Abstract: We study the global structure of the fibration by the invariant two-dimensional tori of the Euler-Poinsot top-the rigid body with a fixed point and no torques. We base our analysis on the notion of bifibration (or dual pair) which, as results from the approach based on the so-called non-commutative integrability, provides a thorough description of the geometry of integrable degenerate Hamiltonian systems. In this way, we get a global geometric picture of the Euler-Poinsot system which fully accounts for its degeneracy through the (Poisson) structure? of the base of the fibration by the two-dimensional invariant tori. In particular, we explain in this way why this system does not possess global 'generalized' action-angle coordinates: the obstructions are the topological non-triviality of the fibration by the invariant two-dimensional tori and the compactness of the symplectic leaves of its base manifold. We also compare this description with the usual description based on the notion of complete integrability, and we remark that, as a general fact, such a common approach fails to provide a natural, thorough description of degenerate systems.

The Euler-Poinsot top: A non-commutatively integrable system without global action-angle coordinates

FASSO', FRANCESCO
1996

Abstract

Abstract: We study the global structure of the fibration by the invariant two-dimensional tori of the Euler-Poinsot top-the rigid body with a fixed point and no torques. We base our analysis on the notion of bifibration (or dual pair) which, as results from the approach based on the so-called non-commutative integrability, provides a thorough description of the geometry of integrable degenerate Hamiltonian systems. In this way, we get a global geometric picture of the Euler-Poinsot system which fully accounts for its degeneracy through the (Poisson) structure? of the base of the fibration by the two-dimensional invariant tori. In particular, we explain in this way why this system does not possess global 'generalized' action-angle coordinates: the obstructions are the topological non-triviality of the fibration by the invariant two-dimensional tori and the compactness of the symplectic leaves of its base manifold. We also compare this description with the usual description based on the notion of complete integrability, and we remark that, as a general fact, such a common approach fails to provide a natural, thorough description of degenerate systems.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/128559
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