• DOI:10.1134/S1061920822010095
• Corpus ID: 235435974

Integrable Hamiltonian Systems on the Symplectic Realizations of $$\textbf{e}(3)^*$$

@article{Odzijewicz2021IntegrableHS,
  title={Integrable Hamiltonian Systems on the Symplectic Realizations of 
 
 
 
 \$\$\textbf\{e\}(3)^*\$\$
},
  author={A. Odzijewicz and E. Wawreniuk},
  journal={Russian Journal of Mathematical Physics},
  year={2021},
  volume={29},
  pages={91-114}
}

• A. Odzijewicz, E. Wawreniuk
• Published 15 June 2021
• Mathematics
• Russian Journal of Mathematical Physics

Abstract The phase space of a gyrostat with a fixed point and a heavy top is the Lie–Poisson space $$\textbf{e}(3)^*\cong \mathbb{R}^3\times \mathbb{R}^3$$ dual to the Lie algebra $$\textbf{e}(3)$$ of the Euclidean group $$E(3)$$ . One has three naturally distinguished Poisson submanifolds of $$\textbf{e}(3)^*$$ : (i) the dense open submanifold $$\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*$$ 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 $$\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}$$ , $$(\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*$$ and $$\nu < 0 $$ , $$\mu$$ are some fixed real parameters. Using 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 $$\textbf{e}(3)^*$$ lift to Hamiltonian systems on the above symplectic realizations. In this way, after lifting the integrable cases of a gyrostat with a fixed point and of a heavy top, we obtain a large family of integrable Hamiltonian systems on the phase spaces defined by these symplectic realizations.