipschitz mechanical systems | Authors: | VANPOUCKE, Danny E.P. WENMACKERS, Sylvia |
Issue Date: | 2021 | Publisher: | AMER INST PHYSICS | Source: | Chaos (Woodbury, N.Y.), 31 (12) (Art N° 123131) | Abstract: | We present a method for assigning probabilities to the solutions of initial value problems that have a Lipschitz singularity. To illustrate the method, we focus on the following toy example: d(2)r(t)/dt(2) = r(a), r ( t = 0 ) = 0, and d r ( t )/d t divide r ( t = 0 ) = 0, with a & ISIN; ] 0 , 1 [. This example has a physical interpretation as a mass in a uniform gravitational field on a frictionless, rigid dome of a particular shape; the case with a = 1 / 2 is known as Norton's dome. Our approach is based on (1) finite difference equations, which are deterministic; (2) elementary techniques from alpha-theory, a simplified framework for non-standard analysis that allows us to study infinitesimal perturbations; and (3) a uniform prior on the canonical phase space. Our deterministic, hyperfinite grid model allows us to assign probabilities to the solutions of the initial value problem in the original, indeterministic model. | Notes: | Wenmackers, S (corresponding author), Katholieke Univ Leuven, Inst Philosophy, Ctr Log & Philosophy Sci CLPS, Kardinaal Mercierpl 2,Bus 3200, B-3000 Leuven, Belgium. sylvia.wenmackers@kuleuven.be |
Document URI: | http://hdl.handle.net/1942/36495 | ISSN: | 1054-1500 | e-ISSN: | 1089-7682 | DOI: | 10.1063/5.0063388 | ISI #: | 000739120300004 | Rights: | 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0063388 Open access | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2023 |
Appears in Collections: | Research publications |
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Assigning probabilities to non-Lipschitz mechanical systems.pdf | Published version | 3.85 MB | Adobe PDF | View/Open |
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