{"id":318,"date":"2017-02-05T16:13:36","date_gmt":"2017-02-05T16:13:36","guid":{"rendered":"http:\/\/www.briansutton.uk\/?p=318"},"modified":"2017-02-05T16:51:16","modified_gmt":"2017-02-05T16:51:16","slug":"the-brachistochrone-problem","status":"publish","type":"post","link":"https:\/\/www.briansutton.uk\/?p=318","title":{"rendered":"The Brachistochrone problem"},"content":{"rendered":"<p>Brachistochrone<\/p>\n<p>I was struck by this GiF recently, at:<\/p>\n<p><a href=\"http:\/\/imgur.com\/5t32VJU\">http:\/\/imgur.com\/5t32VJU<\/a><\/p>\n<div style=\"width: 580px;\" class=\"wp-video\"><!--[if lt IE 9]><script>document.createElement('video');<\/script><![endif]-->\n<video class=\"wp-video-shortcode\" id=\"video-318-1\" width=\"580\" height=\"325\" preload=\"metadata\" controls=\"controls\"><source type=\"video\/mp4\" src=\"http:\/\/www.briansutton.uk\/wp-content\/uploads\/2017\/02\/5t32VJU_Brachistochrone.mp4?_=1\" \/><a href=\"http:\/\/www.briansutton.uk\/wp-content\/uploads\/2017\/02\/5t32VJU_Brachistochrone.mp4\">http:\/\/www.briansutton.uk\/wp-content\/uploads\/2017\/02\/5t32VJU_Brachistochrone.mp4<\/a><\/video><\/div>\n<p>&nbsp;<\/p>\n<p>This little video shows three possible paths &#8211; the straight line (the shortest), a path that descends quickly and then becomes flat as it gets to the destination, and one in between, looking more elliptical* in shape.<\/p>\n<p>Ever wondered what downhill profile of road will get you most quickly to the bottom of the hill freewheeling from a given point T (the top) to a given point P (the pub)? Now you know &#8211; it&#8217;s a cycloid, appropriately enough. Now I just need to think about the best shape of road from the pub to the top for the fastest ascent for a given a) wattage, and b) for a given pedal pressure&#8230;<\/p>\n<p>&#8230;The classical problem is well-known as the difficult to solve Brachistochrone problem. See <a href=\"http:\/\/www.math.utk.edu\/~freire\/teaching\/m231f08\/m231f08brachistochrone.pdf\">http:\/\/www.math.utk.edu\/\u2026\/m231f08\/m231f08brachistochrone.pdf<\/a> for a solution &#8211; minimising a quantity over a fami\u0142y of functions, or <a href=\"http:\/\/www.osaka-ue.ac.jp\/zemi\/nishiyama\/math2010\/cycloid.pdf\">http:\/\/www.osaka-ue.ac.jp\/ze\u2026\/nishiyama\/math2010\/cycloid.pdf<\/a> which is perhaps a little more accessible and talks a little more about the cycloid (the resolution of the problem).<\/p>\n<p>Even more arcanely (I was going to say more interestingly, but I&#8217;m realistic about reading endurance!) the cycloid, as the second article says, is isochronous. It means that even if you start freewheeling from a rest position R half way down the hill, it still takes the same time to get to the pub. Why? Briefly, because starting higher up, at the intermediate point R you are moving, whereas starting from R you are at rest. Weird, huh? It&#8217;s the same for any point R, no matter how close to the pub.<\/p>\n<p>*the elliptical path is nearly as fast as the cycloid &#8211; but not quite!!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Brachistochrone I was struck by this GiF recently, at: http:\/\/imgur.com\/5t32VJU &nbsp; This little video shows three possible paths &#8211; the straight line (the shortest), a path that descends quickly and then becomes flat as it gets to the destination, and one in between, looking more elliptical* in shape. Ever wondered what downhill profile of road [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","jetpack_publicize_message":"","jetpack_is_tweetstorm":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/paDNTg-58","jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/posts\/318"}],"collection":[{"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=318"}],"version-history":[{"count":3,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/posts\/318\/revisions"}],"predecessor-version":[{"id":322,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=\/wp\/v2\/posts\/318\/revisions\/322"}],"wp:attachment":[{"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=318"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=318"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.briansutton.uk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}