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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.

Please note that this resource requires at least version 1.5 of Java (JRE).
View the supplemental document attached to this resource
  • 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.
View the source code document attached to this resource
  • 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.
Subjects Levels Resource Types
Motion, Forces, and Energy
- General
- Gravity
- Motion in One Dimension
= Gravitational Acceleration
- Work and Energy
= Conservation of Energy
- Upper Undergraduate
- Lower Undergraduate
- Instructional Material
= Interactive Simulation
Intended Users Formats Ratings
- Learners
- Educators
- General Publics
- application/java
  • Currently 0.0/5

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Access Rights: Free access
License: This material is released under a GNU General Public License Version 3 license.
Rights Holder: Todd Timberlake
Keywords: EJS, Easy Java Simulations, Lagrangian, OSP, Open Source Physics, free particle, gravity, least action, least action
Record Cloner: Metadata instance created September 3, 2012 by Todd Timberlake
Record Updated: Jun 10, 2014 by Andreu Glasmann
Last Update
when Cataloged:
September 3, 2012
Other Collections:
ComPADRE is beta testing Citation Styles!

Record Link
AIP Format
T. Timberlake, Computer Program LEAST ACTION (GRAVITY/FREE), Version 1.0 (2012), WWW Document, (
T. Timberlake, Computer Program LEAST ACTION (GRAVITY/FREE), Version 1.0 (2012), <>.
APA Format
Timberlake, T. (2012). Least Action (Gravity/Free) (Version 1.0) [Computer software]. Retrieved September 23, 2017, from
Chicago Format
Timberlake, Todd. "Least Action (Gravity/Free)." Version 1.0. (accessed 23 September 2017).
MLA Format
Timberlake, Todd. Least Action (Gravity/Free). Vers. 1.0. Computer software. 2012. Java (JRE) 1.5. 23 Sep. 2017 <>.
BibTeX Export Format
@misc{ Author = "Todd Timberlake", Title = {Least Action (Gravity/Free)}, Month = {September}, Year = {2012} }
Refer Export Format

%A Todd Timberlake
%T Least Action (Gravity/Free)
%D September 3, 2012
%O 1.0
%O application/java

EndNote Export Format

%0 Computer Program
%A Timberlake, Todd
%D September 3, 2012
%T Least Action (Gravity/Free)
%7 1.0
%8 September 3, 2012

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The AIP Style presented is based on information from the AIP Style Manual.

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Least Action (Gravity/Free):

Is Based On Easy Java Simulations Modeling and Authoring Tool

The Easy Java Simulations Modeling and Authoring Tool is needed to explore the computational model used in the Least Action (Gravity/Free).

relation by Wolfgang Christian

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