Integrable Hamiltonian systems on the symplectic realizations of $\textbf{e}(3)^*$

Abstract

The phase space of a gyrostat with a fixed point and a heavy top is the Lie-Poisson space e(3) * ∼ = R 3 × R 3 dual to the Lie algebra e(3) of Euclidean group E(3). One has three naturally distinguished Poisson submanifolds of e(3) * : (i) the dense open submanifold R 3 × Ṙ3 ⊂ e(3) * which consists of all 4-dimensional symplectic leaves ( Γ 2 > 0 3) * and ν < 0, µ are some fixed real parameters. Basing on the U (2, 2)-invariant symplectic structure of Penrose twistor space we find full and complete E(3)-equivariant symplectic realizations of these Poisson submanifolds which are 8-dimensional for (i) and 6-dimensional for (ii) and (iii). As a consequence of the above Hamiltonian systems on e(3) * lift to the ones on the above symplectic realizations. In such a way after lifting integrable cases of gyrostat with a fixed point, as well as of heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.