Inducing Stability or Instability in the Special Cases of the Swing and Pendulum

Date: 
Tuesday, May 2, 2017 - 2:00pm
Location: 
312 Math Science Building
Speaker: 
Bryan Novak (advisor Konstantin Makarov)
(MU Math)

Abstract: 

 

In this project, we discuss the stability of systems of differential equations with periodic coefficients with emphasis on those that model the behavior of a person on a swing and the inverted pendulum with oscillating point of suspension. Based on the study of the Poincare' map via Liouville’s Theorem (from classical mechanics), we determine an optimal strategy for a person on a swing to follow to make it unstable and enjoy the swing. To the contrary, in the case of the inverted pendulum, we determine how to make it stable. We also briefly discuss the nonlinear inverted pendulum with oscillating point of suspension and how Kolmogorov-Arnol’d-Moser (KAM) Methods can be used to determine when it is stable.