A primer on elliptic functions with applications in classical mechanics

TL;DR

This paper reviews elliptic functions, especially Jacobi and Weierstrass types, highlighting their historical importance and applications in solving classical mechanics problems like pendulums and tops, advocating for their inclusion in physics education.

Contribution

It provides a comprehensive overview of elliptic functions and demonstrates their relevance to classical mechanics, emphasizing their pedagogical value for undergraduate physics courses.

Findings

Elliptic functions solve key classical mechanics problems.

They connect different types of pendulum and top motions.

Mathematical software makes elliptic functions accessible for students.

Abstract

The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a spherical pendulum (Weierstrass), and the motion of a heavy symmetric top with one fixed point (Weierstrass). The problem of the planar pendulum, in fact, can be used to construct the general connection between the Jacobi and Weierstrass elliptic functions. The easy access to mathematical software by physics students suggests that they might reappear as useful tools in the undergraduate curriculum.