# 13.19: Hamiltonian equations of motion for rigid-body rotation

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The Hamiltonian equations of motion are expressed in terms of the Euler angles plus their corresponding canonical angular momenta \((\phi , \theta , \psi , p_{\phi} , p_{ \theta }, p_{\psi} )\) in contrast to Lagrangian mechanics which is based on the Euler angles plus their corresponding angular velocities \((\phi , \theta , \psi , \dot{\phi} , \dot{\theta} , \dot{\psi})\). The Hamiltonian approach is conveniently expressed in terms of a set of Andoyer-Deprit action-angle coordinates that include the three Euler angles, specifying the orientation of the body-fixed frame, plus the corresponding three angles specifying the orientation of the spin frame of reference. This phase space approach[Dep67] can be employed for calculations of rotational motion in celestial mechanics that can include spin-orbit coupling. This Hamiltonian approach is beyond the scope of the present textbook.