Why do we use constants of motion while studying the motion of a heavy symmetric top?

TL;DR

This paper explains the importance of using constants of motion in analyzing the dynamics of a heavy symmetric top, illustrating the equivalence with torque-angular momentum relations and aiding student understanding of rigid body rotation.

Contribution

It demonstrates the equivalence of torque-angular momentum relations and Euler equations for a heavy symmetric top, and explores simple motions to clarify the role of constants of motion.

Findings

Torque-angular momentum relation is equivalent to Euler equations.

Simple motions illustrate the necessity of constants of motion.

Historical perspective on precession and top rise is discussed.

Abstract

While studying the motion of a heavy symmetric top, in general, constants of motion are used. Some students may want to understand the motion in terms of torque, which can lie on their routine based on the usage of Newton's second law. However, this is not easy, and examples of this work will illustrate this situation. In this work, first, we will show the equivalence of torque-angular momentum relation and Euler equations for a heavy symmetric top which give the description of the motion in terms of torque. Then, we will study some simple motions of a heavy symmetric top in terms of torque, angular momentum, angular velocities and accelerations, which can help students in understanding rigid body rotations and the necessity of considering the motion of a heavy symmetric top in terms of constants. We will also study Perry's historical observational principle on the relation between precession and the rise of the top.