Hamilton–Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque

We give sufficient conditions for the rigid body in the presence of an axisymmetric force field and a gyroscopic torque to admit a Hamilton–Poisson formulation. Even if by adding a gyroscopic torque we initially lose one of the conserved Casimirs, we recover another conservation law as a Casimir function for a modified Poisson structure. We apply this frame to several well-known results in the literature. ► Gyroscopic torques and sufficient conditions for a Hamilton–Poisson formulation. ► An algorithm to find a third constant of motion as a Casimir function. ► A unitary framework that encompasses well-known gyroscopic torques from literature.