https://en.wikipedia.org/wiki/Norton%27s_dome Jump to content [ ] Toggle sidebar [wikipe] Wikipedia The Free Encyclopedia Search [ ] [Search] [Go] * Create account * Log in [ ] Personal tools * Create account * Log in Pages for logged out editors learn more * Talk * Contributions Navigation * Main page * Contents * Current events * Random article * About Wikipedia * Contact us * Donate Contribute * Help * Learn to edit * Community portal * Recent changes * Upload file Tools * What links here * Related changes * Upload file * Special pages * Permanent link * Page information * Cite this page * Wikidata item Print/export * Download as PDF * Printable version Languages On this Wikipedia the language links are at the top of the page across from the article title. Go to top. [ ] Contents move to sidebar hide * (Top) * 1Description + 1.1Solutions to the equations of motion * 2Resolutions to the paradox + 2.1Indeterminate derivatives * 3See also * 4References * 5External links Toggle the table of contents [ ] Norton's dome [ ] 3 languages * Deutsch * Francais * Portugues Edit links * Article * Talk [ ] English * Read * Edit * View history [ ] More * Read * Edit * View history From Wikipedia, the free encyclopedia Nondeterministic Newtonian mechanical system [300px-Cross_section_of_Norton] Cross section of Norton's dome, where h and x are measured in units of 2 g 2 / ( 3 b 4 ) {\displaystyle 2g^{2}/(3b^{4})} {\displaystyle 2g^{2}/(3b^{4})}. Norton's dome is a thought experiment that exhibits a non-deterministic system within the bounds of Newtonian mechanics. It was devised by John D. Norton in 2003.^[1]^[2] It is a special limiting case of a more general class of examples from 1997 due to Sanjay Bhat and Dennis Bernstein.^[3] The Norton's dome problem can be regarded as a problem in physics, mathematics, or philosophy.^[4]^ [5]^[6]^[7] Description[edit] The model consists of an idealized particle initially sitting motionless at the apex of an idealized radially symmetrical frictionless dome described by the equation^[6]^[7] h = 2 b 2 3 g r 3 2 ; 0 <= r < g 2 b 4 , {\displaystyle h={\frac {2b^{2}}{3g}}r^{\frac {3}{2}}\;;\;0\leq r<{\frac {g^{2}}{b^ {4}}},} {\displaystyle h={\frac {2b^{2}}{3g}}r^{\frac {3}{2}}\;; \;0\leq r<{\frac {g^{2}}{b^{4}}},} where h is the vertical displacement from the top of the dome to a point on the dome, r is the geodesic distance from the dome's apex to that point (in other words, a radial coordinate r is "inscribed" on the surface), g is acceleration due to gravity and b is a proportionality constant.^[6] From Newton's second law, the tangent component of the acceleration on a point mass resting frictionlessly on the surface is a [?] = b 2 r {\displaystyle a_{\parallel }=b^{2}{\sqrt {r}}} {\displaystyle a_{\ parallel }=b^{2}{\sqrt {r}}}.^[6] Solutions to the equations of motion[edit] Norton shows that there are two classes of mathematical solutions to this equation. In the first, the particle stays sitting at the apex of the dome forever. In the second, the particle sits at the apex of the dome for a while, and then after an arbitrary period of time starts to slide down the dome in an arbitrary direction. The apparent paradox in this second case is that this would seem to occur for no discernible reason, and without any radial force being exerted on it by any other entity, apparently contrary to both physical intuition and normal intuitive concepts of cause and effect, yet the motion is still entirely consistent with the mathematics of Newton's laws of motion.^[citation needed] To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction. However, the same argument applied to the usual kinds of domes (e.g., a hemisphere) fails, because a ball launched with just the right energy to reach the top and stay there would actually take infinite time to do so.