{"id":440466,"date":"2017-08-17T02:00:00","date_gmt":"2017-08-16T23:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=3c85e21cfb4f7335c60cbd2382318e13"},"modified":"2017-08-17T02:00:00","modified_gmt":"2017-08-16T23:00:00","slug":"dynamics-of-a-particle-on-an-axisymmetric-surface-under-gravitational-force-a-geometric-approach","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=440466","title":{"rendered":"Dynamics of a particle on an axisymmetric surface under gravitational force: a geometric approach"},"content":{"rendered":"<p>In this paper the problem of a particle constrained to move on an axisymmetric surface embedded in<br \/>\nthree-dimensional Euclidean space, under the influence of a gravitational field, is addressed from a<br \/>\ngeometrical point of view. Using a covariant geometrical approach at undergraduate level, a<br \/>\nvariational framework is implemented in order to obtain the particle\u2019s motion equation, which is<br \/>\nsolved in an analytical way for the case where the particle moves on a cylinder and numerically when<br \/>\nthe particle moves on a catenoid. The trajectories obtained are expressed as functions of conserved<br \/>\nquantities such as total energy, E , and the z component of the particle\u2019s angular momentum, L 3 .<br \/>\nMoreover, expressions for normal and geodesic curvatures as well as the speed, tangential<br \/>\nacceleration and normal force constraining the particle on the surface along the trajectories are<br \/>\nobtained.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper the problem of a particle constrained to move on an axisymmetric surface embedded in<br \/>\nthree-dimensional Euclidean space, under the influence of a gravitational field, is addressed from a<br \/>\ngeometrical point of view. Using a covariant geome&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[178],"tags":[],"class_list":{"0":"post-440466","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-rss-fuusikaharidus","7":"entry"},"_links":{"self":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/440466","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=440466"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/440466\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=440466"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=440466"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=440466"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}