13.19: Hamiltonian equations of motion for rigid-body rotation
( \newcommand{\kernel}{\mathrm{null}\,}\)
The Hamiltonian equations of motion are expressed in terms of the Euler angles plus their corresponding canonical angular momenta (ϕ,θ,ψ,pϕ,pθ,pψ) in contrast to Lagrangian mechanics which is based on the Euler angles plus their corresponding angular velocities (ϕ,θ,ψ,˙ϕ,˙θ,˙ψ). 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.