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We find, generally, that Kepler's Laws do not hold. What is more, even Newton's First Law (the law of inertia) fails on closed surfaces with variable curvature (e.g. the ellipsoid).\n                  <\/jats:p>","DOI":"10.1098\/rspa.2016.0020","type":"journal-article","created":{"date-parts":[[2016,8,10]],"date-time":"2016-08-10T05:08:23Z","timestamp":1470805703000},"page":"20160020","update-policy":"https:\/\/doi.org\/10.1098\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["N-body dynamics on closed surfaces: the axioms of mechanics"],"prefix":"10.1098","volume":"472","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5708-0281","authenticated-orcid":false,"given":"Stefanella","family":"Boatto","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada, Instituto de Matem\u00e1tica, Universidade Federal de Rio de Janeiro, Rio de Janeiro, CEP 21941-909, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David G.","family":"Dritschel","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rodrigo G.","family":"Schaefer","sequence":"additional","affiliation":[{"name":"Department de Matem\u00e0tica Aplicada, Facultat de Matem\u00e0tiques i Estad\u00edstica, Universitat Polit\u00e8cnica de Catalunya Pau Gargallo, 5. 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PhD thesis University of California San Diego CA USA."},{"key":"e_1_3_7_18_2","unstructured":"Shchepetilov AV. 2006 Calculus and mechanics on two-point homogeneous Riemannian spaces . Lecture Notes in Physics vol. 707. Berlin Germany: Springer."},{"key":"e_1_3_7_19_2","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-04-12913-6"},{"key":"e_1_3_7_20_2","volume-title":"Mathematical methods of classical mechanics","author":"Arnold VI","year":"1997"},{"key":"e_1_3_7_21_2","volume-title":"Classical mechanics","author":"Goldstein H","year":"2014"},{"key":"e_1_3_7_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(88)90028-9"},{"key":"e_1_3_7_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2009.05.003"},{"key":"e_1_3_7_24_2","unstructured":"Boatto S Dos Santos Loiola R Schaefer R Duarte G Stuchi T Dritschel D. In preparation. Dynamic of masses on surfaces: a geometry point of view."},{"key":"e_1_3_7_25_2","unstructured":"Wheeler J. 2008 Bertrand's theorem. 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