Integrable systems with linear periodic integral for the Lie algebra e(3)
The phase space topology of integrable systems is of primary interest in mechanics and has been studied by many authors over the years. The present paper provides a nice overview of the topological properties of integrable Hamiltonian systems with \(S^1\) symmetry and rigid body equations for the Euclidean Lie algebra \(e(3)\). Let us mention here that this setting covers the Lagrange and Kirchhoff integrable cases of rigid body dynamics, the dynamics of spin in the superfluid \(^3\mathrm{He}\) and the integrable case of the motion of a heavy ellipsoid on the horizontal plane. Starting with the definition of the Lie-Poisson brackets for the Euclidean Lie algebra \(e(3)\) the authors describe the above mentioned integrable systems and observe that all of them possess an additional integral which is linear function on the dual space \(e(3)^*\). The also provide the corresponding bifurcation analysis. The paper is clearly written and quite informative.
scientific article; zbMATH DE number 6846086