SYSTEMS OF HESS-APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY

@article{Dragovic2009SYSTEMSOH,
title={SYSTEMS OF HESS-APPEL'ROT TYPE AND ZHUKOVSKII PROPERTY},
author={V. Dragovic and B. Gaji{\'c} and B. Jovanovic},
journal={International Journal of Geometric Methods in Modern Physics},
year={2009},
volume={06},
pages={1253-1304}
}
• Published 2009
• Mathematics
• International Journal of Geometric Methods in Modern Physics
We start with a review of a class of systems with invariant relations, so called systems of Hess–Appel'rot type that generalizes the classical Hess–Appel'rot rigid body case. The systems of Hess–Appel'rot type have remarkable property: there exists a pair of compatible Poisson structures, such that a system is certain Hamiltonian perturbation of an integrable bi-Hamiltonian system. The invariant relations are Casimir functions of the second structure. The systems of Hess–Appel'rot type carry an… Expand
8 Citations
The rigid body dynamics: classical and algebro-geometric integration
The basic notion for a motion of a heavy rigid body fixed at a point in three-dimensional space as well as its higher-dimensional generalizations are presented. On a basis of Lax representation, theExpand
Contact flows and integrable systems
• Mathematics, Physics
• 2015
Abstract We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold–Liouville theorem: the system need not be integrable on the wholeExpand
INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS
Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and G˜$$\tilde{G}$$ be the Lie group of fixed points of σ. We find aExpand
Note on free symmetric rigid body motion
• Mathematics
• 2015
We consider the Euler equations of motion of a free symmetric rigid body around a fixed point, restricted to the invariant subspace given by the zero values of the corresponding linear NoetherExpand
On the cases of Kirchhoff and Chaplygin of the Kirchhoff equations
• Mathematics, Physics
• 2012
It is proven that the completely integrable general Kirchhoff case of the Kirchhoff equations for B ≠ 0 is not an algebraic complete integrable system. Similar analytic behavior of the generalExpand
On the Completeness of the Manakov Integrals
• Mathematics, Physics
• 2015
The aim of this note is to present a simple proof of the completeness of the Manakov integrals for a motion of a rigid body fixed at a point in ℝn, as well as for geodesic flows on a class ofExpand
Some Recent Generalizations of the Classical Rigid Body Systems
• Mathematics
• 2016
Some recent generalizations of the classical rigid body systems are reviewed. The cases presented include dynamics of a heavy rigid body fixed at a point in three-dimensional space, the KirchhoffExpand
Four-dimensional generalization of the Grioli precession
• Physics
• 2014
A particular solution of the four-dimensional Lagrange top on e(4) representing a four-dimensional regular precession is constructed. Using it, a four-dimensional analogue of the Grioli nonverticalExpand

References

SHOWING 1-10 OF 84 REFERENCES
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 levelExpand
Systems of Hess-Appel’rot Type
• Mathematics, Physics
• 2006
We construct higher-dimensional generalizations of the classical Hess- Appel’rot rigid body system. We give a Lax pair with a spectral parameter leading to an algebro-geometric integration of thisExpand
Types of integrability on a submanifold and generalizations of Gordon’s theorem
At the beginning of the paper the concept of Liouville integrability is analysed for systems of general form, that is, ones that are not necessarily Hamiltonian. On this simple basis HamiltonianExpand
Bi-Hamiltonian structures with symmetries, Lie pencils and integrable systems
There are two classical ways of constructing integrable systems by means of bi-Hamiltonian structures. The first one supposes nondegeneracy of one of the Poisson structures generating the pencil andExpand
Linearization of Hamiltonian systems, Jacobi varieties and representation theory☆
• Mathematics
• 1980
The purpose of this paper is twofold: first we show that all the systems discussed in Adler and van Moerbeke [2] (paper I) in connection with KacMoody Lie algebras can be linearized according to aExpand
Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems
• Mathematics
• 1994
The present survey is devoted to a general group-theoretic scheme which allows to construct integrable Hamiltonian systems and their solutions in a systematic way. This scheme originates from theExpand
Non-commutative integrability, paths and quasi-determinants
• Mathematics, Physics
• 2011
In previous work, we showed that the solution of certain systems of discrete integrable equations, notably Q and T-systems, is given in terms of partition functions of positively weighted paths,Expand
Elliptic curves and a new construction of integrable systems
• Mathematics, Physics
• 2009
A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with the above Lax matrices are constructed. Five cases fromExpand
An L–A pair for the Hess–Apel'rot system and a new integrable case for the Euler–Poisson equations on so(4) × so(4)
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2001
We present an L–A pair for the Hess–Apel'rot case of a heavy rigid three-dimensional body. Using it, we give an algebro-geometric integration procedure. Generalizing this L–A pair, we obtain a newExpand
Necessary conditions for partial and super-integrability of Hamiltonian systems with homogeneous potentia
• Mathematics, Physics
• 2007
We consider a natural Hamiltonian system of $n$ degrees of freedom with a homogeneous potential. Such system is called partially integrable if it admits $1<l<n$ independent and commuting firstExpand