Elwira Wawreniuk

Symplectic realizations of $\textbf{e}(3)^*$

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 $E(3)$ is the phase space of a heavy top system. We consider the dense open submanifold $\mathbb{R}^3\times \dot{\mathbb{R}}^3\subset \textbf{e}(3)^*$ of $\textbf{e}(3)^*$ consisting of all $4$-dimensional symplectic leaves ($\vec{\Gamma}^2>0$) and its two $5$-dimensional submanifolds:

submanifold of $\mathbb{R}^3\times \dot{\mathbb{R}}^3$ defined by $\vec{J}\cdot \vec{\Gamma} = \mu ||\vec{\Gamma}||$,
submanifold of $\mathbb{R}^3\times \dot{\mathbb{R}}^3$ defined by $\vec{\Gamma}^2 = \nu^2$,

where $(\vec{J}, \vec{\Gamma})\in \mathbb{R}^3\times \mathbb{R}^3\cong \textbf{e}(3)^*$, $\mu , \nu $ are some real fixed parameters and $\dot{\mathbb{R}}^3:= \mathbb{R}^3\backslash \{0\}$. Basing on $U(2,2)$-invariant symplectic structure of the Penrose twistor space we find full and complete $E(3)$-equivariant symplectic realizations of these submanifolds. Lifts of the integrable Hamiltonian systems on $\textbf{e}(3)^*$ to these symplectic realizations give a large family of integrable Hamiltonian systems.

XXXIX Workshop on Geometric Methods in Physics	19.06–25.06.2022
XI School on Geometry and Physics	27.06–1.07.2022

Event sponsored by:
University of Bialystok