On bi-hamiltonian geometry of the Lagrange top

TL;DR

This paper explores three distinct bi-Hamiltonian structures for the Lagrange top, revealing their shared foliation and connection to Poisson-Lichnerowicz cohomology, advancing understanding of integrable systems.

Contribution

It introduces and analyzes three incompatible bi-Hamiltonian structures for the Lagrange top with the same symplectic foliation, linked to different 2-coboundaries in Poisson cohomology.

Findings

Identified three incompatible bi-Hamiltonian structures for the Lagrange top.

Showed these structures share the same foliation by symplectic leaves.

Connected the bivectors to different 2-coboundaries in Poisson-Lichnerowicz cohomology.

Abstract

We consider three different incompatible bi-Hamiltonian structures for the Lagrange top, which have the same foliation by symplectic leaves. These bivectors may be associated with the different 2-coboundaries in the Poisson-Lichnerowicz cohomology defined by canonical bivector on $e^*(3)$.