Abstract
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 gives 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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Communicated by A. Bloch.
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Birtea, P., Caşu, I., Ratiu, T.S. et al. Stability of Equilibria for the \(\mathfrak{so}(4)\) Free Rigid Body. J Nonlinear Sci 22, 187–212 (2012). https://doi.org/10.1007/s00332-011-9113-2
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DOI: https://doi.org/10.1007/s00332-011-9113-2