Two equivalent problems: gyrostats in free motion and parametric quadratic Hamiltonians

A gyrostat G is a mechanical system made up of a rigid body P (platform) and other bodies R (rotors), connected to the platform in such a way that the motion of the rotors does not modify the distribution of mass of the gyrostat. The present study demonstrates that the gyrostat in free motion may be formulated as a multi parametric quadratic Hamiltonian on the unit sphere. Each case corresponds on the case of the parametric quadratic Hamiltonian. As a result, a real physical problem is identified with a theoretical one.