# The Rigid Body Dynamics in an ideal fluid: Clebsch Top and Kummer surfaces

## TL;DR

This paper explores the integrable Hamiltonian system of the Clebsch top in an ideal fluid, highlighting its geometric structure, connection to Kummer surfaces, and explicit action-angle coordinate computation.

## Contribution

It provides a detailed geometric and algebraic analysis of the Clebsch top system, including its relation to Kummer surfaces and explicit solution methods.

## Key findings

- Connection between Clebsch top and Kummer surfaces
- Explicit action-angle coordinates derived
- System linearized on Jacobian of genus two curve

## Abstract

This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account on the geometric setting of the system, the structure of the Poisson commuting first integrals is discussed following the methods by Magri and Skrypnyk. Introducing supplementary coordinates, a geometric connection to Kummer surfaces, a typical class of K3 surfaces, is mentioned and also the system is linearized on the Jacobian of a hyperelliptic curve of genus two determined by the system. Further some special solutions contained in some vector subspace are discussed. Finally, an explicit computation of the action-angle coordinates is introduced.