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written by
Todd Timberlake
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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 *August 20, 2013*

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This file has previous versions.*