{"id":46,"date":"2013-05-29T19:53:36","date_gmt":"2013-05-29T19:53:36","guid":{"rendered":"http:\/\/homepages.se.edu\/cmoretti\/?page_id=46"},"modified":"2019-10-28T15:42:41","modified_gmt":"2019-10-28T15:42:41","slug":"mathematica-notebooks-differential-equations","status":"publish","type":"page","link":"https:\/\/www.se.edu\/cmoretti\/main\/mathematicalibrarymain\/mathematica-notebooks-differential-equations\/","title":{"rendered":"Dr. Moretti&#8217;s Mathematica notebooks &#8211; Differential Equations"},"content":{"rendered":"<h1 style=\"text-align: center\">Mathematica Notebooks for Differential Equations<\/h1>\n<p><span style=\"text-decoration: underline\">Important Note:<\/span>\u00a0The links for the notebooks open a new window or tab with a Google Drive page &#8211; the current settings for our homepages won&#8217;t allow me to host mathematica notebooks locally.<\/p>\n<h2>A Real and Linearized Pendulum<\/h2>\n<p>When simulating the motion of a pendulum the true equation is one which is not solvable exactly as it is nonlinear (for a unit mass it is\u00a0<em>y&#8221;+d y&#8217;+k sin(y)<\/em>=0, where <em>y<\/em> is the angle of the pendulum,\u00a0<em>d<\/em> is a friction constant, and\u00a0<em>k<\/em> is a &#8220;spring constant&#8221; related to the length of the pendulum and gravity). \u00a0When the angle\u00a0<em>y<\/em> is small in size \u00a0you can use linearization <em>sin(y)~y<\/em>\u00a0to approximate the pendulum with the equation\u00a0<em>y&#8221;+d y&#8217;+k y<\/em>=0, which is solvable exactly (this is why when talking about pendulums most high school physics textbooks have a small note in the margin along the lines &#8220;this is accurate is the angle of the swing is small, say less than 8\u00b0&#8221;). \u00a0This notebook lets you see the graphs of the solutions to the real and linearized equations of the pendulum in the phase plane. \u00a0You can move the starting point for both curves (linearized in blue\/green, ideal in red) which corresponds to the initial values for the system &#8211; and as the initial angle\/angular velocity gets larger, you can see the curves begin to separate (in fact if the initial angle is large enough the two curves can converge to different points!). \u00a0The manipulation will let you change the values for the friction constant, spring constant, the viewing range for the graphs, and the time ranges of the solution curves.<em><br \/>\n<\/em><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2.jpg\"><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter  wp-image-141\" alt=\"reallinearpenduluum\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2.jpg\" width=\"434\" height=\"668\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2.jpg 868w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2-195x300.jpg 195w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2-768x1182.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/reallinearpenduluum2-665x1024.jpg 665w\" sizes=\"(max-width: 434px) 100vw, 434px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImUTlKQk5sWkZPNzg&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other of my Differential Equations notebooks from Google Drive<\/a><\/p>\n<h2>A Mass-Spring System<\/h2>\n<p>This notebook simulates the motion of a mass bouncing back and forth on a spring, possibly with an external force acting on it. \u00a0You can change the mass, friction constant, spring constant, initial position and velocity, graphing range, time range, and external force (which is represented by a red arrow acting from the center of mass). \u00a0You can use this to explore underdamping, critical damping, overdamping, and resonance.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2.jpg\"><img decoding=\"async\" class=\"aligncenter  wp-image-142\" alt=\"mass-spring\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2.jpg\" width=\"343\" height=\"272\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2.jpg 1372w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2-300x238.jpg 300w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2-768x609.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/mass-spring2-1024x812.jpg 1024w\" sizes=\"(max-width: 343px) 100vw, 343px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImUTlKQk5sWkZPNzg&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other of my Differential Equations notebooks from Google Drive<\/a><\/p>\n<h2>A Double Pendulum<\/h2>\n<p>This manipulation simulates an ideal double pendulum (i.e. frictionless with massless pendulum arms). \u00a0You can change the pendulum lengths, bob masses, and initial angles and angular speeds. \u00a0You can also control the time range and number of frames per second in the animation. \u00a0Please note that I got the differential equations from the <a href=\"http:\/\/scienceworld.wolfram.com\/physics\/DoublePendulum.html\">Double Pendulum<\/a> page at Wolfram Research&#8217;s excellent Scienceworld site. \u00a0Because of the animation this notebook is much larger than most of the ones I have posted.<\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1.jpg\"><img decoding=\"async\" class=\"aligncenter  wp-image-143\" alt=\"doublependulum\" src=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1.jpg\" width=\"410\" height=\"323\" srcset=\"https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1.jpg 1640w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1-300x236.jpg 300w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1-768x605.jpg 768w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1-1024x807.jpg 1024w, https:\/\/www.se.edu\/cmoretti\/wp-content\/uploads\/sites\/81\/2013\/05\/doublependulum1-1568x1235.jpg 1568w\" sizes=\"(max-width: 410px) 100vw, 410px\" \/><\/a><\/p>\n<p style=\"text-align: center\"><a href=\"https:\/\/drive.google.com\/folderview?id=0BxSwUt0i0mImUTlKQk5sWkZPNzg&amp;usp=sharing\" target=\"_blank\" rel=\"noopener noreferrer\">Download this or other of my Differential Equations notebooks from Google Drive<\/a><\/p>\n<p><a title=\"Mathematica Notebook Library\" href=\"http:\/\/homepages.se.edu\/cmoretti\/main\/mathematicalibrarymain\/\">Back to my Mathematica Library<\/a><\/p>\n<p><a title=\"Dr. Christopher Moretti\u2019s Homepage\" href=\"http:\/\/homepages.se.edu\/cmoretti\/\">Back to my homepage<\/a><\/p>\n<p><a title=\"Southeastern main page\" href=\"http:\/\/www.se.edu\">Back to the Southeastern Main Page<\/a>\t\t<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Mathematica Notebooks for Differential Equations Important Note:\u00a0The links for the notebooks open a new window or tab with a Google Drive page &#8211; the current settings for our homepages won&#8217;t [&hellip;]<\/p>\n","protected":false},"author":24,"featured_media":0,"parent":19,"menu_order":0,"comment_status":"open","ping_status":"open","template":"page-templates\/page-with-sidebar.php","meta":{"footnotes":""},"class_list":["post-46","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Dr. Moretti&#039;s Mathematica notebooks - Differential Equations -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.se.edu\/cmoretti\/main\/mathematicalibrarymain\/mathematica-notebooks-differential-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Dr. Moretti&#039;s Mathematica notebooks - Differential Equations -\" \/>\n<meta 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