{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T11:17:44Z","timestamp":1753874264482,"version":"3.41.2"},"reference-count":31,"publisher":"AIP Publishing","issue":"12","license":[{"start":{"date-parts":[[2021,12,30]],"date-time":"2021-12-30T00:00:00Z","timestamp":1640822400000},"content-version":"vor","delay-in-days":29,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2021,12,30]],"date-time":"2021-12-30T00:00:00Z","timestamp":1640822400000},"content-version":"tdm","delay-in-days":29,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003130","name":"Fonds Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["G066918N"],"award-info":[{"award-number":["G066918N"]}],"id":[{"id":"10.13039\/501100003130","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,12,1]]},"abstract":"<jats:p>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: d2r(t)dt2=ra, r(t=0)=0, and dr(t)dt\u2223r(t=0)=0, with a\u2208]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\u2019s 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.<\/jats:p>","DOI":"10.1063\/5.0063388","type":"journal-article","created":{"date-parts":[[2021,12,30]],"date-time":"2021-12-30T13:41:12Z","timestamp":1640871672000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Assigning probabilities to non-Lipschitz mechanical systems"],"prefix":"10.1063","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5919-7336","authenticated-orcid":false,"given":"Danny E. 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