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

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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.

Event sponsored by:
University of Bialystok

Webpage by: Tomasz Golinski

Tomasz Golinski