{"id":13534,"date":"2026-02-01T18:16:32","date_gmt":"2026-02-01T23:16:32","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=13534"},"modified":"2026-02-02T16:29:28","modified_gmt":"2026-02-02T21:29:28","slug":"double-pendulum","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2026\/02\/01\/double-pendulum\/","title":{"rendered":"Double Pendulum"},"content":{"rendered":"<div class=\"content\"><!--introduction-->\r\n<p>I was amazed by two recent YouTube videos about the double pendulum.<\/p>\r\n<!--\/introduction-->\r\n<h3>Contents<\/h3>\r\n<div>\r\n<ul>\r\n<li>\r\n<a href=\"#f998dacd-0007-401d-863f-d12f12440170\">Double Pendulum<\/a>\r\n<\/li>\r\n<li>\r\n<a href=\"#c10a74d0-ab37-4c8c-a6d8-4f9afd2df083\">Cleve's Lab<\/a>\r\n<\/li>\r\n<li>\r\n<a href=\"#1a2c8c2d-ff64-46b5-a4a5-3dc0ba5f556a\">Drew's Campfire<\/a>\r\n<\/li>\r\n<li>\r\n<a href=\"#4230ae8d-ecc2-4f3d-b7b9-bcc9769df0b7\">2Swap<\/a>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h4>Double Pendulum<a name=\"f998dacd-0007-401d-863f-d12f12440170\"><\/a>\r\n<\/h4>\r\n<p>The classic double pendulum consists of one pendulum attached to a fixed pivot and a second pendulum attached to the end of the first. The system is governed by Newton's laws, and its motion is completely determined by the initial conditions. There is no randomness. However, its behavior is sometimes difficult to predict. It may be <i>chaotic<\/i>.<\/p>\r\n<h4>Cleve's Lab<a name=\"c10a74d0-ab37-4c8c-a6d8-4f9afd2df083\"><\/a>\r\n<\/h4>\r\n<p>Ten years ago, I included the double pendulum in <a href=\"https:\/\/www.mathworks.com\/company\/technical-articles\/introducing-cleves-laboratory.html\">Cleve's Lab<\/a>.<\/p>\r\n<p>\r\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/swinger.gif\" alt=\"\"> <\/p>\r\n<h4>Drew's Campfire<a name=\"1a2c8c2d-ff64-46b5-a4a5-3dc0ba5f556a\"><\/a>\r\n<\/h4>\r\n<p>Drew's Campfire is one recent video about the double pendulum. Here is a <a href=\"https:\/\/www.youtube.com\/watch?v=8jVogdTJESw&amp;t=355s,\" target=\"_blank\" rel=\"noopener\">link to the video<\/a>.<\/p>\r\n<p>Here is a sample frame.<\/p>\r\n<p>\r\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/Drew.png\" alt=\"\"> <\/p>\r\n<h4>2Swap<a name=\"4230ae8d-ecc2-4f3d-b7b9-bcc9769df0b7\"><\/a>\r\n<\/h4>\r\n<p>The second video comes from 2Swap. Here is a <a href=\"https:\/\/www.youtube.com\/watch?v=dtjb2OhEQcU,\" target=\"_blank\" rel=\"noopener\">link to the video<\/a>.<\/p>\r\n<p>Here is a frame featuring the Pretzel pendulum.<\/p>\r\n<p>\r\n<img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/Pretzel.jpg\" alt=\"\"> <\/p>\r\n<script language=\"JavaScript\"> <!-- \r\n    function grabCode_6d2eb2765ab8436192df2a9fb7512ba8() {\r\n        \/\/ Remember the title so we can use it in the new page\r\n        title = document.title;\r\n\r\n        \/\/ Break up these strings so that their presence\r\n        \/\/ in the Javascript doesn't mess up the search for\r\n        \/\/ the MATLAB code.\r\n        t1='6d2eb2765ab8436192df2a9fb7512ba8 ' + '##### ' + 'SOURCE BEGIN' + ' #####';\r\n        t2='##### ' + 'SOURCE END' + ' #####' + ' 6d2eb2765ab8436192df2a9fb7512ba8';\r\n    \r\n        b=document.getElementsByTagName('body')[0];\r\n        i1=b.innerHTML.indexOf(t1)+t1.length;\r\n        i2=b.innerHTML.indexOf(t2);\r\n \r\n        code_string = b.innerHTML.substring(i1, i2);\r\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\r\n\r\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \r\n        \/\/ in the XML parser.