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Geodesic flows on real forms of complex semi-simple Lie groups of rigid body type. (English) Zbl 1460.53074

Free rigid body dynamics has been generalized to semi-simple Lie groups in [A. S. Mishchenko and A. T. Fomenko, Math. USSR, Izv. 12, 371–389 (1978; Zbl 0405.58031)] and [A. S. Mishchenko and A. T. Fomenko, Tr. Semin. Vektorn. Tenzorn. Anal. 19, 3–94 (1979; Zbl 0452.58015)].
The motions of the generalized free rigid bodies can be formulated as geodesic flows for certain left-invariant metrics on semi-simple Lie groups, called of rigid body type.
The Williamson type of an isolated equilibrium point for a Hamiltonian system is a triple giving the numbers of elliptic, hyperbolic, and focus-focus components.
The main result of the article under review characterizes the Williamson types of the isolated relative equilibria on generic adjoint orbits in terms of the root system of the complexified Lie-algebra.

MSC:

53D25 Geodesic flows in symplectic geometry and contact geometry
70E15 Free motion of a rigid body
70E45 Higher-dimensional generalizations in rigid body dynamics
22E46 Semisimple Lie groups and their representations
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