The complex geometry of Lagrange top

TL;DR

This paper provides a detailed geometric and algebraic analysis of the Lagrange top, revealing its linearization on a generalized Jacobian and deriving explicit solutions using Baker-Akhiezer functions.

Contribution

It introduces a novel geometric framework for the Lagrange top using generalized Jacobians and derives explicit solutions with Baker-Akhiezer functions.

Findings

Linearization on a generalized Jacobian of an elliptic curve.

Explicit formulas for the general solution of Lagrange top.

Transparent description of invariant level sets.

Abstract

We prove that the heavy symmetric top (Lagrange, 1788) linearizes on a two-dimensional non-compact algebraic group -- the generalized Jacobian of an elliptic curve with two points identified. This leads to a transparent description of its complex and real invariant level sets. We also deduce, by making use of a Baker-Akhiezer function, simple explicit formulae for the general solution of Lagrange top.