The Lagrange Applet

The applet implemented here computes a solution of the Swinging Spring dynamical system, integrating the equations numerically with a fourth-order Runge-Kutta method, and displays the results graphically. A vertical cross-section (in the x-z plane) is shown on the left and a horizontal projection (the x-y plane) on the right. The size of the bob indicates distance from the viewer. Physical parameters and initial conditions are set in the loading page. Default values of the parameters ensure that the spring is in resonance (spring frequency is twice swing frequency).

Some examples of the applet are given here [you must have a Java-enabled browser]:

• Regular Motion: Star-Pattern.
  This shows a case of low-energy motion, in which the bob traces out a regular pattern whose horizontal projection is a twelve-pointed star.
• Chaotic Motion.
  The energy is much larger in this case and the bob no longer traces out a regular pattern, but follows a chaotic and unpredictable trajectory.
• Progressive Ellipse.
  Here the exchange of energy between springing and swinging modes vanishes. We get a trajectory whose horizontal projection is a precessing ellipse.
• Retrogressing Ellipse.
  In this case the ellipse moves in the opposite direction to that in which the bob is revolving.

Physical parameters can be changed using the right-hand button and entering new values in the table. However, if you want to do serious experiments using the applet, and have control of the initial conditions, you should download the Class Files in a gzipped tar file, and also a loading page such as this page, in which default initial conditions are specified and may be changed. Many beautiful patterns can be generated by experimenting.

The Java Source Code of the Swinging Spring Applet, which includes the Runge-Kutta solver, is yours for a click. However, you may prefer to work from the original source code of Peter Selinger, using the Documentation provided by him.

Here are some other applications of the Lagrange Applet originally written by Peter Selinger. The Java Source Code of the original applet is available. There is also an excellent description of the applet, and clear instructions on how to modify it, at Peter Selinger's site.

Double Pendulum

Double Spring

Spring Pendulum with Toggle

Updated December 3rd, 2002.