The Lie symmetries of a Top

The existence of symmetries in differential equations can generate transformations of dependent and independent variables that facilitate the integration of these equations. 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. In order to illustrate the approach, we consider the classical problem of a spinning top with stationary precession. From the equations of motion, the Lie symmetries are identified and used in a transformation that results in an order reduction. The first integrals are obtained from this result using the Noether theorem, and we illustrate that the Lie symmetries for this problem are also Noether symmetries. Finally, the solution to the equations of motion are written using Jacobi elliptic functions, and from this we obtain the precession, nutation and spin angles under the presented conditions.