Challenge Team Interim Report
Team Number: 072
School Name: Sandia Preparatory School
Area of Science: Mathematics
Project Title: Examination of Chaos and Non-Linearity Through Modeling of the Levitron
Our project this year is to study patterns in solutions of dynamic systems using non-linear mathematics. We are modeling the Levitron, a small magnetic top that is suspended above a larger base magnet, as a chaotic system. The Levitron demonstrates Earnshaw's theorem, which proves that stable levitation cannot be achieved in a static system using permanent magnets. By spinning the smaller magnet, and precisely controlling its weight, the base magnet exerts a torque on the top, creating a gyroscopic moment and thus allowing temporary suspension to occur. The gyroscopic motion causes the magnetic top to precess, or gradually change its axis of rotation, around the vertical axis. We plan to show the chaotic patterns that emerge from this precession of the top.
A dynamic system is a system that is constantly changing. Modeling these systems is apparently difficult because of their seemingly random nature. Consequently, most of the concentration has been given to describing linear systems because answers are easily obtained and do not change each time it is modeled. Unfortunately, we do not live in a linear world and find that nearly everything in our lives is dynamic. From weather, to stocks, to rush hour traffic-- it is extremely important to be able to model dynamic systems. Chaos theory emerged as a method of predicting the behavior of such dynamic systems.
In chaos theory, dynamic systems demonstrate order, although they never repeat themselves. To be considered chaotic, a dynamic system must be deterministic; there must be something determining the outcome. Such systems are extremely sensitive to initial conditions. By modeling dynamic systems in time, with slight changes in initial conditions, we can see that certain patterns emerge. Repetitive calculation such as this necessitates high computational power since it is inefficient to solve the same equation repeatedly by hand. As the equations get more complex, we need to move to supercomputers in order to solve a sufficient number of cases in a reasonable period of time.
These chaotic patterns in non-linear math easily lend themselves to computer science due to the nature of the equations involved. The precession of the Levitron demonstrates chaotic behavior as it spirals away from the axis. The equations that determine its motion are known, however the model is extremely sensitive to initial conditions. The Levitron is an interesting way to study the patterns that emerge in chaotic systems as well as characteristics of sub-atomic particles. Trapped sub-atomic particles behave in a similar manner as the levitating top and the same chaotic patterns emerge in both cases.
Presently, we have learned a great deal of the math associated with physics and magnetism. We will use the Runge-Kutta method for solving the first-order differential equations derived for the Levitron. We are using Runge-Kutta because it is a sound and fairly accurate method. Other methods such as Euler's method are more efficient but the improved accuracy gained with Runge-Kutta is worth the added computational time. We will graph the results of the equation over time and with changes in the initial conditions in order to determine which chaotic patterns are exhibited by the precession of the Levitron.