Symplectic Realizations of e(3)^∗

Abstract

Using the U(2, 2)-invariant symplectic structure of the Penrose twistor space we find full and complete E(3)-equivariant symplectic realizations of some Poisson submanifolds of the Lie-Poisson space $\mathbf {e}(3)^*\cong \mathbb {R}^3\times \mathbb {R}^3$ dual to the Lie algebra e(3) of the Euclidean group E(3), which is an underlying space in the rigid body theory. Considering concrete integrable cases of gyrostat systems on e(3)^∗ we can take their liftings to the ones on the constructed symplectic realizations. This way we obtain integrable systems on the phase spaces given by the symplectic realizations. As examples we compute integrable Hamiltonian systems obtained through symplectic realizations of the Lagrange, Kovalevskaya-Yahia, and Clebsh integrable cases.