Least Action (Gravity/Free) Documents

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Main Document

Least Action (Gravity/Free) 

written by Todd Timberlake

The EJS Least Action (Gravity/Free) model illustrates the principle of least action for the one dimensional motion of a free particle or a particle subject to a constant gravitational force. The simulation displays height versus time, with the path broken into equally spaced time intervals. The user can set the initial and final heights, as well as the number of time intervals to be used. The user can then adjust the intermediate heights in order to minimize the action along the path, or else allow the computer to implement an algorithm for finding the path of least action. Both the action of the current path and the least action so far observed with the current parameters are displayed.

The simulation can also display average values for velocity, change in velocity, acceleration, kinetic energy, potential energy, the Lagrangian function, and total mechanical energy for each segment of the path. This helps to illustrate that the path of least action is also a path of constant acceleration (zero if there is no gravity) and constant total energy.

The algorithm minimizes the action by examining three consecutive points on the path. The outer two points are held fixed and basic calculus can be used to determine the value for the middle height that minimizes the action for this segment of the path. This procedure is repeated for each segment of three points on the path, moving left to right. If this entire process is repeated over and over the path will gradually approach the path of global least action. This approach, and the entire EJS Least Action (Gravity/Free) model, was inspired by the Principle of Least Action Interactive page by Edwin Taylor and Slavomir Tuleja.

Published September 3, 2012
Last Modified June 10, 2014

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Supplemental Documents

Principle of Least Action Tutorial 

This is a tutorial exercise that helps students understand the Principle of Least Action as applied to a particle moving vertically under a constant gravitational force.  Students use basic calculus to derive the condition for minimizing the action along a segment of the path, and then use the Least Action (Gravity/Free) EJS model to examine what happens when that condition is repeatedly applied to all segments of the path.

This file is available in multiple formats: .pdf, .tex

Released under a This activity handout is released under the Creative Commons BY-NC-SA 3.0 license.

Published September 5, 2012
Last Modified September 5, 2012

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Source Code Documents

Least Action (Gravity/Free) Source Code 

The source code zip archive contains an XML representation of the EJS Least Action (Gravity/Free) Model.   Unzip this archive in your EJS workspace to compile and run this model using EJS.

Released under a This material is released under the GNU General Public License Version 3.

Published September 3, 2012
Last Modified June 10, 2014

This file has previous versions.