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

Abstract

The phase space of a gyrostat with a fixed point and a heavy top is the Lie-Poisson space $\text{e}(3{)}^{\ast }\cong {\mathbb{R}}^3\times {\mathbb{R}}^3$ dual to the Lie algebra $\text{e}(3)$ of Euclidean group $E(3)$. One has three naturally distinguished Poisson submanifolds of $\text{e}(3{)}^{\ast }$: (i) the dense open submanifold ${\mathbb{R}}^3\times {\dot{\mathbb{R}}}^3\subset \text{e}(3{)}^{\ast }$ which consists of all $4$-dimensional symplectic leaves (${\vec{\Gamma }}^2>0$); (ii) the $5$-dimensional Poisson submanifold of ${\mathbb{R}}^3\times {\dot{\mathbb{R}}}^3$ defined by $\vec{J}\cdot \vec{\Gamma }=\mu ||\vec{\Gamma }||$; (iii) the $5$-dimensional Poisson submanifold of ${\mathbb{R}}^3\times {\dot{\mathbb{R}}}^3$ defined by ${\vec{\Gamma }}^2={\nu }^2$, where ${\dot{\mathbb{R}}}^3:={\mathbb{R}}^3\backslash \{0\}$, $(\vec{J},\vec{\Gamma })\in {\mathbb{R}}^3\times {\mathbb{R}}^3\cong \text{e}(3{)}^{\ast }$ and $\nu <0$, $\mu$ 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 $\text{e}(3{)}^{\ast }$ 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.

References in corpus (2)

• An integrable (classical and quantum) four-wave mixing Hamiltonian system
• A family of integrable perturbed Kepler systems