{"id":100356,"date":"2015-06-26T03:00:00","date_gmt":"2015-06-26T00:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=19a878405632c05e8b6ad5abc5cc17bd"},"modified":"2015-06-26T03:00:00","modified_gmt":"2015-06-26T00:00:00","slug":"stormer-problem-restricted-to-a-spherical-surface","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=100356","title":{"rendered":"St\u00f6rmer problem restricted to a spherical surface"},"content":{"rendered":"<p>In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic<br \/>\ndipole, we propose to study the restricted motion of the particle in a spherical surface with the<br \/>\ndipole at its centre. This model can be considered as the classical non-relativistic St\u00f6rmer problem<br \/>\nwithin a sphere, and although this problem no longer represents the real St\u00f6rmer problem, it shows<br \/>\nthe complex behaviour of this magnetic field through the classical dynamics equations that can be<br \/>\nformally integrated. We start from a Lagrangian approach which allows us to analyse the dynamical<br \/>\nproperties of the system, such as the role of a velocity dependent potential, the symmetries and the<br \/>\nconservation properties. We derive the Hamilton equations of motion, which in this restricted case<br \/>\ncan be reduced to a quadrature. From the Hamiltonian function we find, for the polar angle, an<br \/>\nequivalent one-dimensional system of a particle in the presence of an effective potential. This<br \/>\nequivalent p&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In order to analyse in full detail the dynamics of a charged particle in the field of a magnetic<br \/>\ndipole, we propose to study the restricted motion of the particle in a spherical surface with the<br \/>\ndipole at its centre. This model can be considered as t&#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-100356","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\/100356","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=100356"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/100356\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=100356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=100356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=100356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}