Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body

Details Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body

2008-12-17

arxiv.org/abs/0812.3415v5

Mathematics

Dynamical Systems

Preprint of a paper accepted for publication in Journal of Nonlinear Science

Scientific paper

10.1007/s00332-011-9113-2

The stability for all generic equilibria of the Lie-Poisson dynamics of the $\mathfrak{so}(4)$ rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of $\mathfrak{so}(n)$ are equilibrium points for the rigid body dynamics. In the case of $\mathfrak{so}(4)$ there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit give three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in $\mathfrak{so}(3)$. In addition to these coordinate type Cartan equilibria there are others that come in curves.

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