An asymmetric heavy gyroscope can regularly precess only when its suspension point lies on one of two lines marking where the intermediate and largest moments of inertia exchange order, with exactly two motions.
Poincar\'e-Chetaev equations in the Dirac's formalism of constrained systems
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abstract
We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket. In the case of $SO(3)$\,- manifold, the application of this formalism leads to the Poincar\'e-Chetaev equations. The general solution to these equations is written in terms of exponential of the Hamiltonian vector field.
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An alternative interpretation of the Grioli gyroscope suspension points
An asymmetric heavy gyroscope can regularly precess only when its suspension point lies on one of two lines marking where the intermediate and largest moments of inertia exchange order, with exactly two motions.