\r\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\r\n        \/\/ doesn't go ahead and substitute the less-than character. \r\n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\r\n\r\n        copyright = 'Copyright 2026 The MathWorks, Inc.';\r\n\r\n        w = window.open();\r\n        d = w.document;\r\n        d.write('<pre>\\n');\r\n        d.write(code_string);\r\n\r\n        \/\/ Add copyright line at the bottom if specified.\r\n        if (copyright.length > 0) {\r\n            d.writeln('');\r\n            d.writeln('%%');\r\n            if (copyright.length > 0) {\r\n                d.writeln('% _' + copyright + '_');\r\n            }\r\n        }\r\n\r\n        d.write('<\/pre>\\n');\r\n\r\n        d.title = title + ' (MATLAB code)';\r\n        d.close();\r\n    }   \r\n     --> <\/script>\r\n<p style=\"text-align: right; font-size: xx-small; font-weight:lighter;   font-style: italic; color: gray\">\r\n<br>\r\n<a href=\"javascript:grabCode_6d2eb2765ab8436192df2a9fb7512ba8()\"><span style=\"font-size: x-small;        font-style: italic;\">Get \r\n      the MATLAB code <noscript>(requires JavaScript)<\/noscript>\r\n<\/span><\/a>\r\n<br>\r\n<br>\r\n      Published with MATLAB&reg; R2024b<br>\r\n<\/p>\r\n<\/div>\r\n<!--\r\n6d2eb2765ab8436192df2a9fb7512ba8 ##### SOURCE BEGIN #####\r\n%% Double Pendulum\r\n% I was amazed by two recent YouTube videos about the double pendulum.\r\n\r\n%% Double Pendulum\r\n% The classic double pendulum consists of one pendulum attached to a fixed\r\n% pivot and a second pendulum attached to the end of the first. The system\r\n% is governed by Newton's laws, and its motion is completely determined by\r\n% the initial conditions. There is no randomness.  However, its behavior is\r\n% sometimes difficult to predict.  It may be _chaotic_.\r\n\r\n%% Cleve's Lab\r\n% Ten years ago, I included the double pendulum in\r\n% <https:\/\/www.mathworks.com\/company\/technical-articles\/introducing-cleves-laboratory.html\r\n% Cleve's Lab>.\r\n%\r\n% <<swinger.gif>>\r\n%\r\n\r\n%% Drew's Campfire\r\n% Drew's Campfire is one recent video about the double pendulum.\r\n% Here is a\r\n% <https:\/\/www.youtube.com\/watch?v=8jVogdTJESw&t=355s,\r\n% link to the video>.\r\n%\r\n\r\n%%\r\n%\r\n% Here is a sample frame.\r\n%\r\n% <<Drew.png>>\r\n%\r\n\r\n%% 2Swap\r\n% The second video comes from 2Swap.  Here is a\r\n% <https:\/\/www.youtube.com\/watch?v=dtjb2OhEQcU,\r\n% link to the video>.\r\n%\r\n\r\n%%\r\n%\r\n% Here is a frame featuring the Pretzel pendulum.\r\n%\r\n% <<Pretzel.jpg>>\r\n%\r\n##### SOURCE END ##### 6d2eb2765ab8436192df2a9fb7512ba8\r\n-->\r\n","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img src=\"https:\/\/blogs.mathworks.com\/cleve\/files\/swinger2.gif\" class=\"img-responsive attachment-post-thumbnail size-post-thumbnail wp-post-image\" alt=\"\" decoding=\"async\" loading=\"lazy\" \/><\/div><!--introduction-->\r\n<p>I was amazed by two recent YouTube videos about the double pendulum.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2026\/02\/01\/double-pendulum\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":13540,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[32,24,5,23],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/13534"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/users\/78"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/comments?post=13534"}],"version-history":[{"count":2,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/13534\/revisions"}],"predecessor-version":[{"id":13541,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/13534\/revisions\/13541"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media\/13540"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=13534"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=13534"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=13534"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}