The Lie symmetries of a Top

In the nineteenth century, Sophus Lie developed a method of extracting symmetries that can be used effectively to reveal first integrals of a differential equation. These invariants can in some situations be identified by the Noether theorem or from manipulating the equations themselves with Lie transformations.

Despite the formalism over conservation theorems for energy and linear/angular momentum, initial courses in classical mechanics often do not clearly or objectively highlight the relationship between conservation laws and the existence of possible Lie symmetries. For this reason, we seek to present an introduction to Lie symmetries using a language accessible to a graduate student in physics, mathematics, or engineering with basic mastery of classical mechanics in several variables.