^[8]^[non-primary source needed] Resolutions to the paradox[edit] This section does not cite any sources. Please help improve [50px-Q] this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2020) (Learn how and when to remove this template message) While many criticisms have been made of Norton's thought experiment, such as it being a violation of the principle of Lipschitz continuity (the force that appears in Newton's second law is not a Lipschitz continuous function of the particle's trajectory -- this allows evasion of the local uniqueness theorem for solutions of ordinary differential equations), or in violation of the principles of physical symmetry, or that it is somehow in some other way "unphysical", there is no consensus among its critics as to why they regard it as invalid. Indeterminate derivatives[edit] A simple criticism of the thought experiment is as follows, however: The entire argument hinges on the behavior of the particle at the point r = 0 {\displaystyle r=0} r=0, during a time period where it has zero velocity. Traditional Newtonian mechanics would say that the position of the particle would, infinitesimally be h = F 2 m ( D t ) 2 {\displaystyle h={\frac {F}{2m}}(\Delta t)^ {2}} {\displaystyle h={\frac {F}{2m}}(\Delta t)^{2}} , for some small time D t {\displaystyle \Delta t} \Delta t, but because the second derivative of the surface does not exist at this point, the force is indeterminate. It's therefore completely sensible that the infinitesimal motion of the object is also indeterminate. See also[edit] * Indeterminism * Spontaneous symmetry breaking References[edit] 1. ^ Norton, John D. (November 2003). "Causation as Folk Science". Philosophers' Imprint. 3 (4): 1-22. hdl:2027/spo.3521354.0003.004 . 2. ^ Laraudogoitia, Jon Perez (2013). "On Norton's dome". Synthese. 190 (14): 2925-2941. doi:10.1007/s11229-012-0105-z. S2CID 37756181. 3. ^ Bhat, Sanjay P.; Bernstein, Dennis S. (1997-02-01). "Example of indeterminacy in classical dynamics". International Journal of Theoretical Physics. 36 (2): 545-550. Bibcode:1997IJTP...36..545B . doi:10.1007/BF02435747. ISSN 1572-9575. S2CID 10195818. 4. ^ Reutlinger, Alexander (2013). A Theory of Causation in the Social and Biological Sciences. Palgrave Macmillan. p. 109. ISBN 9781137281043. 5. ^ Wilson, Mark (2009). "Determinism and the Mystery of the Missing Physics" (PDF). The British Journal for the Philosophy of Science. 60 (1): 173-193. doi:10.1093/bjps/axn052. 6. ^ ^a ^b ^c ^d Fletcher, Samuel Craig (2011). "What counts as a Newtonian system? The view from Norton's dome". European Journal for Philosophy of Science. 2 (3): 275-297. CiteSeerX 10.1.1.672.9952. doi:10.1007/s13194-011-0040-8. S2CID 10898530. 7. ^ ^a ^b Malament, David B. (2008). "Norton's Slippery Slope". Philosophy of Science. 75 (5): 799-816. doi:10.1086/594525. ISSN 0031-8248. S2CID 2436612. PhilSci:3195. 8. ^ Norton, John. "The Dome". www.pitt.edu. Retrieved 20 January 2021. External links[edit] * John Norton's webpage for the Norton dome problem * Retrieved from "https://en.wikipedia.org/w/index.php?title= Norton%27s_dome&oldid=1112863687" Categories: * Classical mechanics * Physical paradoxes * Thought experiments * Determinism * Thought experiments in physics Hidden categories: * Articles with short description * Short description matches Wikidata * All articles with unsourced statements * Articles with unsourced statements from December 2020 * Articles needing additional references from December 2020 * All articles needing additional references * This page was last edited on 28 September 2022, at 13:56 (UTC). * Text is available under the Creative Commons Attribution-ShareAlike License 3.0 ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia(r) is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. * Privacy policy * About Wikipedia * Disclaimers * Contact Wikipedia * Mobile view * Developers * Statistics * Cookie statement * Wikimedia Foundation * Powered by MediaWiki