Module for the pendulum, a second order D.E.

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Program (Runge-Kutta Method for second order D.E.'s)  To compute a numerical approximation for the solution of the initial value problem  

      with  , ,  

over the interval    at a discrete set of points.  

Example 1.  Solve the second order  I.V.P.  

      with  ,   .    

Use the Runge-Kutta method to compute the solution over the interval .  

Solution 1.

Example 2.  Find the analytic solution to the second order  I.V.P.  

      with  ,   .    

Plot the solution over the interval .  

Solution 2.

Example 3.  Solve the second order  I.V.P.  

      with  ,   .    

Solution 3.

Example 4.  Compare the graphs of the solutions to  

(i)      with  ,   ,    
and
(ii)      with  ,   .    

From the graph's can you tell which is the solution to (ii)?  How?

The differential equation in example 3 is called the nonlinear pendulum.
Is is a better model that the one given in example 1.

Research Experience for Undergraduates

The Pendulum  The Pendulum  Internet hyperlinks to web sites and a bibliography of articles.  

Downloads (The Pendulum The Pendulum).  Download this Mathematica notebook.  

(c) John H. Mathews 2003