Our first reference showed how to do an experiment that shows this claim is not true. We decided the goal of our project will be to develop a mathematical model that allows us to study how a pendulum's amplitude affects its period. With this plan in mind we introduce the following notation for the input and output variables and parameters for our problem. We think of input variables as quantities that will be provided by the user to the program at run time. Output variables are quantities that the computer program computes and returns to the user.
Auxiliary variables are quantities derived from groupings of input or output variables.
Parameters are problem dependent constants that are set at the time the program is compiled. That is, in our notation, the subscript on v refers to the right end of the time interval over which that velocity applies. Reviewing our Background Research, we recall two important facts we uncovered. At this point we believe these are the principles that govern the behavior of the pendulum. The change of any quantity over a period of time of length is that quantity at the end of the time period minus the quantity at the beginning of the period.