# 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} }

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…

## References

SHOWING 1-10 OF 23 REFERENCES

### Integrable systems of classical mechanics and Lie algebras

- Mathematics, Physics
- 1989

1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment…

### Generalized Liouville method of integration of Hamiltonian systems

- Mathematics
- 1978

In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level…

### Remarks on a paper of Hermann

- Mathematics
- 1968

Let G be a Lie group acting differentiably on a manifold and let p be a point left fixed by G. If G is compact, a well-known result of H. Cartan asserts that the action of G in a neighborhood of p is…

### An integrable (classical and quantum) four-wave mixing Hamiltonian system

- Physics
- 2020

A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form $\omega_0…

### A dynamical group Su(2,2) and its use in the MIC-Kepler problem

- Physics, Mathematics
- 1993

It is widely known that the Kepler problem admits SU(2,2) as a dynamical group. The author aims to show that SU(2,2) is also a dynamical group for the MIC-Kepler problem, a generalization of the…

### Geometric Models for Noncommutative Algebras

- Mathematics
- 1999

UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps…

### A Family of Integrable Perturbed Kepler Systems

- MathematicsRussian Journal of Mathematical Physics
- 2019

In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of these systems can be integrated in…

### Poisson Structures and Their Normal Forms

- Mathematics
- 2005

Generalities on Poisson Structures.- Poisson Cohomology.- Levi Decomposition.- Linearization of Poisson Structures.- Multiplicative and Quadratic Poisson Structures.- Nambu Structures and Singular…