Some elliptic fibrations arising from free rigid body dynamics
The dynamics of free rigid body governed by the Euler equation is studied from a complex algebro-geometric viewpoint. The authors show that the dynamics is expressed by an elliptic fibration over \(\mathbb{P}^3\), which has singularities. By desingularizion and other several algebraic operations to the naive elliptic fibration, they also construct its Weierstrass normal form, which is bimeromorphic to the family of the spectral curves arising from the corresponding Manakov equation. In other words, they present the relation between the Euler and the Manakov equations using a four-to-one meromorphic map from the naive fibration to the Weierstrass normal form.
scientific article; zbMATH DE number 6099400